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A First Course in Linear Algebra

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A First Course in Linear Algebra
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Beezer, Robert A.
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Mathematics, Linear Algebra, Linear Equations, Vectors, Matrices, Vector Spaces, Determinants, Linear Transformations, Representations, Equivalent Systems, Equation Operations, Matrix and Vector Notation, Row Operations, Types of Solution Sets, Consistent Systems, Free Variables, Homogeneous Systems of Equations, Solutions of Homogeneous Systems, Null Space …
Algebra, Matrices
Mathematics / Algebra

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Abstract:
This textbook is designed to college sophomores and juniors the basics of linear algebra and the techniques of formal mathematics. There are no prerequisites other than ordinary algebra. The text has two goals: to teach the fundamental concepts and techniques of matrix algebra and abstract vector spaces, and to teach the techniques associated with understanding the definitions and theorems forming a coherent area of mathematics. There is an emphasis on worked examples and on proving theorems carefully. Contents: 1) Systems of Linear Equations. 2) Vectors. 3) Matrices. 4) VS Vector Spaces. 5) Determinants. 6) Linear Transformations. 7) Representations.
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Community College, Higher Education
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http://www.ogtp-cart.com/product.aspx?ISBN=9781616100049
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Adobe PDF Reader
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Robert A. Beezer
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Textbook
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http://linear.ups.edu/index.html
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MAS 103 - INTRODUCTORY LINEAR ALGEBRA I, MAS 101 - LINEAR ALGEBRA WITH MICROCOMPUTERS, MAS 105 - LINEAR ALGEBRA I (CALC. II PREREQU)
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http://florida.theorangegrove.org/og/file/c871aca9-a06e-6a47-a569-4632fe26998e/1/LinearAlgebra.pdf

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University of Florida
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University Press of Florida
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Copyright 2004 by Robert A. Beezer. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the appendix …
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AFirstCourseinLinearAlgebra

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AFirstCourseinLinearAlgebra by RobertA.Beezer DepartmentofMathematicsandComputerScience UniversityofPugetSound Version2.02

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RobertA.BeezerisaProfessorofMathematicsattheUniversityofPugetSound,wherehehasbeen onthefacultysince1984.HereceivedaB.S.inMathematicswithanEmphasisinComputerScience fromtheUniversityofSantaClarain1978,aM.S.inStatisticsfromtheUniversityofIllinoisatUrbanaChampaignin1982andaPh.D.inMathematicsfromtheUniversityofIllinoisatUrbana-Champaignin 1984.Heteachescalculus,linearalgebraandabstractalgebraregularly,whilehisresearchinterestsinclude theapplicationsoflinearalgebratographtheory.Hisprofessionalwebsiteisat http://buzzard.ups.edu Edition Version2.02. November19,2008. Publisher RobertA.Beezer DepartmentofMathematicsandComputerScience UniversityofPugetSound 1500NorthWarner Tacoma,Washington98416-1043 USA c 2004byRobertA.Beezer. Permissionisgrantedtocopy,distributeand/ormodifythisdocumentunderthetermsoftheGNUFree DocumentationLicense,Version1.2oranylaterversionpublishedbytheFreeSoftwareFoundation;with noInvariantSections,noFront-CoverTexts,andnoBack-CoverTexts.Acopyofthelicenseisincluded intheappendixentitledGNUFreeDocumentationLicense". Themostrecentversionofthisworkcanalwaysbefoundat http://linear.ups.edu .

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Tomywife,Pat.

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Contents TableofContents vi Contributors vii Denitions viii Theorems ix Notation x Diagrams xi Examples xii Preface xiii Acknowledgements xviii PartCCore ChapterSLESystemsofLinearEquations2 WILAWhatisLinearAlgebra?...................................2 LALinear"+Algebra"...................................2 AAAnApplication.......................................3 READReadingQuestions...................................6 EXCExercises.........................................7 SOLSolutions..........................................8 SSLESolvingSystemsofLinearEquations............................9 SLESystemsofLinearEquations...............................9 PSSPossibilitiesforSolutionSets...............................10 ESEOEquivalentSystemsandEquationOperations....................11 READReadingQuestions...................................17 EXCExercises.........................................18 SOLSolutions..........................................21 RREFReducedRow-EchelonForm.................................24 MVNSEMatrixandVectorNotationforSystemsofEquations..............24 RORowOperations......................................27 RREFReducedRow-EchelonForm..............................29 READReadingQuestions...................................39 EXCExercises.........................................40 vi

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CONTENTSvii SOLSolutions..........................................44 TSSTypesofSolutionSets.....................................50 CSConsistentSystems.....................................50 FVFreeVariables.......................................55 READReadingQuestions...................................57 EXCExercises.........................................58 SOLSolutions..........................................60 HSEHomogeneousSystemsofEquations.............................62 SHSSolutionsofHomogeneousSystems...........................62 NSMNullSpaceofaMatrix.................................64 READReadingQuestions...................................66 EXCExercises.........................................67 SOLSolutions..........................................69 NMNonsingularMatrices......................................71 NMNonsingularMatrices...................................71 NSNMNullSpaceofaNonsingularMatrix.........................73 READReadingQuestions...................................75 EXCExercises.........................................76 SOLSolutions..........................................78 SLESystemsofLinearEquations...............................82 ChapterVVectors 83 VOVectorOperations........................................83 VEASMVectorEquality,Addition,ScalarMultiplication.................84 VSPVectorSpaceProperties.................................86 READReadingQuestions...................................87 EXCExercises.........................................88 SOLSolutions..........................................89 LCLinearCombinations.......................................90 LCLinearCombinations....................................90 VFSSVectorFormofSolutionSets..............................94 PSHSParticularSolutions,HomogeneousSolutions.....................105 READReadingQuestions...................................107 EXCExercises.........................................108 SOLSolutions..........................................110 SSSpanningSets...........................................112 SSVSpanofaSetofVectors.................................112 SSNSSpanningSetsofNullSpaces..............................117 READReadingQuestions...................................122 EXCExercises.........................................123 SOLSolutions..........................................126 LILinearIndependence.......................................132 LISVLinearlyIndependentSetsofVectors.........................132 LINMLinearIndependenceandNonsingularMatrices...................137 NSSLINullSpaces,Spans,LinearIndependence......................138 READReadingQuestions...................................141 EXCExercises.........................................142 SOLSolutions..........................................146 LDSLinearDependenceandSpans.................................152 LDSSLinearlyDependentSetsandSpans..........................152 Version2.02

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CONTENTSviii COVCastingOutVectors...................................154 READReadingQuestions...................................161 EXCExercises.........................................162 SOLSolutions..........................................164 OOrthogonality...........................................167 CAVComplexArithmeticandVectors............................167 IPInnerproducts........................................168 NNorm.............................................171 OVOrthogonalVectors....................................172 GSPGram-SchmidtProcedure................................175 READReadingQuestions...................................178 EXCExercises.........................................179 SOLSolutions..........................................180 VVectors............................................181 ChapterMMatrices 182 MOMatrixOperations.......................................182 MEASMMatrixEquality,Addition,ScalarMultiplication.................182 VSPVectorSpaceProperties.................................184 TSMTransposesandSymmetricMatrices..........................185 MCCMatricesandComplexConjugation..........................187 AMAdjointofaMatrix....................................189 READReadingQuestions...................................190 EXCExercises.........................................191 SOLSolutions..........................................193 MMMatrixMultiplication......................................194 MVPMatrix-VectorProduct.................................194 MMMatrixMultiplication...................................197 MMEEMatrixMultiplication,Entry-by-Entry.......................198 PMMPropertiesofMatrixMultiplication..........................200 HMHermitianMatrices....................................204 READReadingQuestions...................................206 EXCExercises.........................................207 SOLSolutions..........................................209 MISLEMatrixInversesandSystemsofLinearEquations....................212 IMInverseofaMatrix.....................................213 CIMComputingtheInverseofaMatrix...........................214 PMIPropertiesofMatrixInverses..............................219 READReadingQuestions...................................221 EXCExercises.........................................222 SOLSolutions..........................................224 MINMMatrixInversesandNonsingularMatrices.........................226 NMINonsingularMatricesareInvertible...........................226 UMUnitaryMatrices.....................................229 READReadingQuestions...................................232 EXCExercises.........................................234 SOLSolutions..........................................235 CRSColumnandRowSpaces....................................236 CSSEColumnSpacesandSystemsofEquations......................236 CSSOCColumnSpaceSpannedbyOriginalColumns...................239 Version2.02

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CONTENTSix CSNMColumnSpaceofaNonsingularMatrix.......................241 RSMRowSpaceofaMatrix.................................243 READReadingQuestions...................................248 EXCExercises.........................................249 SOLSolutions..........................................253 FSFourSubsets...........................................257 LNSLeftNullSpace......................................257 CRSComputingColumnSpaces...............................258 EEFExtendedechelonform..................................261 FSFourSubsets........................................263 READReadingQuestions...................................271 EXCExercises.........................................272 SOLSolutions..........................................274 MMatrices...........................................278 ChapterVSVectorSpaces279 VSVectorSpaces...........................................279 VSVectorSpaces........................................279 EVSExamplesofVectorSpaces................................280 VSPVectorSpaceProperties.................................285 RDRecyclingDenitions...................................288 READReadingQuestions...................................289 EXCExercises.........................................290 SOLSolutions..........................................291 SSubspaces..............................................292 TSTestingSubspaces.....................................293 TSSTheSpanofaSet.....................................297 SCSubspaceConstructions..................................302 READReadingQuestions...................................303 EXCExercises.........................................304 SOLSolutions..........................................305 LISSLinearIndependenceandSpanningSets...........................308 LILinearIndependence....................................308 SSSpanningSets........................................312 VRVectorRepresentation...................................316 READReadingQuestions...................................318 EXCExercises.........................................319 SOLSolutions..........................................321 BBases................................................325 BBases.............................................325 BSCVBasesforSpansofColumnVectors..........................328 BNMBasesandNonsingularMatrices............................330 OBCOrthonormalBasesandCoordinates..........................331 READReadingQuestions...................................336 EXCExercises.........................................337 SOLSolutions..........................................338 DDimension.............................................341 DDimension..........................................341 DVSDimensionofVectorSpaces...............................345 RNMRankandNullityofaMatrix.............................347 Version2.02

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CONTENTSx RNNMRankandNullityofaNonsingularMatrix.....................348 READReadingQuestions...................................350 EXCExercises.........................................351 SOLSolutions..........................................353 PDPropertiesofDimension.....................................355 GTGoldilocks'Theorem....................................355 RTRanksandTransposes...................................358 DFSDimensionofFourSubspaces..............................360 DSDirectSums.........................................361 READReadingQuestions...................................365 EXCExercises.........................................366 SOLSolutions..........................................367 VSVectorSpaces........................................369 ChapterDDeterminants370 DMDeterminantofaMatrix....................................370 EMElementaryMatrices...................................370 DDDenitionoftheDeterminant..............................374 CDComputingDeterminants.................................376 READReadingQuestions...................................380 EXCExercises.........................................381 SOLSolutions..........................................382 PDMPropertiesofDeterminantsofMatrices...........................383 DRODeterminantsandRowOperations...........................383 DROEMDeterminants,RowOperations,ElementaryMatrices..............387 DNMMMDeterminants,NonsingularMatrices,MatrixMultiplication..........389 READReadingQuestions...................................392 EXCExercises.........................................393 SOLSolutions..........................................394 DDeterminants.........................................395 ChapterEEigenvalues 396 EEEigenvaluesandEigenvectors..................................396 EEMEigenvaluesandEigenvectorsofaMatrix.......................396 PMPolynomialsandMatrices.................................398 EEEExistenceofEigenvaluesandEigenvectors.......................399 CEEComputingEigenvaluesandEigenvectors.......................403 ECEEExamplesofComputingEigenvaluesandEigenvectors...............406 READReadingQuestions...................................413 EXCExercises.........................................414 SOLSolutions..........................................415 PEEPropertiesofEigenvaluesandEigenvectors.........................419 MEMultiplicitiesofEigenvalues...............................424 EHMEigenvaluesofHermitianMatrices...........................427 READReadingQuestions...................................428 EXCExercises.........................................429 SOLSolutions..........................................430 SDSimilarityandDiagonalization.................................432 SMSimilarMatrices......................................432 PSMPropertiesofSimilarMatrices.............................433 DDiagonalization.......................................435 Version2.02

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CONTENTSxi FSFibonacciSequences....................................442 READReadingQuestions...................................445 EXCExercises.........................................446 SOLSolutions..........................................447 EEigenvalues..........................................451 ChapterLTLinearTransformations452 LTLinearTransformations.....................................452 LTLinearTransformations..................................452 LTCLinearTransformationCartoons............................456 MLTMatricesandLinearTransformations.........................457 LTLCLinearTransformationsandLinearCombinations..................461 PIPre-Images..........................................465 NLTFONewLinearTransformationsFromOld.......................467 READReadingQuestions...................................471 EXCExercises.........................................472 SOLSolutions..........................................474 ILTInjectiveLinearTransformations................................477 EILTExamplesofInjectiveLinearTransformations.....................477 KLTKernelofaLinearTransformation...........................481 ILTLIInjectiveLinearTransformationsandLinearIndependence.............485 ILTDInjectiveLinearTransformationsandDimension...................486 CILTCompositionofInjectiveLinearTransformations...................487 READReadingQuestions...................................487 EXCExercises.........................................488 SOLSolutions..........................................490 SLTSurjectiveLinearTransformations...............................492 ESLTExamplesofSurjectiveLinearTransformations....................492 RLTRangeofaLinearTransformation...........................496 SSSLTSpanningSetsandSurjectiveLinearTransformations...............500 SLTDSurjectiveLinearTransformationsandDimension..................502 CSLTCompositionofSurjectiveLinearTransformations..................503 READReadingQuestions...................................503 EXCExercises.........................................504 SOLSolutions..........................................506 IVLTInvertibleLinearTransformations..............................508 IVLTInvertibleLinearTransformations...........................508 IVInvertibility.........................................511 SIStructureandIsomorphism.................................515 RNLTRankandNullityofaLinearTransformation....................517 SLELTSystemsofLinearEquationsandLinearTransformations.............520 READReadingQuestions...................................521 EXCExercises.........................................522 SOLSolutions..........................................524 LTLinearTransformations..................................528 ChapterRRepresentations530 VRVectorRepresentations.....................................530 CVSCharacterizationofVectorSpaces............................535 CPCoordinatizationPrinciple................................536 READReadingQuestions...................................539 Version2.02

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CONTENTSxii EXCExercises.........................................540 SOLSolutions..........................................541 MRMatrixRepresentations.....................................542 NRFONewRepresentationsfromOld............................548 PMRPropertiesofMatrixRepresentations.........................552 IVLTInvertibleLinearTransformations...........................557 READReadingQuestions...................................561 EXCExercises.........................................562 SOLSolutions..........................................565 CBChangeofBasis.........................................574 EELTEigenvaluesandEigenvectorsofLinearTransformations..............574 CBMChange-of-BasisMatrix.................................575 MRSMatrixRepresentationsandSimilarity.........................581 CELTComputingEigenvectorsofLinearTransformations.................587 READReadingQuestions...................................595 EXCExercises.........................................596 SOLSolutions..........................................597 ODOrthonormalDiagonalization..................................601 TMTriangularMatrices....................................601 UTMRUpperTriangularMatrixRepresentation......................602 NMNormalMatrices......................................606 ODOrthonormalDiagonalization...............................607 NLTNilpotentLinearTransformations...............................610 NLTNilpotentLinearTransformations............................610 PNLTPropertiesofNilpotentLinearTransformations...................615 CFNLTCanonicalFormforNilpotentLinearTransformations...............619 ISInvariantSubspaces........................................627 ISInvariantSubspaces.....................................627 GEEGeneralizedEigenvectorsandEigenspaces.......................630 RLTRestrictionsofLinearTransformations.........................635 JCFJordanCanonicalForm....................................644 GESDGeneralizedEigenspaceDecomposition........................644 JCFJordanCanonicalForm.................................650 CHTCayley-HamiltonTheorem...............................663 RRepresentations.......................................665 AppendixCNComputationNotes667 MMAMathematica.........................................667 ME.MMAMatrixEntry....................................667 RR.MMARowReduce.....................................667 LS.MMALinearSolve.....................................668 VLC.MMAVectorLinearCombinations...........................668 NS.MMANullSpace......................................669 VFSS.MMAVectorFormofSolutionSet..........................669 GSP.MMAGram-SchmidtProcedure.............................670 TM.MMATransposeofaMatrix...............................671 MM.MMAMatrixMultiplication...............................671 MI.MMAMatrixInverse....................................671 TI86TexasInstruments86.....................................672 ME.TI86MatrixEntry.....................................672 Version2.02

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CONTENTSxiii RR.TI86RowReduce.....................................672 VLC.TI86VectorLinearCombinations............................672 TM.TI86TransposeofaMatrix................................673 TI83TexasInstruments83.....................................673 ME.TI83MatrixEntry.....................................673 RR.TI83RowReduce.....................................673 VLC.TI83VectorLinearCombinations............................674 SAGESAGE:OpenSourceMathematicsSoftware........................674 R.SAGERings.........................................674 ME.SAGEMatrixEntry....................................675 RR.SAGERowReduce....................................675 LS.SAGELinearSolve.....................................676 VLC.SAGEVectorLinearCombinations...........................677 MI.SAGEMatrixInverse...................................677 TM.SAGETransposeofaMatrix...............................677 E.SAGEEigenspaces......................................677 AppendixPPreliminaries679 CNOComplexNumberOperations.................................679 CNAArithmeticwithcomplexnumbers...........................679 CCNConjugatesofComplexNumbers............................681 MCNModulusofaComplexNumber............................682 SETSets...............................................683 SCSetCardinality.......................................684 SOSetOperations.......................................685 PTProofTechniques.........................................687 DDenitions..........................................687 TTheorems...........................................688 LLanguage...........................................688 GSGettingStarted.......................................689 CConstructiveProofs.....................................690 EEquivalences.........................................690 NNegation...........................................691 CPContrapositives.......................................691 CVConverses..........................................691 CDContradiction........................................692 UUniqueness..........................................693 MEMultipleEquivalences...................................693 PIProvingIdentities......................................693 DCDecompositions......................................694 IInduction...........................................694 PPractice............................................695 LCLemmasandCorollaries..................................696 AppendixAArchetypes698 A...................................................702 B...................................................707 C...................................................712 D...................................................716 E...................................................720 F...................................................724 Version2.02

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CONTENTSxiv G...................................................729 H...................................................733 I...................................................737 J...................................................741 K...................................................746 L...................................................750 M...................................................754 N...................................................757 O...................................................760 P...................................................763 Q...................................................765 R...................................................769 S...................................................772 T...................................................775 U...................................................777 V...................................................779 W..................................................781 X...................................................783 AppendixGFDLGNUFreeDocumentationLicense786 1.APPLICABILITYANDDEFINITIONS............................786 2.VERBATIMCOPYING.....................................787 3.COPYINGINQUANTITY...................................787 4.MODIFICATIONS........................................788 5.COMBININGDOCUMENTS..................................789 6.COLLECTIONSOFDOCUMENTS..............................790 7.AGGREGATIONWITHINDEPENDENTWORKS.....................790 8.TRANSLATION.........................................790 9.TERMINATION.........................................790 10.FUTUREREVISIONSOFTHISLICENSE.........................790 ADDENDUM:HowtousethisLicenseforyourdocuments...................791 PartTTopics FFields................................................793 FFields.............................................793 FFFiniteFields........................................794 EXCExercises.........................................799 SOLSolutions..........................................801 TTrace................................................802 EXCExercises.........................................806 SOLSolutions..........................................807 HPHadamardProduct.......................................808 DMHPDiagonalMatricesandtheHadamardProduct...................810 EXCExercises.........................................813 VMVandermondeMatrix......................................814 PSMPositiveSemi-deniteMatrices................................818 PSMPositiveSemi-DeniteMatrices.............................818 EXCExercises.........................................821 Version2.02

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CONTENTSxv ChapterMDMatrixDecompositions822 RODRankOneDecomposition...................................822 TDTriangularDecomposition....................................827 TDTriangularDecomposition.................................827 TDSSETriangularDecompositionandSolvingSystemsofEquations...........830 CTDComputingTriangularDecompositions........................831 SVDSingularValueDecomposition.................................835 MAPMatrix-AdjointProduct.................................835 SVDSingularValueDecomposition..............................838 SRSquareRoots...........................................840 SRMSquareRootofaMatrix................................840 PODPolarDecomposition......................................844 PartAApplications CFCurveFitting...........................................847 DFDataFitting........................................848 EXCExercises.........................................851 SASSharingASecret........................................852 Version2.02

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Contributors Beezer,David.BelarminePreparatorySchool,Tacoma Beezer,Robert.UniversityofPugetSoundhttp://buzzard.ups.edu/ Braithwaite,David.Chicago,Illinois Bucht,Sara.UniversityofPugetSound Caneld,Steve.UniversityofPugetSound Hubert,Dupont.Creteil,France Fellez,Sarah.UniversityofPugetSound Fickenscher,Eric.UniversityofPugetSound Jackson,Martin.UniversityofPugetSoundhttp://www.math.ups.edu/~martinj Hamrick,Mark.St.LouisUniversity Linenthal,Jacob.UniversityofPugetSound Million,Elizabeth.UniversityofPugetSound Osborne,Travis.UniversityofPugetSound Riegsecker,Joe.Middlebury,Indianajoepyeatpoboxdotcom Phelps,Douglas.UniversityofPugetSound Shoemaker,Mark.UniversityofPugetSound Zimmer,Andy.UniversityofPugetSound xvi

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Denitions SectionWILA SectionSSLE SLESystemofLinearEquations................................9 ESYSEquivalentSystems.....................................11 EOEquationOperations....................................11 SectionRREF MMatrix............................................24 CVColumnVector........................................24 ZCVZeroColumnVector.....................................25 CMCoecientMatrix......................................25 VOCVectorofConstants.....................................25 SOLVSolutionVector.......................................26 MRLSMatrixRepresentationofaLinearSystem........................26 AMAugmentedMatrix.....................................27 RORowOperations.......................................28 REMRow-EquivalentMatrices..................................28 RREFReducedRow-EchelonForm................................30 RRRow-Reducing........................................39 SectionTSS CSConsistentSystem......................................50 IDVIndependentandDependentVariables...........................52 SectionHSE HSHomogeneousSystem....................................62 TSHSETrivialSolutiontoHomogeneousSystemsofEquations.................62 NSMNullSpaceofaMatrix...................................64 SectionNM SQMSquareMatrix........................................71 NMNonsingularMatrix.....................................71 IMIdentityMatrix.......................................72 SectionVO VSCVVectorSpaceofColumnVectors..............................83 CVEColumnVectorEquality..................................84 CVAColumnVectorAddition..................................84 CVSMColumnVectorScalarMultiplication...........................85 xvii

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DEFINITIONSxviii SectionLC LCCVLinearCombinationofColumnVectors..........................90 SectionSS SSCVSpanofaSetofColumnVectors..............................112 SectionLI RLDCVRelationofLinearDependenceforColumnVectors...................132 LICVLinearIndependenceofColumnVectors..........................132 SectionLDS SectionO CCCVComplexConjugateofaColumnVector.........................167 IPInnerProduct........................................168 NVNormofaVector......................................171 OVOrthogonalVectors.....................................172 OSVOrthogonalSetofVectors.................................173 SUVStandardUnitVectors...................................173 ONSOrthoNormalSet......................................177 SectionMO VSMVectorSpaceof m n Matrices..............................182 MEMatrixEquality.......................................182 MAMatrixAddition.......................................182 MSMMatrixScalarMultiplication................................183 ZMZeroMatrix.........................................185 TMTransposeofaMatrix....................................185 SYMSymmetricMatrix......................................186 CCMComplexConjugateofaMatrix..............................187 AAdjoint............................................189 SectionMM MVPMatrix-VectorProduct...................................194 MMMatrixMultiplication....................................197 HMHermitianMatrix......................................205 SectionMISLE MIMatrixInverse........................................213 SectionMINM UMUnitaryMatrices......................................229 SectionCRS CSMColumnSpaceofaMatrix.................................236 RSMRowSpaceofaMatrix...................................243 SectionFS LNSLeftNullSpace.......................................257 EEFExtendedEchelonForm...................................261 Version2.02

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DEFINITIONSxix SectionVS VSVectorSpace.........................................279 SectionS SSubspace...........................................292 TSTrivialSubspaces......................................296 LCLinearCombination.....................................297 SSSpanofaSet.........................................298 SectionLISS RLDRelationofLinearDependence...............................308 LILinearIndependence....................................308 TSVSToSpanaVectorSpace...................................313 SectionB BBasis.............................................325 SectionD DDimension..........................................341 NOMNullityOfaMatrix.....................................347 ROMRankOfaMatrix......................................347 SectionPD DSDirectSum..........................................361 SectionDM ELEMElementaryMatrices....................................370 SMSubMatrix..........................................375 DMDeterminantofaMatrix..................................375 SectionPDM SectionEE EEMEigenvaluesandEigenvectorsofaMatrix.........................396 CPCharacteristicPolynomial.................................403 EMEigenspaceofaMatrix...................................404 AMEAlgebraicMultiplicityofanEigenvalue..........................406 GMEGeometricMultiplicityofanEigenvalue..........................406 SectionPEE SectionSD SIMSimilarMatrices.......................................432 DIMDiagonalMatrix.......................................435 DZMDiagonalizableMatrix....................................435 SectionLT LTLinearTransformation...................................452 PIPre-Image..........................................465 LTALinearTransformationAddition..............................467 LTSMLinearTransformationScalarMultiplication.......................468 Version2.02

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DEFINITIONSxx LTCLinearTransformationComposition............................469 SectionILT ILTInjectiveLinearTransformation..............................477 KLTKernelofaLinearTransformation.............................481 SectionSLT SLTSurjectiveLinearTransformation.............................492 RLTRangeofaLinearTransformation.............................496 SectionIVLT IDLTIdentityLinearTransformation...............................508 IVLTInvertibleLinearTransformations.............................508 IVSIsomorphicVectorSpaces..................................515 ROLTRankOfaLinearTransformation.............................517 NOLTNullityOfaLinearTransformation............................517 SectionVR VRVectorRepresentation....................................530 SectionMR MRMatrixRepresentation...................................542 SectionCB EELTEigenvalueandEigenvectorofaLinearTransformation.................574 CBMChange-of-BasisMatrix...................................575 SectionOD UTMUpperTriangularMatrix..................................601 LTMLowerTriangularMatrix..................................601 NRMLNormalMatrix........................................606 SectionNLT NLTNilpotentLinearTransformation..............................610 JBJordanBlock.........................................612 SectionIS ISInvariantSubspace.....................................627 GEVGeneralizedEigenvector...................................631 GESGeneralizedEigenspace...................................631 LTRLinearTransformationRestriction.............................635 IEIndexofanEigenvalue...................................641 SectionJCF JCFJordanCanonicalForm...................................650 SectionCNO CNEComplexNumberEquality.................................680 CNAComplexNumberAddition.................................680 CNMComplexNumberMultiplication..............................680 Version2.02

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DEFINITIONSxxi CCNConjugateofaComplexNumber.............................681 MCNModulusofaComplexNumber..............................682 SectionSET SETSet..............................................683 SSETSubset............................................683 ESEmptySet..........................................683 SESetEquality.........................................684 CCardinality..........................................684 SUSetUnion..........................................685 SISetIntersection.......................................685 SCSetComplement.......................................685 SectionPT SectionF FField.............................................793 IMPIntegersModuloaPrime..................................794 SectionT TTrace.............................................802 SectionHP HPHadamardProduct.....................................808 HIDHadamardIdentity.....................................809 HIHadamardInverse......................................809 SectionVM VMVandermondeMatrix....................................814 SectionPSM PSMPositiveSemi-DeniteMatrix...............................818 SectionROD SectionTD SectionSVD SVSingularValues.......................................839 SectionSR SRMSquareRootofaMatrix..................................843 SectionPOD SectionCF LSSLeastSquaresSolution...................................848 SectionSAS Version2.02

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Theorems SectionWILA SectionSSLE EOPSSEquationOperationsPreserveSolutionSets.......................12 SectionRREF REMESRow-EquivalentMatricesrepresentEquivalentSystems.................28 REMEFRow-EquivalentMatrixinEchelonForm.........................30 RREFUReducedRow-EchelonFormisUnique..........................32 SectionTSS RCLSRecognizingConsistencyofaLinearSystem.......................53 ISRNInconsistentSystems, r and n ...............................54 CSRNConsistentSystems, r and n ................................54 FVCSFreeVariablesforConsistentSystems...........................55 PSSLSPossibleSolutionSetsforLinearSystems.........................55 CMVEIConsistent,MoreVariablesthanEquations,Innitesolutions..............56 SectionHSE HSCHomogeneousSystemsareConsistent...........................62 HMVEIHomogeneous,MoreVariablesthanEquations,Innitesolutions............64 SectionNM NMRRINonsingularMatricesRowReducetotheIdentitymatrix................72 NMTNSNonsingularMatriceshaveTrivialNullSpaces......................74 NMUSNonsingularMatricesandUniqueSolutions........................74 NME1NonsingularMatrixEquivalences,Round1........................75 SectionVO VSPCVVectorSpacePropertiesofColumnVectors........................86 SectionLC SLSLCSolutionstoLinearSystemsareLinearCombinations..................93 VFSLSVectorFormofSolutionstoLinearSystems.......................99 PSPHSParticularSolutionPlusHomogeneousSolutions.....................105 SectionSS SSNSSpanningSetsforNullSpaces...............................118 SectionLI xxii

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THEOREMSxxiii LIVHSLinearlyIndependentVectorsandHomogeneousSystems................134 LIVRNLinearlyIndependentVectors, r and n ..........................136 MVSLDMoreVectorsthanSizeimpliesLinearDependence...................137 NMLICNonsingularMatriceshaveLinearlyIndependentColumns...............138 NME2NonsingularMatrixEquivalences,Round2........................138 BNSBasisforNullSpaces....................................139 SectionLDS DLDSDependencyinLinearlyDependentSets.........................152 BSBasisofaSpan.......................................157 SectionO CRVAConjugationRespectsVectorAddition..........................167 CRSMConjugationRespectsVectorScalarMultiplication...................167 IPVAInnerProductandVectorAddition............................169 IPSMInnerProductandScalarMultiplication.........................170 IPACInnerProductisAnti-Commutative............................170 IPNInnerProductsandNorms.................................171 PIPPositiveInnerProducts...................................172 OSLIOrthogonalSetsareLinearlyIndependent........................174 GSPGram-SchmidtProcedure..................................175 SectionMO VSPMVectorSpacePropertiesofMatrices............................184 SMSSymmetricMatricesareSquare..............................186 TMATransposeandMatrixAddition..............................186 TMSMTransposeandMatrixScalarMultiplication.......................187 TTTransposeofaTranspose..................................187 CRMAConjugationRespectsMatrixAddition..........................188 CRMSMConjugationRespectsMatrixScalarMultiplication...................188 CCMConjugateoftheConjugateofaMatrix..........................188 MCTMatrixConjugationandTransposes............................189 AMAAdjointandMatrixAddition................................189 AMSMAdjointandMatrixScalarMultiplication.........................189 AAAdjointofanAdjoint....................................190 SectionMM SLEMMSystemsofLinearEquationsasMatrixMultiplication..................195 EMMVPEqualMatricesandMatrix-VectorProducts.......................196 EMPEntriesofMatrixProducts.................................198 MMZMMatrixMultiplicationandtheZeroMatrix........................200 MMIMMatrixMultiplicationandIdentityMatrix........................200 MMDAAMatrixMultiplicationDistributesAcrossAddition....................201 MMSMMMatrixMultiplicationandScalarMatrixMultiplication.................201 MMAMatrixMultiplicationisAssociative...........................202 MMIPMatrixMultiplicationandInnerProducts........................202 MMCCMatrixMultiplicationandComplexConjugation.....................203 MMTMatrixMultiplicationandTransposes...........................203 MMADMatrixMultiplicationandAdjoints............................204 AIPAdjointandInnerProduct.................................204 Version2.02

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THEOREMSxxiv HMIPHermitianMatricesandInnerProducts..........................205 SectionMISLE TTMITwo-by-TwoMatrixInverse................................214 CINMComputingtheInverseofaNonsingularMatrix.....................217 MIUMatrixInverseisUnique..................................219 SSSocksandShoes.......................................219 MIMIMatrixInverseofaMatrixInverse.............................220 MITMatrixInverseofaTranspose...............................220 MISMMatrixInverseofaScalarMultiple............................221 SectionMINM NPNTNonsingularProducthasNonsingularTerms.......................226 OSISOne-SidedInverseisSucient...............................227 NINonsingularityisInvertibility...............................228 NME3NonsingularMatrixEquivalences,Round3........................228 SNCMSolutionwithNonsingularCoecientMatrix.......................229 UMIUnitaryMatricesareInvertible...............................230 CUMOSColumnsofUnitaryMatricesareOrthonormalSets...................230 UMPIPUnitaryMatricesPreserveInnerProducts........................231 SectionCRS CSCSColumnSpacesandConsistentSystems..........................237 BCSBasisoftheColumnSpace.................................239 CSNMColumnSpaceofaNonsingularMatrix..........................242 NME4NonsingularMatrixEquivalences,Round4........................242 REMRSRow-EquivalentMatriceshaveequalRowSpaces....................244 BRSBasisfortheRowSpace..................................245 CSRSTColumnSpace,RowSpace,Transpose...........................247 SectionFS PEEFPropertiesofExtendedEchelonForm...........................262 FSFourSubsets.........................................263 SectionVS ZVUZeroVectorisUnique....................................285 AIUAdditiveInversesareUnique................................286 ZSSMZeroScalarinScalarMultiplication............................286 ZVSMZeroVectorinScalarMultiplication............................286 AISMAdditiveInversesfromScalarMultiplication.......................287 SMEZVScalarMultiplicationEqualstheZeroVector.......................287 SectionS TSSTestingSubsetsforSubspaces...............................293 NSMSNullSpaceofaMatrixisaSubspace...........................296 SSSSpanofaSetisaSubspace.................................298 CSMSColumnSpaceofaMatrixisaSubspace.........................302 RSMSRowSpaceofaMatrixisaSubspace...........................303 LNSMSLeftNullSpaceofaMatrixisaSubspace.........................303 Version2.02

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THEOREMSxxv SectionLISS VRRBVectorRepresentationRelativetoaBasis.........................317 SectionB SUVBStandardUnitVectorsareaBasis.............................325 CNMBColumnsofNonsingularMatrixareaBasis........................330 NME5NonsingularMatrixEquivalences,Round5........................331 COBCoordinatesandOrthonormalBases............................332 UMCOBUnitaryMatricesConvertOrthonormalBases......................334 SectionD SSLDSpanningSetsandLinearDependence..........................341 BISBaseshaveIdenticalSizes..................................344 DCMDimensionof C m ......................................345 DPDimensionof P n .......................................345 DMDimensionof M mn .....................................345 CRNComputingRankandNullity................................347 RPNCRankPlusNullityisColumns...............................348 RNNMRankandNullityofaNonsingularMatrix........................349 NME6NonsingularMatrixEquivalences,Round6........................349 SectionPD ELISExtendingLinearlyIndependentSets...........................355 GGoldilocks..........................................355 PSSDProperSubspaceshaveSmallerDimension........................358 EDYESEqualDimensionsYieldsEqualSubspaces........................358 RMRTRankofaMatrixistheRankoftheTranspose......................359 DFSDimensionsofFourSubspaces...............................360 DSFBDirectSumFromaBasis..................................361 DSFOSDirectSumFromOneSubspace..............................362 DSZVDirectSumsandZeroVectors...............................362 DSZIDirectSumsandZeroIntersection.............................363 DSLIDirectSumsandLinearIndependence...........................364 DSDDirectSumsandDimension................................364 RDSRepeatedDirectSums...................................365 SectionDM EMDROElementaryMatricesDoRowOperations.........................372 EMNElementaryMatricesareNonsingular...........................374 NMPEMNonsingularMatricesareProductsofElementaryMatrices...............374 DMSTDeterminantofMatricesofSizeTwo...........................376 DERDeterminantExpansionaboutRows............................376 DTDeterminantoftheTranspose...............................377 DECDeterminantExpansionaboutColumns..........................378 SectionPDM DZRCDeterminantwithZeroRoworColumn..........................383 DRCSDeterminantforRoworColumnSwap..........................383 DRCMDeterminantforRoworColumnMultiples........................384 DERCDeterminantwithEqualRowsorColumns........................385 Version2.02

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THEOREMSxxvi DRCMADeterminantforRoworColumnMultiplesandAddition................385 DIMDeterminantoftheIdentityMatrix............................387 DEMDeterminantsofElementaryMatrices...........................388 DEMMMDeterminants,ElementaryMatrices,MatrixMultiplication...............389 SMZDSingularMatriceshaveZeroDeterminants........................389 NME7NonsingularMatrixEquivalences,Round7........................390 DRMMDeterminantRespectsMatrixMultiplication.......................391 SectionEE EMHEEveryMatrixHasanEigenvalue..............................400 EMRCPEigenvaluesofaMatrixareRootsofCharacteristicPolynomials............404 EMSEigenspaceforaMatrixisaSubspace...........................404 EMNSEigenspaceofaMatrixisaNullSpace..........................405 SectionPEE EDELIEigenvectorswithDistinctEigenvaluesareLinearlyIndependent............419 SMZESingularMatriceshaveZeroEigenvalues.........................420 NME8NonsingularMatrixEquivalences,Round8........................420 ESMMEigenvaluesofaScalarMultipleofaMatrix.......................421 EOMPEigenvaluesOfMatrixPowers...............................421 EPMEigenvaluesofthePolynomialofaMatrix........................421 EIMEigenvaluesoftheInverseofaMatrix...........................422 ETMEigenvaluesoftheTransposeofaMatrix.........................423 ERMCPEigenvaluesofRealMatricescomeinConjugatePairs..................423 DCPDegreeoftheCharacteristicPolynomial..........................424 NEMNumberofEigenvaluesofaMatrix............................425 MEMultiplicitiesofanEigenvalue...............................425 MNEMMaximumNumberofEigenvaluesofaMatrix......................427 HMREHermitianMatriceshaveRealEigenvalues........................427 HMOEHermitianMatriceshaveOrthogonalEigenvectors....................428 SectionSD SERSimilarityisanEquivalenceRelation...........................433 SMEESimilarMatriceshaveEqualEigenvalues.........................434 DCDiagonalizationCharacterization..............................436 DMFEDiagonalizableMatriceshaveFullEigenspaces......................438 DEDDistinctEigenvaluesimpliesDiagonalizable........................440 SectionLT LTTZZLinearTransformationsTakeZerotoZero........................456 MBLTMatricesBuildLinearTransformations..........................459 MLTCVMatrixofaLinearTransformation,ColumnVectors...................460 LTLCLinearTransformationsandLinearCombinations....................462 LTDBLinearTransformationDenedonaBasis........................462 SLTLTSumofLinearTransformationsisaLinearTransformation...............467 MLTLTMultipleofaLinearTransformationisaLinearTransformation............468 VSLTVectorSpaceofLinearTransformations..........................469 CLTLTCompositionofLinearTransformationsisaLinearTransformation..........470 SectionILT Version2.02

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THEOREMSxxvii KLTSKernelofaLinearTransformationisaSubspace.....................482 KPIKernelandPre-Image....................................483 KILTKernelofanInjectiveLinearTransformation.......................484 ILTLIInjectiveLinearTransformationsandLinearIndependence...............485 ILTBInjectiveLinearTransformationsandBases........................486 ILTDInjectiveLinearTransformationsandDimension.....................486 CILTICompositionofInjectiveLinearTransformationsisInjective..............487 SectionSLT RLTSRangeofaLinearTransformationisaSubspace.....................497 RSLTRangeofaSurjectiveLinearTransformation.......................498 SSRLTSpanningSetforRangeofaLinearTransformation...................500 RPIRangeandPre-Image....................................501 SLTBSurjectiveLinearTransformationsandBases.......................501 SLTDSurjectiveLinearTransformationsandDimension....................502 CSLTSCompositionofSurjectiveLinearTransformationsisSurjective.............503 SectionIVLT ILTLTInverseofaLinearTransformationisaLinearTransformation.............511 IILTInverseofanInvertibleLinearTransformation......................511 ILTISInvertibleLinearTransformationsareInjectiveandSurjective.............511 CIVLTCompositionofInvertibleLinearTransformations....................514 ICLTInverseofaCompositionofLinearTransformations...................514 IVSEDIsomorphicVectorSpaceshaveEqualDimension.....................516 ROSLTRankOfaSurjectiveLinearTransformation.......................517 NOILTNullityOfanInjectiveLinearTransformation......................517 RPNDDRankPlusNullityisDomainDimension.........................517 SectionVR VRLTVectorRepresentationisaLinearTransformation....................530 VRIVectorRepresentationisInjective.............................534 VRSVectorRepresentationisSurjective............................535 VRILTVectorRepresentationisanInvertibleLinearTransformation..............535 CFDVSCharacterizationofFiniteDimensionalVectorSpaces..................535 IFDVSIsomorphismofFiniteDimensionalVectorSpaces....................536 CLICoordinatizationandLinearIndependence........................536 CSSCoordinatizationandSpanningSets............................537 SectionMR FTMRFundamentalTheoremofMatrixRepresentation.....................544 MRSLTMatrixRepresentationofaSumofLinearTransformations...............548 MRMLTMatrixRepresentationofaMultipleofaLinearTransformation............548 MRCLTMatrixRepresentationofaCompositionofLinearTransformations..........549 KNSIKernelandNullSpaceIsomorphism............................552 RCSIRangeandColumnSpaceIsomorphism..........................555 IMRInvertibleMatrixRepresentations.............................557 IMILTInvertibleMatrices,InvertibleLinearTransformation..................560 NME9NonsingularMatrixEquivalences,Round9........................560 SectionCB Version2.02

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THEOREMSxxviii CBChange-of-Basis.......................................576 ICBMInverseofChange-of-BasisMatrix.............................576 MRCBMatrixRepresentationandChangeofBasis.......................581 SCBSimilarityandChangeofBasis...............................583 EEREigenvalues,Eigenvectors,Representations........................586 SectionOD PTMTProductofTriangularMatricesisTriangular.......................601 ITMTInverseofaTriangularMatrixisTriangular.......................602 UTMRUpperTriangularMatrixRepresentation.........................602 OBUTROrthonormalBasisforUpperTriangularRepresentation................605 ODOrthonormalDiagonalization................................607 OBNMOrthonormalBasesandNormalMatrices.........................609 SectionNLT NJBNilpotentJordanBlocks..................................614 ENLTEigenvaluesofNilpotentLinearTransformations.....................615 DNLTDiagonalizableNilpotentLinearTransformations.....................616 KPLTKernelsofPowersofLinearTransformations.......................616 KPNLTKernelsofPowersofNilpotentLinearTransformations.................617 CFNLTCanonicalFormforNilpotentLinearTransformations..................619 SectionIS EISEigenspacesareInvariantSubspaces............................629 KPISKernelsofPowersareInvariantSubspaces........................629 GESISGeneralizedEigenspaceisanInvariantSubspace.....................631 GEKGeneralizedEigenspaceasaKernel............................632 RGENRestrictiontoGeneralizedEigenspaceisNilpotent....................640 MRRGEMatrixRepresentationofaRestrictiontoaGeneralizedEigenspace..........643 SectionJCF GESDGeneralizedEigenspaceDecomposition..........................644 DGESDimensionofGeneralizedEigenspaces...........................650 JCFLTJordanCanonicalFormforaLinearTransformation...................651 CHTCayley-HamiltonTheorem.................................663 SectionCNO PCNAPropertiesofComplexNumberArithmetic........................680 CCRAComplexConjugationRespectsAddition.........................681 CCRMComplexConjugationRespectsMultiplication......................682 CCTComplexConjugationTwice................................682 SectionSET SectionPT SectionF FIMPFieldofIntegersModuloaPrime.............................795 SectionT TLTraceisLinear........................................802 TSRMTraceisSymmetricwithRespecttoMultiplication...................803 Version2.02

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THEOREMSxxix TISTTraceisInvariantUnderSimilarityTransformations...................803 TSETraceistheSumoftheEigenvalues............................803 SectionHP HPCHadamardProductisCommutative............................808 HPHIDHadamardProductwiththeHadamardIdentity.....................809 HPHIHadamardProductwithHadamardInverses.......................809 HPDAAHadamardProductDistributesAcrossAddition.....................810 HPSMMHadamardProductandScalarMatrixMultiplication..................810 DMHPDiagonalizableMatricesandtheHadamardProduct...................810 DMMPDiagonalMatricesandMatrixProducts..........................811 SectionVM DVMDeterminantofaVandermondeMatrix..........................814 NVMNonsingularVandermondeMatrix.............................817 SectionPSM CPSMCreatingPositiveSemi-DeniteMatrices.........................818 EPSMEigenvaluesofPositiveSemi-deniteMatrices......................819 SectionROD RODRankOneDecomposition..................................823 SectionTD TDTriangularDecomposition.................................827 TDEETriangularDecomposition,EntrybyEntry........................831 SectionSVD EEMAPEigenvaluesandEigenvectorsofMatrix-AdjointProduct................835 SVDSingularValueDecomposition...............................839 SectionSR PSMSRPositiveSemi-DeniteMatricesandSquareRoots....................840 EESREigenvaluesandEigenspacesofaSquareRoot......................841 USRUniqueSquareRoot.....................................843 SectionPOD PDMPolarDecompositionofaMatrix..............................844 SectionCF IPInterpolatingPolynomial..................................847 LSMRLeastSquaresMinimizesResiduals............................848 SectionSAS Version2.02

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Notation M A :Matrix..........................................24 MC[ A ] ij :MatrixComponents.................................24 CV v :ColumnVector......................................25 CVC[ v ] i :ColumnVectorComponents.............................25 ZCV 0 :ZeroColumnVector...................................25 MRLS LS A; b :MatrixRepresentationofaLinearSystem..................26 AM[ A j b ]:AugmentedMatrix.................................27 RO R i $ R j R i R i + R j :RowOperations........................28 RREFA r D F :ReducedRow-EchelonFormAnalysis......................30 NSM N A :NullSpaceofaMatrix...............................64 IM I m :IdentityMatrix.....................................72 VSCV C m :VectorSpaceofColumnVectors...........................83 CVE u = v :ColumnVectorEquality..............................84 CVA u + v :ColumnVectorAddition..............................85 CVSM u :ColumnVectorScalarMultiplication.........................85 SSV h S i :SpanofaSetofVectors................................112 CCCV u :ComplexConjugateofaColumnVector........................167 IP h u ; v i :InnerProduct....................................168 NV k v k :NormofaVector...................................171 SUV e i :StandardUnitVectors..................................173 VSM M mn :VectorSpaceofMatrices..............................182 ME A = B :MatrixEquality...................................182 MA A + B :MatrixAddition..................................183 MSM A :MatrixScalarMultiplication.............................183 ZM O :ZeroMatrix.......................................185 TM A t :TransposeofaMatrix.................................185 CCM A :ComplexConjugateofaMatrix............................187 A A :Adjoint..........................................189 MVPA u :Matrix-VectorProduct.................................194 MI A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 :MatrixInverse.....................................213 CSM C A :ColumnSpaceofaMatrix..............................236 RSM R A :RowSpaceofaMatrix...............................243 LNS L A :LeftNullSpace....................................257 Ddim V :Dimension.....................................341 NOM n A :NullityofaMatrix..................................347 ROM r A :RankofaMatrix...................................347 DS V = U W :DirectSum..................................361 ELEM E i;j E i E i;j :ElementaryMatrix.........................371 SM A i j j :SubMatrix......................................375 xxx

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NOTATIONxxxi DMdet A j A j :DeterminantofaMatrix...........................375 AME A :AlgebraicMultiplicityofanEigenvalue......................406 GME A :GeometricMultiplicityofanEigenvalue.....................406 LT T : U 7! V :LinearTransformation.............................452 KLT K T :KernelofaLinearTransformation.........................481 RLT R T :RangeofaLinearTransformation.........................496 ROLT r T :RankofaLinearTransformation..........................517 NOLT n T :NullityofaLinearTransformation.........................517 VR B w :VectorRepresentation...............................530 MR M T B;C :MatrixRepresentation...............................542 JB J n :JordanBlock....................................612 GES G T :GeneralizedEigenspace...............................631 LTR T j U :LinearTransformationRestriction..........................635 IE T :IndexofanEigenvalue...............................641 CNE = :ComplexNumberEquality.............................680 CNA + :ComplexNumberAddition.............................680 CNM :ComplexNumberMultiplication...........................680 CCN c :ConjugateofaComplexNumber............................681 SETM x 2 S :SetMembership...................................683 SSET S T :Subset........................................683 ES ; :EmptySet.........................................683 SE S = T :SetEquality.....................................684 C j S j :Cardinality.......................................684 SU S [ T :SetUnion......................................685 SI S T :SetIntersection...................................685 SC S :SetComplement.....................................685 T t A :Trace..........................................802 HP A B :HadamardProduct.................................808 HID J mn :HadamardIdentity..................................809 HI b A :HadamardInverse....................................809 SRM A 1 = 2 :SquareRootofaMatrix...............................843 Version2.02

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Diagrams DTSLSDecisionTreeforSolvingLinearSystems.........................56 CSRSTColumnSpaceandRowSpaceTechniques........................271 DLTADenitionofLinearTransformation,Additive......................453 DLTMDenitionofLinearTransformation,Multiplicative...................453 GLTGeneralLinearTransformation...............................457 NILTNon-InjectiveLinearTransformation...........................478 ILTInjectiveLinearTransformation..............................480 FTMRFundamentalTheoremofMatrixRepresentations....................545 FTMRAFundamentalTheoremofMatrixRepresentationsAlternate.............546 MRCLTMatrixRepresentationandCompositionofLinearTransformations..........552 xxxii

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Examples SectionWILA TMPTrailMixPackaging.....................................3 SectionSSLE STNESolvingtwononlinearequations.............................9 NSENotationforasystemofequations.............................10 TTSThreetypicalsystems....................................10 USThreeequations,onesolution...............................14 ISThreeequations,innitelymanysolutions........................15 SectionRREF AMAmatrix...........................................24 NSLENotationforsystemsoflinearequations..........................26 AMAAAugmentedmatrixforArchetypeA............................27 TREMTworow-equivalentmatrices................................28 USRThreeequations,onesolution,reprised..........................29 RREFAmatrixinreducedrow-echelonform...........................30 NRREFAmatrixnotinreducedrow-echelonform........................30 SABSolutionsforArchetypeB.................................36 SAASolutionsforArchetypeA.................................37 SAESolutionsforArchetypeE.................................38 SectionTSS RREFNReducedrow-echelonformnotation............................50 ISSIDescribinginnitesolutionsets,ArchetypeI.......................51 FDVFreeanddependentvariables................................52 CFVCountingfreevariables...................................55 OSGMDOnesolutiongivesmany,ArchetypeD..........................56 SectionHSE AHSACArchetypeCasahomogeneoussystem..........................62 HUSABHomogeneous,uniquesolution,ArchetypeB.......................63 HISAAHomogeneous,innitesolutions,ArchetypeA......................63 HISADHomogeneous,innitesolutions,ArchetypeD......................63 NSEAINullspaceelementsofArchetypeI............................65 CNS1Computinganullspace,#1................................65 CNS2Computinganullspace,#2................................66 SectionNM xxxiii

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EXAMPLESxxxiv SAsingularmatrix,ArchetypeA..............................71 NMAnonsingularmatrix,ArchetypeB............................72 IMAnidentitymatrix.....................................72 SRRSingularmatrix,row-reduced................................73 NSRNonsingularmatrix,row-reduced..............................73 NSSNullspaceofasingularmatrix...............................73 NSNMNullspaceofanonsingularmatrix.............................74 SectionVO VESEVectorequalityforasystemofequations.........................84 VAAdditionoftwovectorsin C 4 ...............................85 CVSMScalarmultiplicationin C 5 .................................86 SectionLC TLCTwolinearcombinationsin C 6 ...............................90 ABLCArchetypeBasalinearcombination...........................91 AALCArchetypeAasalinearcombination...........................92 VFSADVectorformofsolutionsforArchetypeD.........................95 VFSVectorformofsolutions...................................96 VFSAIVectorformofsolutionsforArchetypeI..........................102 VFSALVectorformofsolutionsforArchetypeL.........................103 PSHSParticularsolutions,homogeneoussolutions,ArchetypeD................106 SectionSS ABSAbasicspan.........................................112 SCAASpanofthecolumnsofArchetypeA............................114 SCABSpanofthecolumnsofArchetypeB............................116 SSNSSpanningsetofanullspace................................118 NSDSNullspacedirectlyasaspan................................119 SCADSpanofthecolumnsofArchetypeD............................120 SectionLI LDSLinearlydependentsetin C 5 ................................132 LISLinearlyindependentsetin C 5 ...............................133 LIHSLinearlyindependent,homogeneoussystem........................134 LDHSLinearlydependent,homogeneoussystem.........................135 LDRNLinearlydependent, r
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EXAMPLESxxxv CSIPComputingsomeinnerproducts..............................168 CNSVComputingthenormofsomevectors...........................171 TOVTwoorthogonalvectors...................................172 SUVOSStandardUnitVectorsareanOrthogonalSet......................173 AOSAnorthogonalset......................................173 GSTVGram-Schmidtofthreevectors...............................176 ONTVOrthonormalset,threevectors...............................177 ONFVOrthonormalset,fourvectors...............................178 SectionMO MAAdditionoftwomatricesin M 23 ..............................183 MSMScalarmultiplicationin M 32 ................................183 TMTransposeofa3 4matrix................................185 SYMAsymmetric5 5matrix.................................186 CCMComplexconjugateofamatrix...............................187 SectionMM MTVAmatrixtimesavector...................................194 MNSLEMatrixnotationforsystemsoflinearequations......................195 MBCMoney'sbestcities.....................................195 PTMProductoftwomatrices..................................197 MMNCMatrixmultiplicationisnotcommutative.........................198 PTMEEProductoftwomatrices,entry-by-entry..........................199 SectionMISLE SABMISolutionstoArchetypeBwithamatrixinverse.....................212 MWIAAAmatrixwithoutaninverse,ArchetypeA........................213 MIMatrixinverse........................................214 CMIComputingamatrixinverse................................216 CMIABComputingamatrixinverse,ArchetypeB........................218 SectionMINM UM3Unitarymatrixofsize3...................................229 UPMUnitarypermutationmatrix................................229 OSMCOrthonormalsetfrommatrixcolumns...........................231 SectionCRS CSMCSColumnspaceofamatrixandconsistentsystems....................236 MCSMMembershipinthecolumnspaceofamatrix.......................237 CSTWColumnspace,twoways..................................239 CSOCDColumnspace,originalcolumns,ArchetypeD......................240 CSAAColumnspaceofArchetypeA...............................241 CSABColumnspaceofArchetypeB...............................241 RSAIRowspaceofArchetypeI..................................243 RSREMRowspacesoftworow-equivalentmatrices........................245 IASImprovingaspan......................................246 CSROIColumnspacefromrowoperations,ArchetypeI.....................247 SectionFS LNSLeftnullspace........................................257 Version2.02

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EXAMPLESxxxvi CSANSColumnspaceasnullspace.................................258 SEEFSubmatricesofextendedechelonform...........................261 FS1Foursubsets,#1.......................................267 FS2Foursubsets,#2.......................................268 FSAGFoursubsets,ArchetypeG.................................269 SectionVS VSCVThevectorspace C m ....................................281 VSMThevectorspaceofmatrices, M mn ............................281 VSPThevectorspaceofpolynomials, P n ............................281 VSISThevectorspaceofinnitesequences...........................282 VSFThevectorspaceoffunctions................................282 VSSThesingletonvectorspace.................................283 CVSThecrazyvectorspace...................................283 PCVSPropertiesfortheCrazyVectorSpace...........................288 SectionS SC3Asubspaceof C 3 ......................................292 SP4Asubspaceof P 4 ......................................294 NSC2ZAnon-subspacein C 2 ,zerovector.............................295 NSC2AAnon-subspacein C 2 ,additiveclosure..........................295 NSC2SAnon-subspacein C 2 ,scalarmultiplicationclosure...................296 RSNSRecastingasubspaceasanullspace............................297 LCMAlinearcombinationofmatrices..............................297 SSPSpanofasetofpolynomials................................299 SM32Asubspaceof M 32 ......................................300 SectionLISS LIP4Linearindependencein P 4 .................................308 LIM32Linearindependencein M 32 ................................310 LICLinearlyindependentsetinthecrazyvectorspace....................312 SSP4Spanningsetin P 4 ......................................313 SSM22Spanningsetin M 22 .....................................314 SSCSpanningsetinthecrazyvectorspace..........................315 AVRAvectorrepresentation...................................316 SectionB BPBasesfor P n .........................................326 BMAbasisforthevectorspaceofmatrices..........................326 BSP4Abasisforasubspaceof P 4 ................................326 BSM22Abasisforasubspaceof M 22 ...............................327 BCBasisforthecrazyvectorspace..............................328 RSBRowspacebasis.......................................328 RSReducingaspan.......................................329 CABAKColumnsasBasis,ArchetypeK..............................330 CROB4Coordinatizationrelativetoanorthonormalbasis, C 4 ..................332 CROB3Coordinatizationrelativetoanorthonormalbasis, C 3 ..................333 SectionD LDP4Linearlydependentsetin P 4 ................................344 Version2.02

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EXAMPLESxxxvii DSM22Dimensionofasubspaceof M 22 ..............................345 DSP4Dimensionofasubspaceof P 4 ...............................346 DCDimensionofthecrazyvectorspace............................346 VSPUDVectorspaceofpolynomialswithunboundeddegree...................346 RNMRankandnullityofamatrix................................347 RNSMRankandnullityofasquarematrix............................348 SectionPD BPRBasesfor P n ,reprised....................................356 BDM22Basisbydimensionin M 22 .................................357 SVP4Setsofvectorsin P 4 .....................................357 RRTIRank,rankoftranspose,ArchetypeI...........................359 SDSSimpledirectsum......................................361 SectionDM EMROElementarymatricesandrowoperations.........................371 SSSomesubmatrices......................................375 D33MDeterminantofa3 3matrix...............................375 TCSDTwocomputations,samedeterminant...........................379 DUTMDeterminantofanuppertriangularmatrix........................379 SectionPDM DRODeterminantbyrowoperations..............................386 ZNDABZeroandnonzerodeterminant,ArchetypesAandB...................390 SectionEE SEESomeeigenvaluesandeigenvectors.............................396 PMPolynomialofamatrix...................................398 CAEHWComputinganeigenvaluethehardway..........................401 CPMS3Characteristicpolynomialofamatrix,size3.......................403 EMS3Eigenvaluesofamatrix,size3...............................404 ESMS3Eigenspacesofamatrix,size3...............................405 EMMS4Eigenvaluemultiplicities,matrixofsize4.........................406 ESMS4Eigenvalues,symmetricmatrixofsize4..........................407 HMEM5Highmultiplicityeigenvalues,matrixofsize5......................408 CEMS6Complexeigenvalues,matrixofsize6...........................409 DEMS5Distincteigenvalues,matrixofsize5...........................411 SectionPEE BDEBuildingdesiredeigenvalues................................422 SectionSD SMS5Similarmatricesofsize5..................................432 SMS3Similarmatricesofsize3..................................433 EENSEqualeigenvalues,notsimilar...............................435 DABDiagonalizationofArchetypeB..............................435 DMS3Diagonalizingamatrixofsize3..............................437 NDMS4Anon-diagonalizablematrixofsize4...........................440 DEHDDistincteigenvalues,hencediagonalizable.........................440 HPDMHighpowerofadiagonalizablematrix...........................441 Version2.02

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EXAMPLESxxxviii FSCFFibonaccisequence,closedform..............................442 SectionLT ALTAlineartransformation...................................453 NLTNotalineartransformation.................................454 LTPMLineartransformation,polynomialstomatrices.....................455 LTPPLineartransformation,polynomialstopolynomials...................455 LTMLineartransformationfromamatrix...........................457 MFLTMatrixfromalineartransformation............................459 MOLTMatrixofalineartransformation.............................461 LTDB1Lineartransformationdenedonabasis.........................463 LTDB2Lineartransformationdenedonabasis.........................464 LTDB3Lineartransformationdenedonabasis.........................464 SPIASSamplepre-images,ArchetypeS..............................465 STLTSumoftwolineartransformations.............................468 SMLTScalarmultipleofalineartransformation.........................469 CTLTCompositionoftwolineartransformations........................470 SectionILT NIAQNotinjective,ArchetypeQ.................................477 IARInjective,ArchetypeR...................................478 IAVInjective,ArchetypeV...................................480 NKAONontrivialkernel,ArchetypeO...............................481 TKAPTrivialkernel,ArchetypeP.................................482 NIAQRNotinjective,ArchetypeQ,revisited...........................484 NIAONotinjective,ArchetypeO.................................485 IAPInjective,ArchetypeP...................................485 NIDAUNotinjectivebydimension,ArchetypeU.........................486 SectionSLT NSAQNotsurjective,ArchetypeQ................................492 SARSurjective,ArchetypeR...................................493 SAVSurjective,ArchetypeV...................................494 RAORange,ArchetypeO.....................................496 FRANFullrange,ArchetypeN..................................497 NSAQRNotsurjective,ArchetypeQ,revisited...........................499 NSAONotsurjective,ArchetypeO................................499 SANSurjective,ArchetypeN...................................500 BRLTAbasisfortherangeofalineartransformation.....................501 NSDATNotsurjectivebydimension,ArchetypeT........................502 SectionIVLT AIVLTAninvertiblelineartransformation............................508 ANILTAnon-invertiblelineartransformation...........................509 CIVLTComputingtheInverseofaLinearTransformations...................512 IVSAVIsomorphicvectorspaces,ArchetypeV..........................515 SectionVR VRC4Vectorrepresentationin C 4 .................................531 VRP2Vectorrepresentationsin P 2 ................................533 Version2.02

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EXAMPLESxxxix TIVSTwoisomorphicvectorspaces...............................536 CVSRCrazyvectorspacerevealed................................536 ASCAsubspacecharacterized..................................536 MIVSMultipleisomorphicvectorspaces.............................536 CP2Coordinatizingin P 2 ....................................537 CM32Coordinatizationin M 32 ..................................538 SectionMR OLTTROnelineartransformation,threerepresentations.....................542 ALTMMAlineartransformationasmatrixmultiplication.....................546 MPMRMatrixproductofmatrixrepresentations.........................549 KVMRKernelviamatrixrepresentation..............................553 RVMRRangeviamatrixrepresentation..............................556 ILTVRInverseofalineartransformationviaarepresentation..................559 SectionCB ELTBMEigenvectorsoflineartransformationbetweenmatrices.................574 ELTBPEigenvectorsoflineartransformationbetweenpolynomials...............575 CBPChangeofbasiswithpolynomials.............................576 CBCVChangeofbasiswithcolumnvectors...........................579 MRCMMatrixrepresentationsandchange-of-basismatrices...................581 MRBEMatrixrepresentationwithbasisofeigenvectors.....................584 ELTTEigenvectorsofalineartransformation,twice......................587 CELTComplexeigenvectorsofalineartransformation.....................592 SectionOD ANMAnormalmatrix.......................................606 SectionNLT NM64Nilpotentmatrix,size6,index4..............................610 NM62Nilpotentmatrix,size6,index2..............................611 JB4Jordanblock,size4.....................................612 NJB5NilpotentJordanblock,size5...............................613 NM83Nilpotentmatrix,size8,index3..............................614 KPNLTKernelsofpowersofanilpotentlineartransformation..................618 CFNLTCanonicalformforanilpotentlineartransformation...................623 SectionIS TISTwoinvariantsubspaces..................................627 EISEigenspacesasinvariantsubspaces.............................629 ISJBInvariantsubspacesandJordanblocks..........................630 GE4Generalizedeigenspaces,dimension4domain.......................632 GE6Generalizedeigenspaces,dimension6domain.......................633 LTRGELineartransformationrestrictionongeneralizedeigenspace...............635 ISMR4Invariantsubspaces,matrixrepresentation,dimension4domain............638 ISMR6Invariantsubspaces,matrixrepresentation,dimension6domain............639 GENR6Generalizedeigenspacesandnilpotentrestrictions,dimension6domain........641 SectionJCF JCF10Jordancanonicalform,size10...............................652 Version2.02

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EXAMPLESxl SectionCNO ACNArithmeticofcomplexnumbers..............................679 CSCNConjugateofsomecomplexnumbers...........................681 MSCNModulusofsomecomplexnumbers............................682 SectionSET SETMSetmembership.......................................683 SSETSubset............................................683 CSCardinalityandSize.....................................684 SUSetunion...........................................685 SISetintersection.......................................685 SCSetcomplement.......................................686 SectionPT SectionF IM11Integersmod11.......................................795 VSIM5Vectorspaceoverintegersmod5..............................795 SM2Z7Symmetricmatricesofsize2over Z 7 ...........................796 FF8Finiteeldofsize8.....................................796 SectionT SectionHP HPHadamardProduct.....................................808 SectionVM VM4Vandermondematrixofsize4...............................814 SectionPSM SectionROD ROD2Rankonedecomposition,size2..............................824 ROD4Rankonedecomposition,size4..............................825 SectionTD TD4Triangulardecomposition,size4..............................829 TDSSETriangulardecompositionsolvesasystemofequations.................830 TDEE6Triangulardecomposition,entrybyentry,size6.....................833 SectionSVD SectionSR SectionPOD SectionCF PTFPPolynomialthroughvepoints...............................847 SectionSAS SS6WSharingasecret6ways...................................853 Version2.02

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Preface Thistextbookisdesignedtoteachtheuniversitymathematicsstudentthebasicsoflinearalgebraand thetechniquesofformalmathematics.Therearenoprerequisitesotherthanordinaryalgebra,butitis probablybestusedbyastudentwhohasthemathematicalmaturity"ofasophomoreorjunior.Thetext hastwogoals:toteachthefundamentalconceptsandtechniquesofmatrixalgebraandabstractvector spaces,andtoteachthetechniquesassociatedwithunderstandingthedenitionsandtheoremsforming acoherentareaofmathematics.Sothereisanemphasisonworkedexamplesofnontrivialsizeandon provingtheoremscarefully. Thisbookiscopyrighted.Thismeansthatgovernmentshavegrantedtheauthoramonopoly|the exclusiverighttocontrolthemakingofcopiesandderivativeworksformanyyearstoomanyyearsin somecases.Italsogivesotherslimitedrights,generallyreferredtoasfairuse,"suchastherightto quotesectionsinareviewwithoutseekingpermission.However,theauthorlicensesthisbooktoanyone underthetermsoftheGNUFreeDocumentationLicenseGFDL,whichgivesyoumorerightsthanmost copyrightsseeAppendixGFDL[786].Looselyspeaking,youmaymakeasmanycopiesasyoulikeatno cost,andyoumaydistributetheseunmodiedcopiesifyouplease.Youmaymodifythebookforyourown use.Thecatchisthatifyoumakemodicationsandyoudistributethemodiedversion,ormakeuseof portionsinexcessoffairuseinanotherwork,thenyoumustalsolicensethenewworkwiththeGFDL.So thebookhaslotsofinherentfreedom,andnooneisallowedtodistributeaderivativeworkthatrestricts thesefreedoms.Seethelicenseitselfintheappendixfortheexactdetailsoftheadditionalrightsyou havebeengiven. Noticethatinitiallymostpeoplearestruckbythenotionthatthisbookis free theFrenchwouldsay gratuit ,atnocost.Anditis.However,itismoreimportantthatthebookhas freedom theFrench wouldsay liberte ,liberty.Itwillnevergooutofprint"norwillthereeverbetrivialupdatesdesigned onlytofrustratetheusedbookmarket.Thoseconsideringteachingacoursewiththisbookcanexamine itthoroughlyinadvance.Addingnewexercisesornewsectionshasbeenpurposelymadeveryeasy,and thehopeisthatotherswillcontributethesemodicationsbackforincorporationintothebook,forthe benetofall. Dependingonhowyoureceivedyourcopy,youmaywanttocheckforthelatestversionandother newsat http://linear.ups.edu/ Topics ThersthalfofthistextthroughChapterM[182]isbasicallyacourseinmatrixalgebra, thoughthefoundationofsomemoreadvancedideasisalsobeingformedintheseearlysections.Vectors arepresentedexclusivelyascolumnvectorssincewealsohavethetypographicfreedomtoavoidwriting acolumnvectorinlineasthetransposeofarowvector,andlinearcombinationsarepresentedveryearly. Spans,nullspaces,columnspacesandrowspacesarealsopresentedearly,simplyassets,savingmostof theirvectorspacepropertiesforlater,sotheyarefamiliarobjectsbeforebeingscrutinizedcarefully. Youcannotdo everything early,soinparticularmatrixmultiplicationcomeslaterthanusual.However, withadenitionbuiltonlinearcombinationsofcolumnvectors,itshouldseemmorenaturalthanthe morefrequentdenitionusingdotproductsofrowswithcolumns.Andthisdelayemphasizesthatlinear algebraisbuiltuponvectoradditionandscalarmultiplication.Ofcourse,matrixinversesmustwaitfor matrixmultiplication,butthisdoesnotpreventnonsingularmatricesfromoccurringsooner.Vectorspace xli

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PREFACExlii propertiesarehintedatwhenvectorandmatrixoperationsarerstdened,butthenotionofavector spaceissavedforamoreaxiomatictreatmentlaterChapterVS[279].Oncebasesanddimensionhave beenexploredinthecontextofvectorspaces,lineartransformationsandtheirmatrixrepresentations follow.ThegoalofthebookistogoasfarasJordancanonicalformintheCorePartC[2],withless centraltopicscollectedintheTopicsPartT[793].AthirdpartcontainscontributedapplicationsPart A[847],withnotationandtheoremsintegratedwiththeearliertwoparts. Linearalgebraisanidealsubjectforthenovicemathematicsstudenttolearnhowtodevelopatopic precisely,withalltherigormathematicsrequires.Unfortunately,muchofthisrigorseemstohaveescaped thestandardcalculuscurriculum,soformanyuniversitystudentsthisistheirrstexposuretocareful denitionsandtheorems,andtheexpectationthattheyfullyunderstandthem,tosaynothingofthe expectationthattheybecomeprocientinformulatingtheirownproofs.Wehavetriedtomakethistext ashelpfulaspossiblewiththistransition.Everydenitionisstatedcarefully,setapartfromthetext. Likewise,everytheoremiscarefullystated,andalmosteveryonehasacompleteproof.Theoremsusually havejustoneconclusion,sotheycanbereferencedpreciselylater.Denitionsandtheoremsarecataloged inorderoftheirappearanceinthefrontofthebookDenitions[viii],Theorems[ix],andalphabetical orderintheindexattheback.Alongtheway,therearediscussionsofsomemoreimportantideasrelating toformulatingproofsProofTechniques[ ?? ],whichispartadviceandpartlogic. OriginandHistory Thisbookistheresultoftheconuenceofseveralrelatedeventsandtrends. AttheUniversityofPugetSoundweteachaone-semester,post-calculuslinearalgebracourseto studentsmajoringinmathematics,computerscience,physics,chemistryandeconomics.Between January1986andJune2002,Itaughtthiscourseseventeentimes.FortheSpring2003semester,I electedtoconvertmycoursenotestoanelectronicformsothatitwouldbeeasiertoincorporatethe inevitableandnearly-constantrevisions.Centraltomynewnoteswasacollectionofstockexamples thatwouldbeusedrepeatedlytoillustratenewconcepts.ThesewouldbecometheArchetypes, AppendixA[698].Itwasonlyashortleaptothendecidetodistributecopiesofthesenotesand examplestothestudentsinthetwosectionsofthiscourse.Asthesemesterworeon,thenotesbegan tolooklesslikenotesandmorelikeatextbook. IusedthenotesagainintheFall2003semesterforasinglesectionofthecourse.Simultaneously,the textbookIwasusingcameoutinafthedition.Anewchapterwasaddedtowardthestartofthe book,andafewadditionalexerciseswereaddedinotherchapters.Thisdemandedtheannoyance ofreworkingmynotesandlistofsuggestedexercisestoconformwiththechangednumberingofthe chaptersandexercises.IhadanalmostidenticalexperiencewiththethirdcourseIwasteaching thatsemester.IalsolearnedthatinthenextacademicyearIwouldbeteachingacoursewheremy textbookofchoicehadgoneoutofprint.Ifelttherehadtobeabetteralternativetohavingthe organizationofmycoursesbuetedbytheeconomicsoftraditionaltextbookpublishing. IhadusedT E XandtheInternetformanyyears,sotherewaslittletostandinthewayoftypesetting, distributingandmarketing"afreebook.Withrecreationalandprofessionalinterestsinsoftware development,Ihadlongbeenfascinatedbytheopen-sourcesoftwaremovement,asexempliedby thesuccessofGNUandLinux,thoughpublic-domainT E Xmightalsodeservemention.Obviously, thisbookisanattempttocarryoverthatmodelofcreativeendeavortotextbookpublishing. AsasabbaticalprojectduringtheSpring2004semester,Iembarkedonthecurrentprojectofcreating afreely-distributablelinearalgebratextbook.NoticetheimpliednancialsupportoftheUniversity ofPugetSoundtothisproject.Mostofthematerialwaswrittenfromscratchsincechangesin notationandapproachmademuchofmynotesoflittleuse.ByAugust2004Ihadwrittenhalfthe materialnecessaryforourMath232course.TheremaininghalfwaswrittenduringtheFall2004 semesterasItaughtanothertwosectionsofMath232. Version2.02

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PREFACExliii Whileinearly2005thebookwascompleteenoughtobuildacoursearoundandVersion1.0was released.Workhascontinuedsince,llingoutthenarrative,exercisesandsupplements. However,muchofmymotivationforwritingthisbookiscapturedbythesentimentsexpressedbyH.M. CundyandA.P.RolletintheirPrefacetotheFirstEditionof MathematicalModels ,especiallythe nalsentence, Thisbookwasbornintheclassroom,andarosefromthespontaneousinterestofaMathematical Sixthintheconstructionofsimplemodels.Adesiretoshowthateveninmathematicsonecould havefunledtoanexhibitionoftheresultsandattractedconsiderableattentionthroughoutthe school.SincethentheSherbornecollectionhasgrown,ideashavecomefrommanysources,and widespreadinteresthasbeenshown.Itseemsthereforedesirabletogivepermanentformtothe lessonsofexperiencesothatotherscanbenetbythemandbeencouragedtoundertakesimilar work. HowToUseThisBook Chapters,Theorems,etc.arenotnumberedinthisbook,butareinstead referencedbyacronyms.ThismeansthatTheoremXYZwillalwaysbeTheoremXYZ,nomatterif newsectionsareadded,orifanindividualdecidestoremovecertainothersections.Withinsections, thesubsectionsareacronymsthatbeginwiththeacronymofthesection.SoSubsectionXYZ.ABisthe subsectionABinSectionXYZ.Acronymsareuniquewithintheirtype,soforexamplethereisjustone DenitionB[325],butthereisalsoaSectionB[325].Atrst,allthelettersyingaroundmaybeconfusing, butwithtime,youwillbegintorecognizethemoreimportantonesonsight.Furthermore,therearelists oftheorems,examples,etc.inthefrontofthebook,andanindexthatcontainseveryacronym.Ifyou arereadingthisinanelectronicversionPDForXML,youwillseethatallofthecross-referencesare hyperlinks,allowingyoutoclicktoadenitionorexample,andthenusethebackbuttontoreturn.In printedversions,youmustrelyonthepagenumbers.However,notethatpagenumbersarenotpermanent! Dierenteditions,dierentmargins,ordierentsizedpaperwillaectwhatcontentisoneachpage.And intime,theadditionofnewmaterialwillaectthepagenumbering. Chapterdivisionsarenotcriticaltotheorganizationofthebook,asSectionsarethemainorganizational unit.Sectionsaredesignedtobethesubjectofasinglelectureorclassroomsession,thoughthereis frequentlymorematerialthancanbediscussedandillustratedinafty-minutesession.Consequently, theinstructorwillneedtobeselectiveaboutwhichtopicstoillustratewithotherexamplesandwhich topicstoleavetothestudent'sreading.Manyoftheexamplesaremeanttobelarge,suchasusingve orsixvariablesinasystemofequations,sotheinstructormayjustwanttowalk"aclassthroughthese examples.Thebookhasbeenwrittenwiththeideathatsomemayworkthroughitindependently,sothe hopeisthatstudentscanlearnsomeofthemoremechanicalideasontheirown. ThehighestleveldivisionofthebookisthethreeParts:Core,Topics,ApplicationsPartC[2],PartT [793],PartA[847].TheCoreismeanttocarefullydescribethebasicideasrequiredofarstexposureto linearalgebra.InthenalsectionsoftheCore,oneshouldaskthequestion:whichpreviousSectionscould beremovedwithoutdestroyingthelogicaldevelopmentofthesubject?Hopefully,theanswerisnone." ThegoalofthebookistonishtheCorewithaverygeneralrepresentationofalineartransformation Jordancanonicalform,SectionJCF[644].Ofcourse,therewillnotbeuniversalagreementonwhat should,orshouldnot,constitutetheCore,butthemainideaistolimitittoaboutfortysections.Topics PartT[793]ismeanttocontainthosesubjectsthatareimportantinlinearalgebra,andwhichwould makeprotabledetoursfromtheCoreforthoseinterestedinpursuingthem.ApplicationsPartA[847] shouldillustratethepowerandwidespreadapplicabilityoflinearalgebratoasmanyeldsaspossible.The ArchetypesAppendixA[698]covermanyofthecomputationalaspectsofsystemsoflinearequations, matricesandlineartransformations.Thestudentshouldconsultthemoften,andthisisencouragedby exercisesthatsimplysuggesttherightpropertiestoexamineattherighttime.Butwhatismoreimportant, thisarepositorythatcontainsenoughvarietytoprovideabundantexamplesofkeytheorems,whilealso providingcounterexamplestohypothesesorconversesoftheorems.Thesummarytableatthestartofthis appendixshouldbeespeciallyuseful. Version2.02

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PREFACExliv IrequiremystudentstoreadeachSection prior totheday'sdiscussiononthatsection.Forsome studentsthisisanovelidea,butattheendofthesemesterafewalwaysreportonthebenets,bothfor thiscourseandothercourseswheretheyhaveadoptedthehabit.Tomakegoodonthisrequirement,each sectioncontainsthreeReadingQuestions.Thesesometimesonlyrequireparrotingbackakeydenitionor theorem,ortheyrequireperformingasmallexampleofakeycomputation,ortheyaskformusingsonkey ideasornewrelationshipsbetweenoldideas.Answersareemailedtometheeveningbeforethelecture. Giventheavorandpurposeofthesequestions,includingsolutionsseemsfoolish. EverychapterofPartC[2]endswithAnnotatedAcronyms",ashortlistofcriticaltheoremsor denitionsfromthatchapter.Thereareavarietyofreasonsforanyoneofthesetohavebeenchosen, andreadingtheshortparagraphsaftersomeofthesemightprovideinsightintothepossibilities.An end-of-chapterreviewmightusefullyincorporateaclosereadingoftheselists. Formulatinginterestingandeectiveexercisesisasdicult,ormoreso,thanbuildinganarrative. Butitistheplacewhereastudentreallylearnsthematerial.Assuch,forthestudent'sbenet,complete solutionsshouldbegiven.Asthelistofexercisesexpands,theamountwithsolutionsshouldsimilarly expand.Exercisesandtheirsolutionsarereferencedwithasectionname,followedbyadot,thena letterC,M,orTandanumber.Theletter`C'indicatesaproblemthatismostlycomputationalin nature,whiletheletter`T'indicatesaproblemthatismoretheoreticalinnature.Aproblemwithaletter `M'issomewhereinbetweenmiddle,mid-level,median,middling,probablyamixofcomputationand applicationsoftheorems.SoSolutionMO.T13[193]isasolutiontoanexerciseinSectionMO[182]that istheoreticalinnature.Thenumber`13'hasnointrinsicmeaning. MoreonFreedom Thisbookisfreely-distributableunderthetermsoftheGFDL,alongwiththe underlyingT E Xcodefromwhichthebookisbuilt.Thisarrangementprovidesmanybenetsunavailable withtraditionaltexts. Nocost,orlowcost,tostudents.Withnophysicalvesseli.e.paper,binding,notransportation costsInternetbandwidthbeinganegligiblecostandnomarketingcostsevaluationanddeskcopies arefreetoall,anyonewithanInternetconnectioncanobtainit,andateachercouldmakeavailable papercopiesinsucientquantitiesforaclass.Thecosttoprintacopyisnotinsignicant,butis justafractionofthecostofatraditionaltextbookwhenprintingishandledbyaprint-on-demand serviceovertheInternet.Studentswillnotfeeltheneedtosellbacktheirbooknorshouldtherebe muchofamarketforusedcopies,andinfutureyearscanevenpickupanewereditionfreely. Electronicversionsofthebookcontainextensivehyperlinks.Specically,mostlogicalstepsinproofs andexamplesincludelinksbacktothepreviousdenitionsortheoremsthatsupportthatstep.With whatevervieweryoumightbeusingwebbrowser,PDFreadertheback"buttoncanthenreturn youtothemiddleoftheproofyouwerestudying.Soevenifyouarereadingaphysicalcopyofthis book,youcanbenetfromalsoworkingwithanelectronicversion. Atraditionalbook,whichthepublisherisunwillingtodistributeinaneasily-copiedelectronicform, cannotoerthisveryintuitiveandexibleapproachtolearningmathematics. Thebookwillnotgooutofprint.Nomatterwhat,ateachercanmaintaintheirowncopyandusethe bookforasmanyyearsastheydesire.Further,thenamingschemesforchapters,sections,theorems, etc.isdesignedsothattheadditionofnewmaterialwillnotbreakanycoursesyllabiorassignment list. Withmanyeyesreadingthebookandwithfrequentpostingsofupdates,thereliabilityshouldbecome veryhigh.Pleasereportanyerrorsyoundthatpersistintothelatestversion. ForthosewithaworkinginstallationofthepopulartypesettingprogramT E X,thebookhasbeen designedsothatitcanbecustomized.Pagelayouts,presenceofexercises,solutions,sectionsorchapterscanallbeeasilycontrolled.Furthermore,manyvariantsofmathematicalnotationareachieved Version2.02

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PREFACExlv viaT E Xmacros.Sobychangingasinglemacro,one'sfavoritenotationcanbereectedthroughout thetext.Forexample,everytransposeofamatrixiscodedinthesourceas transpose{A} ,which whenprintedwillyield A t .Howeverbychangingthedenitionof transpose{} ,anydesiredalternativenotationsuperscriptt,superscriptT,superscriptprimewillthenappearthroughoutthe textinstead. Thebookhasalsobeendesignedtomakeiteasyforotherstocontributematerial.Wouldyoulike toseeasectiononsymmetricbilinearforms?Considerwritingoneandcontributingittooneofthe Topicschapters.Shouldtherebemoreexercisesaboutthenullspaceofamatrix?Sendmesome. HistoricalNotes?Contactme,andwewillseeaboutaddingthoseinalso. Youhavenolegalobligationtopayforthisbook.Ithasbeenlicensedwithnoexpectationthatyou payforit.Youdonotevenhaveamoralobligationtopayforthebook.ThomasJeerson{ 1826,theauthoroftheUnitedStatesDeclarationofIndependence,wrote, Ifnaturehasmadeanyonethinglesssusceptiblethanallothersofexclusiveproperty,it istheactionofthethinkingpowercalledanidea,whichanindividualmayexclusively possessaslongashekeepsittohimself;butthemomentitisdivulged,itforcesitselfinto thepossessionofeveryone,andthereceivercannotdispossesshimselfofit.Itspeculiar character,too,isthatnoonepossessestheless,becauseeveryotherpossessesthewholeofit. Hewhoreceivesanideafromme,receivesinstructionhimselfwithoutlesseningmine;ashe wholightshistaperatmine,receiveslightwithoutdarkeningme.Thatideasshouldfreely spreadfromonetoanotherovertheglobe,forthemoralandmutualinstructionofman, andimprovementofhiscondition,seemstohavebeenpeculiarlyandbenevolentlydesigned bynature,whenshemadethem,likere,expansibleoverallspace,withoutlesseningtheir densityinanypoint,andliketheairinwhichwebreathe,move,andhaveourphysical being,incapableofconnementorexclusiveappropriation. LettertoIsaacMcPherson August13,1813 However,ifyoufeelaroyaltyisduetheauthor,orifyouwouldliketoencouragetheauthor,orifyou wishtoshowothersthatthisapproachtotextbookpublishingcanalsobringnancialcompensation, thendonationsaregratefullyreceived.Moreover,non-nancialformsofhelpcanoftenbeevenmore valuable.Asimplenoteofencouragement,submittingareportofanerror,orcontributingsome exercisesorperhapsanentiresectionfortheTopicsorApplicationsareallimportantwaysyoucan acknowledgethefreedomsaccordedtothisworkbythecopyrightholderandothercontributors. Conclusion Foremost,Ihopethatstudentsndtheirtimespentwiththisbookprotable.Ihopethat instructorsnditexibleenoughtottheneedsoftheircourse.AndIhopethateveryonewillsendme theircommentsandsuggestions,andalsoconsiderthemyriadwaystheycanhelpaslistedonthebook's websiteat http://linear.ups.edu RobertA.Beezer Tacoma,Washington July2008 Version2.02

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Acknowledgements Manypeoplehavehelpedtomakethisbook,anditsfreedoms,possible. First,thetimetocreate,editanddistributethebookhasbeenprovidedimplicitlyandexplicitlyby theUniversityofPugetSound.AsabbaticalleaveSpring2004andacoursereleaseinSpring2007aretwo obviousexamplesofexplicitsupport.ThelatterwasprovidedbysupportfromtheLind-VanEnkevortFund. Theuniversityhasalsoprovidedclericalsupport,computerhardware,networkserversandbandwidth. ThankstoDeanKrisBartanenandthechairoftheMathematicsandComputerScienceDepartment, ProfessorMartinJackson,fortheirsupport,encouragementandexibility. MycolleaguesintheMathematicsandComputerScienceDepartmenthavegraciouslytaughtour introductorylinearalgebracourseusingpreliminaryversionsandhaveprovidedvaluablesuggestionsthat haveimprovedthebookimmeasurably.ThankstoProfessorMartinJacksonv0.30,ProfessorDavidScott v0.70andProfessorBryanSmithv0.70,0.80,v1.00. UniversityofPugetSoundlibrariansLoriRicigliano,ElizabethKnightandJeanneKimuraprovided valuableadviceonproduction,andinterestingconversationsaboutcopyrights. Manyaspectsofthebookhavebeeninuencedbyinsightfulquestionsandcreativesuggestionsfrom thestudentswhohavelaboredthroughthebookinourcourses.Forexample,theashcardswiththeorems anddenitionsareadirectresultofastudentsuggestion.Iwillsingleoutahandfulofstudentshavebeen especiallyadeptatndingandreportingmathematicallysignicanttypographicalerrors:JakeLinenthal, ChristieSu,KimLe,SarahMcQuate,AndyZimmer,TravisOsborne,AndrewTapay,MarkShoemaker, TashaUnderhill,TimZitzer,ElizabethMillion,andSteveCaneld. Ihavetriedtobeasoriginalaspossibleintheorganizationandpresentationofthisbeautifulsubject. However,Ihavebeeninuencedbymanyyearsofteachingfromanotherexcellenttextbook, Introduction toLinearAlgebra byL.W.Johnson,R.D.ReissandJ.T.Arnold.WhenIhaveneededinspirationfor thecorrectapproachtoparticularlyimportantproofs,Ihavelearnedtoeventuallyconsulttwoother textbooks.SheldonAxler's LinearAlgebraDoneRight isahighlyoriginalexposition,whileBenNoble's AppliedLinearAlgebra frequentlystrikesjusttherightnotebetweenrigorandintuition.Noble'sexcellent bookishighlyrecommended,eventhoughitspublicationdatesto1969. Conversiontovariouselectronicformatshavegreatlydependedonassistancefrom:EitanGurari, authorofthepowerfulL A T E Xtranslator, tex4ht ;DavideCervone,authorof jsMath ;andCarlWitty,who advisedandtestedtheSonyReaderformat.Thankstotheseindividualsfortheircriticalassistance. Generalsupportandencouragementoffreeandaordabletextbooks,inadditiontospecicpromotion ofthistext,wasprovidedbyNicoleAllen,TextbookAdvocateatStudentPublicInterestResearchGroups. Nicolewasanearlyconsumerofthismaterial,backwhenitlookedmorelikelecturenotesthanatextbook. Finally,ineverypossiblecase,theproductionanddistributionofthisbookhasbeenaccomplishedwith open-sourcesoftware.Therangeofindividualsandprojectsisfartoogreattopretendtolistthemall. Thebook'swebsitewillsomedaymaintainpointerstoasmanyoftheseprojectsaspossible. xlvi

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PartC Core 1

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ChapterSLE SystemsofLinearEquations Wewillmotivateourstudyoflinearalgebrabystudyingsolutionstosystemsoflinearequations.While thefocusofthischapterisonthepracticalmatterofhowtond,anddescribe,thesesolutions,wewill alsobesettingourselvesupformoretheoreticalideasthatwillappearlater. SectionWILA WhatisLinearAlgebra? SubsectionLA Linear"+Algebra" Thesubjectoflinearalgebracanbepartiallyexplainedbythemeaningofthetwotermscomprising thetitle.Linear"isatermyouwillappreciatebetterattheendofthiscourse,andindeed,attaining thisappreciationcouldbetakenasoneoftheprimarygoalsofthiscourse.Howeverfornow,youcan understandittomeananythingthatisstraight"orat."Forexampleinthe xy -planeyoumightbe accustomedtodescribingstraightlinesisthereanyotherkind?asthesetofsolutionstoanequation oftheform y = mx + b ,wheretheslope m andthe y -intercept b areconstantsthattogetherdescribe theline.Inmultivariatecalculus,youmayhavediscussedplanes.Livinginthreedimensions,with coordinatesdescribedbytriples x;y;z ,theycanbedescribedasthesetofsolutionstoequationsofthe form ax + by + cz = d ,where a;b;c;d areconstantsthattogetherdeterminetheplane.Whilewemight describeplanesasat,"linesinthreedimensionsmightbedescribedasstraight."Fromamultivariate calculuscourseyouwillrecallthatlinesaresetsofpointsdescribedbyequationssuchas x =3 t )]TJ/F15 10.9091 Tf 11.664 0 Td [(4, y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 t +2, z =9 t ,where t isaparameterthatcantakeonanyvalue. Anotherviewofthisnotionofatness"istorecognizethatthesetsofpointsjustdescribedaresolutions toequationsofarelativelysimpleform.Theseequationsinvolveadditionandmultiplicationonly.We willhaveaneedforsubtraction,andoccasionallywewilldivide,butmostlyyoucandescribelinear" equationsasinvolvingonlyadditionandmultiplication.Herearesomeexamplesoftypicalequationswe willseeinthenextfewsections: 2 x +3 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 z =134 x 1 +5 x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 3 + x 4 + x 5 =09 a )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b +7 c +2 d = )]TJ/F15 10.9091 Tf 8.484 0 Td [(7 Whatwewillnotseeareequationslike: xy +5 yz =13 x 1 + x 3 2 =x 4 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 3 x 4 x 2 5 =0tan ab +log c )]TJ/F21 10.9091 Tf 10.909 0 Td [(d = )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 Theexceptionwillbethatwewillonoccasionneedtotakeasquareroot. 2

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SubsectionWILA.AAAnApplication3 Youhaveprobablyheardthewordalgebra"frequentlyinyourmathematicalpreparationforthis course.Mostlikely,youhavespentagoodtentofteenyearslearningthealgebraoftherealnumbers, alongwithsomeintroductiontotheverysimilaralgebraofcomplexnumbersseeSectionCNO[679]. However,therearemanynewalgebrastolearnanduse,andlikelylinearalgebrawillbeyoursecond algebra.Likelearningasecondlanguage,thenecessaryadjustmentscanbechallengingattimes,butthe rewardsaremany.Anditwillmakelearningyourthirdandfourthalgebraseveneasier.Perhapsyouhave heardofgroups"andrings"ormaybeyouhavestudiedthemalready,whichareexcellentexamplesof otheralgebraswithveryinterestingpropertiesandapplications.Inanyevent,prepareyourselftolearna newalgebraandrealizethatsomeoftheoldrulesyouusedfortherealnumbersmaynolongerapplyto this new algebrayouwillbelearning! Thebriefdiscussionaboveaboutlinesandplanessuggeststhatlinearalgebrahasaninherentlygeometricnature,andthisistrue.Examplesintwoandthreedimensionscanbeusedtoprovidevaluableinsight intoimportantconceptsofthiscourse.However,muchofthepoweroflinearalgebrawillbetheabilityto workwithat"orstraight"objectsinhigherdimensions,withoutconcerningourselveswithvisualizing thesituation.Whilemuchofourintuitionwillcomefromexamplesintwoandthreedimensions,wewill maintainan algebraic approachtothesubject,withthegeometrybeingsecondary.Othersmaywishto switchthisemphasisaround,andthatcanleadtoaveryfruitfulandbenecialcourse,buthereandnow wearelayingourbiasbare. SubsectionAA AnApplication Weconcludethissectionwitharatherinvolvedexamplethatwillhighlightsomeofthepowerandtechniquesoflinearalgebra.Workthroughallofthedetailswithpencilandpaper,untilyoubelieveallthe assertionsmade.However,inthisintroductoryexample,donotconcernyourselfwithhowsomeofthe resultsareobtainedorhowyoumightbeexpectedtosolveasimilarproblem.Wewillcomebackto thisexamplelaterandexposesomeofthetechniquesusedandpropertiesexploited.Fornow,useyour backgroundinmathematicstoconvinceyourselfthateverythingsaidherereallyiscorrect. ExampleTMP TrailMixPackaging Supposeyouaretheproductionmanageratafood-packagingplantandoneofyourproductlinesistrail mix,ahealthysnackpopularwithhikersandbackpackers,containingraisins,peanutsandhard-shelled chocolatepieces.Byadjustingthemixofthesethreeingredients,youareabletosellthreevarietiesofthis item.Thefancyversionissoldinhalf-kilogrampackagesatoutdoorsupplystoresandhasmorechocolate andfewerraisins,thuscommandingahigherprice.Thestandardversionissoldinonekilogrampackages ingrocerystoresandgasstationmini-markets.Sincethestandardversionhasroughlyequalamountsof eachingredient,itisnotasexpensiveasthefancyversion.Finally,abulkversionissoldinbinsatgrocery storesforconsumerstoloadintoplasticbagsinamountsoftheirchoosing.Toappealtotheshoppers thatlikebulkitemsfortheireconomyandhealthfulness,thismixhasmanymoreraisinsattheexpense ofchocolateandthereforesellsforless. Yourproductionfacilitieshavelimitedstoragespaceandearlyeachmorningyouareabletoreceive andstore380kilogramsofraisins,500kilogramsofpeanutsand620kilogramsofchocolatepieces.As productionmanager,oneofyourmostimportantdutiesistodecidehowmuchofeachversionoftrailmix tomakeeveryday.Clearly,youcanhaveupto1500kilogramsofrawingredientsavailableeachday,so tobethemostproductiveyouwilllikelyproduce1500kilogramsoftrailmixeachday.Also,youwould prefernottohaveanyingredientsleftovereachday,sothatyournalproductisasfreshaspossibleand sothatyoucanreceivethemaximumdeliverythenextmorning.Buthowshouldtheseingredientsbe allocatedtothemixingofthebulk,standardandfancyversions? Version2.02

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SubsectionWILA.AAAnApplication4 First,weneedalittlemoreinformationaboutthemixes.Workersmixtheingredientsin15kilogram batches,andeachrowofthetablebelowgivesarecipefora15kilogrambatch.Thereissomeadditional informationonthecostsoftheingredientsandthepricethemanufacturercanchargeforthedierent versionsofthetrailmix. Raisins Peanuts Chocolate Cost SalePrice kg/batch kg/batch kg/batch $/kg $/kg Bulk 7 6 2 3.69 4.99 Standard 6 4 5 3.86 5.50 Fancy 2 5 8 4.45 6.50 Storagekg 380 500 620 Cost$/kg 2.55 4.65 4.80 Asproductionmanager,itisimportanttorealizethatyouonlyhavethreedecisionstomake|theamount ofbulkmixtomake,theamountofstandardmixtomakeandtheamountoffancymixtomake.Everything elseisbeyondyourcontrolorishandledbyanotherdepartmentwithinthecompany.Principally,youare alsolimitedbytheamountofrawingredientsyoucanstoreeachday.Letusdenotetheamountofeach mixtoproduceeachday,measuredinkilograms,bythevariablequantities b s and f .Yourproduction schedulecanbedescribedasvaluesof b s and f thatdoseveralthings.First,wecannotmakenegative quantitiesofeachmix,so b 0 s 0 f 0 Second,ifwewanttoconsumeallofouringredientseachday,thestoragecapacitiesleadtothreelinear equations,oneforeachingredient, 7 15 b + 6 15 s + 2 15 f =380raisins 6 15 b + 4 15 s + 5 15 f =500peanuts 2 15 b + 5 15 s + 8 15 f =620chocolate Ithappensthatthissystemofthreeequationshasjustonesolution.Inotherwords,asproductionmanager, yourjobiseasy,sincethereisbutonewaytouseupallofyourrawingredientsmakingtrailmix.This singlesolutionis b =300kg s =300kg f =900kg : Wedonotyethavethetoolstoexplainwhythissolutionistheonlyone,butitshouldbesimpleforyou toverifythatthisisindeedasolution.Goahead,wewillwait.Determiningsolutionssuchasthis,and establishingthattheyareunique,willbethemainmotivationforourinitialstudyoflinearalgebra. Sowehavesolvedtheproblemofmakingsurethatwemakethebestuseofourlimitedstoragespace, andeachdayuseupalloftherawingredientsthatareshippedtous.Additionally,asproductionmanager, youmustreportweeklytotheCEOofthecompany,andyouknowhewillbemoreinterestedintheprot derivedfromyourdecisionsthanintheactualproductionlevels.Soyoucompute, 300 : 99 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 : 69+300 : 50 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 : 86+900 : 50 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 : 45=2727 : 00 foradailyprotof$2,727fromthisproductionschedule.Thecomputationofthedailyprotisalso beyondourcontrol,thoughitisdenitelyofinterest,andittoolookslikealinear"computation. Asoftenhappens,thingsdonotstaythesameforlong,andnowthemarketingdepartmenthas suggestedthatyourcompany'strailmixproductsstandardizeoneverymixbeingone-thirdpeanuts. Adjustingthepeanutportionofeachrecipebyalsoadjustingthechocolateportionleadstorevisedrecipes, andslightlydierentcostsforthebulkandstandardmixes,asgiveninthefollowingtable. Version2.02

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SubsectionWILA.AAAnApplication5 Raisins Peanuts Chocolate Cost SalePrice kg/batch kg/batch kg/batch $/kg $/kg Bulk 7 5 3 3.70 4.99 Standard 6 5 4 3.85 5.50 Fancy 2 5 8 4.45 6.50 Storagekg 380 500 620 Cost$/kg 2.55 4.65 4.80 Inasimilarfashionasbefore,wedesirevaluesof b s and f sothat b 0 s 0 f 0 and 7 15 b + 6 15 s + 2 15 f =380raisins 5 15 b + 5 15 s + 5 15 f =500peanuts 3 15 b + 4 15 s + 8 15 f =620chocolate Itnowhappensthatthissystemofequationshas innitely manysolutions,aswewillnowdemonstrate. Let f remainavariablequantity.Thenifwemake f kilogramsofthefancymix,wewillmake4 f )]TJ/F15 10.9091 Tf 10.942 0 Td [(3300 kilogramsofthebulkmixand )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 f +4800kilogramsofthestandardmix.Letusnowverifythat,forany choiceof f ,thevaluesof b =4 f )]TJ/F15 10.9091 Tf 10.257 0 Td [(3300and s = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 f +4800willyieldaproductionschedulethatexhausts alloftheday'ssupplyofrawingredientsrightnow,donotbeconcernedabouthowyoumightderive expressionslikethesefor b and s .Grabyourpencilandpaperandplayalong. 7 15 f )]TJ/F15 10.9091 Tf 10.909 0 Td [(3300+ 6 15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 f +4800+ 2 15 f =0 f + 5700 15 =380 5 15 f )]TJ/F15 10.9091 Tf 10.909 0 Td [(3300+ 5 15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 f +4800+ 5 15 f =0 f + 7500 15 =500 3 15 f )]TJ/F15 10.9091 Tf 10.909 0 Td [(3300+ 4 15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 f +4800+ 8 15 f =0 f + 9300 15 =620 Convinceyourselfthattheseexpressionsfor b and s allowustovary f andobtainaninnitenumberof possibilitiesforsolutionstothethreeequationsthatdescribeourstoragecapacities.Asapracticalmatter, therereallyarenotaninnitenumberofsolutions,sinceweareunlikelytowanttoendthedaywitha fractionalnumberofbagsoffancymix,soourallowablevaluesof f shouldprobablybeintegers.More importantly,weneedtorememberthatwecannotmakenegativeamountsofeachmix!Wheredoesthis leadus?Positivequantitiesofthebulkmixrequiresthat b 0 4 f )]TJ/F15 10.9091 Tf 10.909 0 Td [(3300 0 f 825 Similarlyforthestandardmix, s 0 )]TJ/F15 10.9091 Tf 33.334 0 Td [(5 f +4800 0 f 960 So,asproductionmanager,youreallyhavetochooseavalueof f fromtheniteset f 825 ; 826 ;:::; 960 g leavingyouwith136choices,eachofwhichwillexhausttheday'ssupplyofrawingredients.Pausenow andthinkaboutwhich you wouldchoose. Version2.02

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SubsectionWILA.READReadingQuestions6 RecallingyourweeklymeetingwiththeCEOsuggeststhatyoumightwanttochooseaproduction schedulethatyieldsthebiggestpossibleprotforthecompany.Soyoucomputeanexpressionforthe protbasedonyourasyetundetermineddecisionforthevalueof f f )]TJ/F15 10.9091 Tf 10.909 0 Td [(3300 : 99 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 : 70+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 f +4800 : 50 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 : 85+ f : 50 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 : 45= )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 04 f +3663 Since f hasanegativecoecientitwouldappearthatmixingfancymixisdetrimentaltoyourprotand shouldbeavoided.Soyouwillmakethedecisiontosetdailyfancymixproductionat f =825.Thishas theeectofsetting b =4 )]TJ/F15 10.9091 Tf 11.03 0 Td [(3300=0andwestopproducingbulkmixentirely.Sotheremainderof yourdailyproductionisstandardmixatthelevelof s = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5+4800=675kilogramsandtheresulting dailyprotis )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 : 04+3663=2805.Itisapleasantsurprisethatdailyprothasrisento$2,805, butthisisnotthemostimportantpartofthestory.Whatisimportanthereisthattherearealarge numberofwaystoproducetrailmixthatusealloftheday'sworthofrawingredients and youwereable toeasilychoosetheonethatnettedthelargestprot.Noticetoohowalloftheabovecomputationslook linear." Inthefoodindustry,thingsdonotstaythesameforlong,andnowthesalesdepartmentsaysthat increasedcompetitionhasledtothedecisiontostaycompetitiveandchargejust$5.25forakilogramofthe standardmix,ratherthantheprevious$5.50perkilogram.Thisdecisionhasnoeectonthepossibilities fortheproductionschedule,butwillaectthedecisionbasedonprotconsiderations.Soyourevisitjust theprotcomputation,suitablyadjustedforthenewsellingpriceofstandardmix, f )]TJ/F15 10.9091 Tf 10.909 0 Td [(3300 : 99 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 : 70+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 f +4800 : 25 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 : 85+ f : 50 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 : 45=0 : 21 f +2463 Nowitwouldappearthatfancymixisbenecialtothecompany'sprotsincethevalueof f hasapositive coecient.Soyoutakethedecisiontomakeasmuchfancymixaspossible,setting f =960.Thisleads to s = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5+4800=0andtheincreasedcompetitionhasdrivenyououtofthestandardmixmarket alltogether.Theremainderofproductionisthereforebulkmixatadailylevelof b =4 )]TJ/F15 10.9091 Tf 10.313 0 Td [(3300=540 kilogramsandtheresultingdailyprotis0 : 21+2463=2664 : 60.Adailyprotof$2,664.60isless thanitusedtobe,butasproductionmanager,youhavemadethebestofadicultsituationandshown thesalesdepartmentthatthebestcourseistopulloutofthehighlycompetitivestandardmixmarket completely. Thisexampleistakenfromaeldofmathematicsvariouslyknownbynamessuchasoperationsresearch, systemsscience,ormanagementscience.Morespecically,thisisaperfectexampleofproblemsthatare solvedbythetechniquesoflinearprogramming." Thereisalotgoingonunderthehoodinthisexample.Theheartofthematteristhesolutionto systemsoflinearequations,whichisthetopicofthenextfewsections,andarecurrentthemethroughout thiscourse.Wewillreturntothisexampleonseveraloccasionstorevealsomeofthereasonsforits behavior. SubsectionREAD ReadingQuestions 1.Istheequation x 2 + xy +tan y 3 =0linearornot?Whyorwhynot? 2.Findallsolutionstothesystemoftwolinearequations2 x +3 y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(8, x )]TJ/F21 10.9091 Tf 10.909 0 Td [(y =6. 3.Describehowtheproductionmanagermightexplaintheimportanceoftheproceduresdescribedin thetrailmixapplicationSubsectionWILA.AA[3]. Version2.02

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SubsectionWILA.EXCExercises7 SubsectionEXC Exercises C10 InExampleTMP[3]thersttableliststhecostperkilogramtomanufactureeachofthethree varietiesoftrailmixbulk,standard,fancy.Forexample,itcosts$3.69tomakeonekilogramofthebulk variety.Re-computeeachofthesethreecostsandnoticethatthecomputationsarelinearincharacter. ContributedbyRobertBeezer M70 InExampleTMP[3]twodierentpriceswereconsideredformarketingstandardmixwiththe revisedrecipesone-thirdpeanutsineachrecipe.Sellingstandardmixat$5.50resultedinsellingthe minimumamountofthefancymixandnobulkmix.At$5.25itwasbestforprotstosellthemaximum amountoffancymixandthensellnostandardmix.Determineasellingpriceforstandardmixthatallows formaximumprotswhilestillsellingsomeofeachtypeofmix. ContributedbyRobertBeezerSolution[8] Version2.02

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SubsectionWILA.SOLSolutions8 SubsectionSOL Solutions M70 ContributedbyRobertBeezerStatement[7] Ifthepriceofstandardmixissetat$5.292,thentheprotfunctionhasazerocoecientonthevariable quantity f .So,wecanset f tobeanyintegerquantityin f 825 ; 826 ;:::; 960 g .Allbuttheextremevalues f =825, f =960willresultinproductionlevelswheresomeofeverymixismanufactured.Nomatter whatvalueof f ischosen,theresultingprotwillbethesame,at$2,664.60. Version2.02

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SectionSSLESolvingSystemsofLinearEquations9 SectionSSLE SolvingSystemsofLinearEquations Wewillmotivateourstudyoflinearalgebrabyconsideringtheproblemofsolvingseverallinearequations simultaneously.Thewordsolve"tendstogetabusedsomewhat,asinsolvethisproblem."Whentalking aboutequationsweunderstandamoreprecisemeaning:nd all ofthevaluesofsomevariablequantities thatmakeanequation,orseveralequations,true. SubsectionSLE SystemsofLinearEquations ExampleSTNE Solvingtwononlinearequations Supposewedesirethesimultaneoussolutionsofthetwoequations, x 2 + y 2 =1 )]TJ/F21 10.9091 Tf 8.485 0 Td [(x + p 3 y =0 Youcaneasilycheckbysubstitutionthat x = p 3 2 ;y = 1 2 and x = )]TJ/F25 7.9701 Tf 9.68 10.993 Td [(p 3 2 ;y = )]TJ/F19 7.9701 Tf 9.68 4.295 Td [(1 2 arebothsolutions.We needtoalsoconvinceourselvesthatthesearethe only solutions.Toseethis,ploteachequationonthe xy -plane,whichmeanstoplot x;y pairsthatmakeanindividualequationtrue.Inthiscaseweget acirclecenteredattheoriginwithradius1andastraightlinethroughtheoriginwithslope 1 p 3 .The intersectionsofthesetwocurvesareourdesiredsimultaneoussolutions,andsowebelievefromourplot thatthetwosolutionsweknowalreadyareindeedtheonlyones.Weliketowritesolutionsassets,soin thiscasewewritethesetofsolutionsas S = f p 3 2 ; 1 2 ; )]TJ/F25 7.9701 Tf 9.68 10.993 Td [(p 3 2 ; )]TJ/F19 7.9701 Tf 9.681 4.295 Td [(1 2 g Inordertodiscusssystemsoflinearequationscarefully,weneedaprecisedenition.Andbefore wedothat,wewillintroduceourperiodicdiscussionsaboutProofTechniques."Linearalgebraisan excellentsettingforlearninghowtoread,understandandformulateproofs.Butthisisadicultstepin yourdevelopmentasamathematician,sowehaveincludedaseriesofshortessayscontainingadviceand explanationstohelpyoualong.ThesecanbefoundbackinSectionPT[687]ofAppendixP[679],and wewillreferencethemastheybecomeappropriate.Besuretoheadbacktotheappendixtoreadthisas theyareintroduced.Withadenitionnext,nowisthetimefortherstofourprooftechniques.Head backtoSectionPT[687]ofAppendixP[679]andstudyTechniqueD[687].We'llberightherewhenyou getback.Seeyouinabit. DenitionSLE SystemofLinearEquations A systemoflinearequations isacollectionof m equationsinthevariablequantities x 1 ;x 2 ;x 3 ;:::;x n oftheform, a 11 x 1 + a 12 x 2 + a 13 x 3 + + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + a 23 x 3 + + a 2 n x n = b 2 Version2.02

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SubsectionSSLE.PSSPossibilitiesforSolutionSets10 a 31 x 1 + a 32 x 2 + a 33 x 3 + + a 3 n x n = b 3 a m 1 x 1 + a m 2 x 2 + a m 3 x 3 + + a mn x n = b m wherethevaluesof a ij b i and x j arefromthesetofcomplexnumbers, C 4 Don'tletthementionofthecomplexnumbers, C ,rattleyou.Wewillstickwithrealnumbersexclusively formanymoresections,anditwillsometimesseemlikeweonlyworkwithintegers!However,wewantto leavethepossibilityofcomplexnumbersopen,andtherewillbeoccasionsinsubsequentsectionswhere theyarenecessary.YoucanreviewthebasicpropertiesofcomplexnumbersinSectionCNO[679],but thesefactswillnotbecriticaluntilwereachSectionO[167].Fornow,hereisanexampletoillustrate usingthenotationintroducedinDenitionSLE[9]. ExampleNSE Notationforasystemofequations Giventhesystemoflinearequations, x 1 +2 x 2 + x 4 =7 x 1 + x 2 + x 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 4 =3 3 x 1 + x 2 +5 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x 4 =1 wehave n =4variablesand m =3equations.Also, a 11 =1 a 12 =2 a 13 =0 a 14 =1 b 1 =7 a 21 =1 a 22 =1 a 23 =1 a 24 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 b 2 =3 a 31 =3 a 32 =1 a 33 =5 a 34 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 b 3 =1 Additionally,convinceyourselfthat x 1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2, x 2 =4, x 3 =2, x 4 =1isonesolutionbutitisnottheonly one!. Wewilloftenshortenthetermsystemoflinearequations"tosystemofequations"leavingthelinear aspectimplied.Afterall,thisisabookabout linear algebra. SubsectionPSS PossibilitiesforSolutionSets Thenextexampleillustratesthepossibilitiesforthesolutionsetofasystemoflinearequations.Wewill notbetooformalhere,andthenecessarytheoremstobackupourclaimswillcomeinsubsequentsections. Soreadforfeelingandcomebacklatertorevisitthisexample. ExampleTTS Threetypicalsystems Considerthesystemoftwoequationswithtwovariables, 2 x 1 +3 x 2 =3 x 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 2 =4 Ifweplotthesolutionstoeachoftheseequationsseparatelyonthe x 1 x 2 -plane,wegettwolines,onewith negativeslope,theotherwithpositiveslope.Theyhaveexactlyonepointincommon, x 1 ;x 2 = ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1, whichisthesolution x 1 =3, x 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Fromthegeometry,webelievethatthisistheonlysolutiontothe systemofequations,andsowesayitisunique. Version2.02

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SubsectionSSLE.ESEOEquivalentSystemsandEquationOperations11 Nowadjustthesystemwithadierentsecondequation, 2 x 1 +3 x 2 =3 4 x 1 +6 x 2 =6 Aplotofthesolutionstotheseequationsindividuallyresultsintwolines,oneontopoftheother!There areinnitelymanypairsofpointsthatmakebothequationstrue.Wewilllearnshortlyhowtodescribe thisinnitesolutionsetpreciselyseeExampleSAA[37],TheoremVFSLS[99].Noticenowhowthe secondequationisjustamultipleoftherst. Onemoreminoradjustmentprovidesathirdsystemoflinearequations, 2 x 1 +3 x 2 =3 4 x 1 +6 x 2 =10 Aplotnowrevealstwolineswithidenticalslopes,i.e.parallellines.Theyhavenopointsincommon,and sothesystemhasasolutionsetthatisempty, S = ; Thisexampleexhibitsallofthetypicalbehaviorsofasystemofequations.Asubsequenttheoremwill tellusthateverysystemoflinearequationshasasolutionsetthatisempty,containsasinglesolutionor containsinnitelymanysolutionsTheoremPSSLS[55].ExampleSTNE[9]yieldedexactlytwosolutions, butthisdoesnotcontradicttheforthcomingtheorem.TheequationsinExampleSTNE[9]arenotlinear becausetheydonotmatchtheformofDenitionSLE[9],andsowecannotapplyTheoremPSSLS[55] inthiscase. SubsectionESEO EquivalentSystemsandEquationOperations Withallthistalkaboutndingsolutionsetsforsystemsoflinearequations,youmightbereadytobegin learninghowtondthesesolutionsetsyourself.Webeginwithourrstdenitionthattakesacommon wordandgivesitaveryprecisemeaninginthecontextofsystemsoflinearequations. DenitionESYS EquivalentSystems Twosystemsoflinearequationsare equivalent iftheirsolutionsetsareequal. 4 Noticeherethatthetwosystemsofequationscould look verydierenti.e.notbeequal,butstillhave equalsolutionsets,andwewouldthencallthesystemsequivalent.Twolinearequationsintwovariables mightbeplottedastwolinesthatintersectinasinglepoint.Adierentsystem,withthreeequationsin twovariablesmighthaveaplotthatisthreelines,allintersectingatacommonpoint,withthiscommon pointidenticaltotheintersectionpointfortherstsystem.Byourdenition,wecouldthensaythese twoverydierentlookingsystemsofequationsareequivalent,sincetheyhaveidenticalsolutionsets.Itis reallylikeaweakerformofequality,whereweallowthesystemstobedierentinsomerespects,butwe usethetermequivalenttohighlightthesituationwhentheirsolutionsetsareequal. Withthisdenition,wecanbegintodescribeourstrategyforsolvinglinearsystems.Givenasystem oflinearequationsthatlooksdiculttosolve,wewouldliketohavean equivalent systemthatiseasyto solve.Sincethesystemswillhaveequalsolutionsets,wecansolvetheeasy"systemandgetthesolution settothedicult"system.Herecomethetoolsformakingthisstrategyviable. DenitionEO EquationOperations Givenasystemoflinearequations,thefollowingthreeoperationswilltransformthesystemintoadierent one,andeachoperationisknownasan equationoperation Version2.02

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SubsectionSSLE.ESEOEquivalentSystemsandEquationOperations12 1.Swapthelocationsoftwoequationsinthelistofequations. 2.Multiplyeachtermofanequationbyanonzeroquantity. 3.Multiplyeachtermofoneequationbysomequantity,andaddthesetermstoasecondequation,on bothsidesoftheequality.Leavetherstequationthesameafterthisoperation,butreplacethe secondequationbythenewone. 4 Thesedescriptionsmightseemabitvague,buttheproofortheexamplesthatfollowshouldmakeit clearwhatismeantbyeach.Wewillshortlyproveakeytheoremaboutequationoperationsandsolutions tolinearsystemsofequations.Weareabouttogivearatherinvolvedproof,soadiscussionaboutjust whatatheoremreallyiswouldbetimely.HeadbackandreadTechniqueT[688].Inthetheoremweare abouttoprove,theconclusionisthattwosystemsareequivalent.ByDenitionESYS[11]thistranslates torequiringthatsolutionsetsbeequalforthetwosystems.Sowearebeingaskedtoshow thattwosets areequal .Howdowedothis?Well,thereisaverystandardtechnique,andwewilluseitrepeatedly throughthecourse.Ifyouhavenotdonesoalready,headtoSectionSET[683]andfamiliarizeyourself withsets,theiroperations,andespeciallythenotionofsetequality,DenitionSE[684]andthenearby discussionaboutitsuse. TheoremEOPSS EquationOperationsPreserveSolutionSets IfweapplyoneofthethreeequationoperationsofDenitionEO[11]toasystemoflinearequations DenitionSLE[9],thentheoriginalsystemandthetransformedsystemareequivalent. Proof Wetakeeachequationoperationinturnandshowthatthesolutionsetsofthetwosystemsare equal,usingthedenitionofsetequalityDenitionSE[684]. 1.Itwillnotbeourhabitinproofstoresorttosayingstatementsareobvious,"butinthiscase,it shouldbe.Thereisnothingaboutthe order inwhichwewritelinearequationsthataectstheir solutions,sothesolutionsetwillbeequalifthesystemsonlydierbyarearrangementoftheorder oftheequations. 2.Suppose 6 =0isanumber.Let'schoosetomultiplythetermsofequation i by tobuildthenew systemofequations, a 11 x 1 + a 12 x 2 + a 13 x 3 + + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + a 23 x 3 + + a 2 n x n = b 2 a 31 x 1 + a 32 x 2 + a 33 x 3 + + a 3 n x n = b 3 a i 1 x 1 + a i 2 x 2 + a i 3 x 3 + + a in x n = b i a m 1 x 1 + a m 2 x 2 + a m 3 x 3 + + a mn x n = b m Let S denotethesolutionstothesysteminthestatementofthetheorem,andlet T denotethe solutionstothetransformedsystem. aShow S T .Suppose x 1 ;x 2 ;x 3 ;:::;x n = 1 ; 2 ; 3 ;:::; n 2 S isasolutiontotheoriginal system.Ignoringthe i -thequationforamoment,weknowitmakesalltheotherequationsof thetransformedsystemtrue.Wealsoknowthat a i 1 1 + a i 2 2 + a i 3 3 + + a in n = b i Version2.02

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SubsectionSSLE.ESEOEquivalentSystemsandEquationOperations13 whichwecanmultiplyby toget a i 1 1 + a i 2 2 + a i 3 3 + + a in n = b i Thissaysthatthe i -thequationofthetransformedsystemisalsotrue,sowehaveestablished that 1 ; 2 ; 3 ;:::; n 2 T ,andtherefore S T bNowshow T S .Suppose x 1 ;x 2 ;x 3 ;:::;x n = 1 ; 2 ; 3 ;:::; n 2 T isasolutiontothe transformedsystem.Ignoringthe i -thequationforamoment,weknowitmakesalltheother equationsoftheoriginalsystemtrue.Wealsoknowthat a i 1 1 + a i 2 2 + a i 3 3 + + a in n = b i whichwecanmultiplyby 1 ,since 6 =0,toget a i 1 1 + a i 2 2 + a i 3 3 + + a in n = b i Thissaysthatthe i -thequationoftheoriginalsystemisalsotrue,sowehaveestablishedthat 1 ; 2 ; 3 ;:::; n 2 S ,andtherefore T S .Locatethekeypointwherewerequiredthat 6 =0,andconsiderwhatwouldhappenif =0. 3.Suppose isanumber.Let'schoosetomultiplythetermsofequation i by andaddthemto equation j inordertobuildthenewsystemofequations, a 11 x 1 + a 12 x 2 + + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2 n x n = b 2 a 31 x 1 + a 32 x 2 + + a 3 n x n = b 3 a i 1 + a j 1 x 1 + a i 2 + a j 2 x 2 + + a in + a jn x n = b i + b j a m 1 x 1 + a m 2 x 2 + + a mn x n = b m Let S denotethesolutionstothesysteminthestatementofthetheorem,andlet T denotethe solutionstothetransformedsystem. aShow S T .Suppose x 1 ;x 2 ;x 3 ;:::;x n = 1 ; 2 ; 3 ;:::; n 2 S isasolutiontothe originalsystem.Ignoringthe j -thequationforamoment,weknowthissolutionmakesallthe otherequationsofthetransformedsystemtrue.Usingthefactthatthesolutionmakesthe i -th and j -thequationsoftheoriginalsystemtrue,wend a i 1 + a j 1 1 + a i 2 + a j 2 2 + + a in + a jn n = a i 1 1 + a i 2 2 + + a in n + a j 1 1 + a j 2 2 + + a jn n = a i 1 1 + a i 2 2 + + a in n + a j 1 1 + a j 2 2 + + a jn n = b i + b j : Thissaysthatthe j -thequationofthetransformedsystemisalsotrue,sowehaveestablished that 1 ; 2 ; 3 ;:::; n 2 T ,andtherefore S T Version2.02

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SubsectionSSLE.ESEOEquivalentSystemsandEquationOperations14 bNowshow T S .Suppose x 1 ;x 2 ;x 3 ;:::;x n = 1 ; 2 ; 3 ;:::; n 2 T isasolutiontothe transformedsystem.Ignoringthe j -thequationforamoment,weknowitmakesalltheother equationsoftheoriginalsystemtrue.Wethennd a j 1 1 + a j 2 2 + + a jn n = a j 1 1 + a j 2 2 + + a jn n + b i )]TJ/F21 10.9091 Tf 10.909 0 Td [(b i = a j 1 1 + a j 2 2 + + a jn n + a i 1 1 + a i 2 2 + + a in n )]TJ/F21 10.9091 Tf 10.909 0 Td [(b i = a j 1 1 + a i 1 1 + a j 2 2 + a i 2 2 + + a jn n + a in n )]TJ/F21 10.9091 Tf 10.909 0 Td [(b i = a i 1 + a j 1 1 + a i 2 + a j 2 2 + + a in + a jn n )]TJ/F21 10.9091 Tf 10.909 0 Td [(b i = b i + b j )]TJ/F21 10.9091 Tf 10.909 0 Td [(b i = b j Thissaysthatthe j -thequationoftheoriginalsystemisalsotrue,sowehaveestablishedthat 1 ; 2 ; 3 ;:::; n 2 S ,andtherefore T S Whydidn'tweneedtorequirethat 6 =0forthisrowoperation?Inotherwords,howdoesthe thirdstatementofthetheoremreadwhen =0?Doesourproofrequiresomeextracarewhen =0?Compareyouranswerswiththesimilarsituationforthesecondrowoperation.SeeExercise SSLE.T20[20]. TheoremEOPSS[12]isthenecessarytooltocompleteourstrategyforsolvingsystemsofequations. Wewilluseequationoperationstomovefromonesystemtoanother,allthewhilekeepingthesolutionset thesame.Withtherightsequenceofoperations,wewillarriveatasimplerequationtosolve.Thenext twoexamplesillustratethisidea,whilesavingsomeofthedetailsforlater. ExampleUS Threeequations,onesolution Wesolvethefollowingsystembyasequenceofequationoperations. x 1 +2 x 2 +2 x 3 =4 x 1 +3 x 2 +3 x 3 =5 2 x 1 +6 x 2 +5 x 3 =6 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1timesequation1,addtoequation2: x 1 +2 x 2 +2 x 3 =4 0 x 1 +1 x 2 +1 x 3 =1 2 x 1 +6 x 2 +5 x 3 =6 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2timesequation1,addtoequation3: x 1 +2 x 2 +2 x 3 =4 0 x 1 +1 x 2 +1 x 3 =1 0 x 1 +2 x 2 +1 x 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2timesequation2,addtoequation3: x 1 +2 x 2 +2 x 3 =4 Version2.02

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SubsectionSSLE.ESEOEquivalentSystemsandEquationOperations15 0 x 1 +1 x 2 +1 x 3 =1 0 x 1 +0 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1timesequation3: x 1 +2 x 2 +2 x 3 =4 0 x 1 +1 x 2 +1 x 3 =1 0 x 1 +0 x 2 +1 x 3 =4 whichcanbewrittenmoreclearlyas x 1 +2 x 2 +2 x 3 =4 x 2 + x 3 =1 x 3 =4 Thisisnowaveryeasysystemofequationstosolve.Thethirdequationrequiresthat x 3 =4tobetrue. Makingthissubstitutionintoequation2wearriveat x 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3,andnally,substitutingthesevaluesof x 2 and x 3 intotherstequation,wendthat x 1 =2.Notetoothatthisistheonlysolutiontothisnal systemofequations,sincewewereforcedtochoosethesevaluestomaketheequationstrue.Sincewe performedequationoperationsoneachsystemtoobtainthenextoneinthelist,allofthesystemslisted hereareallequivalenttoeachotherbyTheoremEOPSS[12].Thus x 1 ;x 2 ;x 3 = ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 4istheunique solutiontothe original systemofequationsandalloftheotherintermediatesystemsofequationslisted aswetransformedoneintoanother. ExampleIS Threeequations,innitelymanysolutions ThefollowingsystemofequationsmadeanappearanceearlierinthissectionExampleNSE[10],where welisted one ofitssolutions.Now,wewilltrytondallofthesolutionstothissystem.Don'tconcern yourselftoomuchaboutwhywechoosethisparticularsequenceofequationoperations,justbelievethat theworkwedoisallcorrect. x 1 +2 x 2 +0 x 3 + x 4 =7 x 1 + x 2 + x 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 4 =3 3 x 1 + x 2 +5 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x 4 =1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1timesequation1,addtoequation2: x 1 +2 x 2 +0 x 3 + x 4 =7 0 x 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 2 + x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 x 1 + x 2 +5 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x 4 =1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3timesequation1,addtoequation3: x 1 +2 x 2 +0 x 3 + x 4 =7 0 x 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 2 + x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 0 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 2 +5 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(10 x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5timesequation2,addtoequation3: x 1 +2 x 2 +0 x 3 + x 4 =7 Version2.02

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SubsectionSSLE.ESEOEquivalentSystemsandEquationOperations16 0 x 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 2 + x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 0 x 1 +0 x 2 +0 x 3 +0 x 4 =0 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1timesequation2: x 1 +2 x 2 +0 x 3 + x 4 =7 0 x 1 + x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 3 +2 x 4 =4 0 x 1 +0 x 2 +0 x 3 +0 x 4 =0 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2timesequation2,addtoequation1: x 1 +0 x 2 +2 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 x 1 + x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 3 +2 x 4 =4 0 x 1 +0 x 2 +0 x 3 +0 x 4 =0 whichcanbewrittenmoreclearlyas x 1 +2 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 3 +2 x 4 =4 0=0 Whatdoestheequation0=0mean?Wecanchoose any valuesfor x 1 ;x 2 ;x 3 ;x 4 andthisequationwill betrue,soweonlyneedtoconsiderfurtherthersttwoequations,sincethethirdistruenomatterwhat. Wecananalyzethesecondequationwithoutconsiderationofthevariable x 1 .Itwouldappearthatthere isconsiderablelatitudeinhowwecanchoose x 2 ;x 3 ;x 4 andmakethisequationtrue.Let'schoose x 3 and x 4 tobe anything weplease,say x 3 = a and x 4 = b Nowwecantakethesearbitraryvaluesfor x 3 and x 4 ,substitutetheminequation1,toobtain x 1 +2 a )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 b = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 x 1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 a +3 b Similarly,equation2becomes x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(a +2 b =4 x 2 =4+ a )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b Soourarbitrarychoicesofvaluesfor x 3 and x 4 a and b translateintospecicvaluesof x 1 and x 2 .The lonesolutiongiveninExampleNSE[10]wasobtainedbychoosing a =2and b =1.Nowwecaneasily andquicklyndmanymoreinnitelymore.Supposewechoose a =5and b = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2,thenwecompute x 1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 10.91 0 Td [(2+3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2= )]TJ/F15 10.9091 Tf 8.485 0 Td [(17 x 2 =4+5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2=13 andyoucanverifythat x 1 ;x 2 ;x 3 ;x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(17 ; 13 ; 5 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2makesallthreeequationstrue.Theentire solutionsetiswrittenas S = f )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 a +3 b; 4+ a )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b;a;b j a 2 C ;b 2 C g Itwouldbeinstructivetonishoyourstudyofthisexamplebytakingthegeneralformofthesolutions giveninthissetandsubstitutingthemintoeachofthethreeequationsandverifythattheyaretruein eachcaseExerciseSSLE.M40[19]. Inthenextsectionwewilldescribehowtouseequationoperationstosystematicallysolveanysystem oflinearequations.Butrst,readoneofourmoreimportantpiecesofadviceaboutspeakingandwriting mathematics.SeeTechniqueL[688]. Version2.02

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SubsectionSSLE.READReadingQuestions17 Beforeattackingtheexercisesinthissection,itwillbehelpfultoreadsomeadviceongettingstarted ontheconstructionofaproof.SeeTechniqueGS[689]. SubsectionREAD ReadingQuestions 1.Howmanysolutionsdoesthesystemofequations3 x +2 y =4,6 x +4 y =8have?Explainyour answer. 2.Howmanysolutionsdoesthesystemofequations3 x +2 y =4,6 x +4 y = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2have?Explainyour answer. 3.Whatdowemeanwhenwesaymathematicsisalanguage? Version2.02

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SubsectionSSLE.EXCExercises18 SubsectionEXC Exercises C10 FindasolutiontothesysteminExampleIS[15]where x 3 =6and x 4 =2.Findtwoothersolutions tothesystem.Findasolutionwhere x 1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(17and x 2 =14.Howmanypossibleanswersaretheretoeach ofthesequestions? ContributedbyRobertBeezer C20 EacharchetypeAppendixA[698]thatisasystemofequationsbeginsbylistingsomespecic solutions.Verifythespecicsolutionslistedinthefollowingarchetypesbyevaluatingthesystemof equationswiththesolutionslisted. ArchetypeA[702] ArchetypeB[707] ArchetypeC[712] ArchetypeD[716] ArchetypeE[720] ArchetypeF[724] ArchetypeG[729] ArchetypeH[733] ArchetypeI[737] ArchetypeJ[741] ContributedbyRobertBeezer C50 Athree-digitnumberhastwoproperties.Thetens-digitandtheones-digitaddupto5.Ifthe numberiswrittenwiththedigitsinthereverseorder,andthensubtractedfromtheoriginalnumber,the resultis792.Useasystemofequationstondallofthethree-digitnumberswiththeseproperties. ContributedbyRobertBeezerSolution[21] C51 Findallofthesix-digitnumbersinwhichtherstdigitisonelessthanthesecond,thethirddigitis halfthesecond,thefourthdigitisthreetimesthethirdandthelasttwodigitsformanumberthatequals thesumofthefourthandfth.Thesumofallthedigitsis24.From TheMENSAPuzzleCalendar for January9,2006. ContributedbyRobertBeezerSolution[21] C52 Drivingalong,Terrynoticesthatthelastfourdigitsonhiscar'sodometerarepalindromic.Amile later,thelastvedigitsarepalindromic.Afterdrivinganothermile,themiddlefourdigitsarepalindromic. Onemoremile,andallsixarepalindromic.WhatwastheodometerreadingwhenTerryrstlookedat it?Formalinearsystemofequationsthatexpressestherequirementsofthispuzzle. CarTalk Puzzler, NationalPublicRadio,WeekofJanuary21,2008Acarodometerdisplayssixdigitsandasequenceisa palindrome ifitreadsthesameleft-to-rightasright-to-left. ContributedbyRobertBeezerSolution[22] M10 Eachsentencebelowhasatleasttwomeanings.Identifythesourceofthedoublemeaning,and rewritethesentenceatleasttwicetoclearlyconveyeachmeaning. 1.Theyarebakingpotatoes. 2.Heboughtmanyripepearsandapricots. 3.Shelikeshissculpture. 4.Idecidedonthebus. Version2.02

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SubsectionSSLE.EXCExercises19 ContributedbyRobertBeezerSolution[22] M11 Discussthedierenceinmeaningofeachofthefollowingthreealmostidenticalsentences,which allhavethesamegrammaticalstructure.TheseareduetoKeithDevlin. 1.Shesawhimintheparkwithadog. 2.Shesawhimintheparkwithafountain. 3.Shesawhimintheparkwithatelescope. ContributedbyRobertBeezerSolution[22] M12 Thefollowingsentence,duetoNoamChomsky,hasacorrectgrammaticalstructure,butismeaningless.Critiqueitsfaults.Colorlessgreenideassleepfuriously."Chomsky,Noam. SyntacticStructures TheHague/Paris:Mouton,1957.p.15. ContributedbyRobertBeezerSolution[22] M13 Readthefollowingsentenceandformamentalpictureofthesituation. Thebabycriedandthemotherpickeditup. What assumptions didyoumakeaboutthesituation? ContributedbyRobertBeezerSolution[22] M30 Thisproblemappearsinamiddle-schoolmathematicstextbook:TogetherDanandDianehave $20.TogetherDianeandDonnahave$15.Howmuchdothethreeofthemhaveintotal? Transition Mathematics ,SecondEdition,ScottForesmanAddisonWesley,1998.Problem5{1.19. ContributedbyDavidBeezerSolution[22] M40 SolutionstothesysteminExampleIS[15]aregivenas x 1 ;x 2 ;x 3 ;x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 a +3 b; 4+ a )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b;a;b Evaluatethethreeequationsoftheoriginalsystemwiththeseexpressionsin a and b andverifythateach equationistrue,nomatterwhatvaluesarechosenfor a and b ContributedbyRobertBeezer M70 Wehaveseeninthissectionthatsystemsoflinearequationshavelimitedpossibilitiesforsolution sets,andwewillshortlyproveTheoremPSSLS[55]thatdescribesthesepossibilitiesexactly.Thisexercise willshowthatifwerelaxtherequirementthatourequationsbelinear,thenthepossibilitiesexpandgreatly. Considerasystemoftwoequationsinthetwovariables x and y ,wherethedeparturefromlinearityinvolves simplysquaringthevariables. x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(y 2 =1 x 2 + y 2 =4 Aftersolvingthissystemof non-linear equations,replacethesecondequationinturnby x 2 +2 x + y 2 =3, x 2 + y 2 =1, x 2 )]TJ/F21 10.9091 Tf 11.557 0 Td [(x + y 2 =0,4 x 2 +4 y 2 =1andsolveeachresultingsystemoftwoequationsintwo variables. ContributedbyRobertBeezerSolution[23] T10 TechniqueD[687]asksyoutoformulateadenitionofwhatitmeansforawholenumbertobe odd.Whatisyourdenition?Don'tsaytheoppositeofeven."Is6odd?Is11odd?Justifyyour Version2.02

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SubsectionSSLE.EXCExercises20 answersbyusingyourdenition. ContributedbyRobertBeezerSolution[23] T20 ExplainwhythesecondequationoperationinDenitionEO[11]requiresthatthescalarbenonzero, whileinthethirdequationoperationthisrestrictiononthescalarisnotpresent. ContributedbyRobertBeezerSolution[23] Version2.02

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SubsectionSSLE.SOLSolutions21 SubsectionSOL Solutions C50 ContributedbyRobertBeezerStatement[18] Let a bethehundredsdigit, b thetensdigit,and c theonesdigit.Thentherstconditionsaysthat b + c =5.Theoriginalnumberis100 a +10 b + c ,whilethereversednumberis100 c +10 b + a .Sothe secondconditionis 792= a +10 b + c )]TJ/F15 10.9091 Tf 10.909 0 Td [( c +10 b + a =99 a )]TJ/F15 10.9091 Tf 10.91 0 Td [(99 c Sowearriveatthesystemofequations b + c =5 99 a )]TJ/F15 10.9091 Tf 10.909 0 Td [(99 c =792 Usingequationoperations,wearriveattheequivalentsystem a )]TJ/F21 10.9091 Tf 10.909 0 Td [(c =8 b + c =5 Wecanvary c andobtaininnitelymanysolutions.However, c mustbeadigit,restrictingustotenvalues {9.Furthermore,if c> 1,thentherstequationforces a> 9,animpossibility.Setting c =0,yields 850asasolution,andsetting c =1yields941asanothersolution. C51 ContributedbyRobertBeezerStatement[18] Let abcdef denoteanysuchsix-digitnumberandconverteachrequirementintheproblemstatementinto anequation. a = b )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 c = 1 2 b d =3 c 10 e + f = d + e 24= a + b + c + d + e + f Inamorestandardformthisbecomes a )]TJ/F21 10.9091 Tf 10.909 0 Td [(b = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F21 10.9091 Tf 8.485 0 Td [(b +2 c =0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 c + d =0 )]TJ/F21 10.9091 Tf 8.485 0 Td [(d +9 e + f =0 a + b + c + d + e + f =24 UsingequationoperationsorthetechniquesoftheupcomingSectionRREF[24],thissystemcanbe convertedtotheequivalentsystem a + 16 75 f =5 b + 16 75 f =6 c + 8 75 f =3 Version2.02

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SubsectionSSLE.SOLSolutions22 d + 8 25 f =9 e + 11 75 f =1 Clearly,choosing f =0willyieldthesolution abcde =563910.Furthermore,tohavethevariablesresult insingle-digitnumbers,noneoftheotherchoicesfor f ; 2 ;:::; 9willyieldasolution. C52 ContributedbyRobertBeezerStatement[18] 198888isonesolution,andDavidBraithwaitefound199999asanother. M10 ContributedbyRobertBeezerStatement[18] 1.Isbaking"averboranadjective? Potatoesarebeingbaked. Thosearebakingpotatoes. 2.Aretheapricotsripe,orjustthepears?Parenthesescouldindicatejustwhattheadjectiveripe"is meanttomodify.Weretheremanyapricotsaswell,orjustmanypears? Heboughtmanypearsandmanyripeapricots. Heboughtapricotsandmanyripepears. 3.Issculpture"asinglephysicalobject,orthesculptor'sstyleexpressedovermanypiecesandmany years? Shelikeshissculptureofthegirl. Shelikeshissculpturalstyle. 4.Wasadecisionmadewhileinthebus,orwastheoutcomeofadecisiontochoosethebus.Wouldthe sentenceIdecidedonthecar,"haveasimilardoublemeaning? Imademydecisionwhileonthebus. Idecidedtoridethebus. M11 ContributedbyRobertBeezerStatement[19] Weknowthedogbelongstotheman,andthefountainbelongstothepark.Itisnotclearifthetelescope belongstotheman,thewoman,orthepark. M12 ContributedbyRobertBeezerStatement[19] Inadjacentpairsthewordsarecontradictoryorinappropriate.Somethingcannotbebothgreenand colorless,ideasdonothavecolor,ideasdonotsleep,anditishardtosleepfuriously. M13 ContributedbyRobertBeezerStatement[19] Didyouassumethatthebabyandmotherarehuman? Didyouassumethatthebabyisthechildofthemother? Didyouassumethatthemotherpickedupthebabyasanattempttostopthecrying? M30 ContributedbyRobertBeezerStatement[19] If x y and z representthemoneyheldbyDan,DianeandDonna,then y =15 )]TJ/F21 10.9091 Tf 11.412 0 Td [(z and x =20 )]TJ/F21 10.9091 Tf 11.413 0 Td [(y = 20 )]TJ/F15 10.9091 Tf 10.287 0 Td [( )]TJ/F21 10.9091 Tf 10.287 0 Td [(z =5+ z .Wecanlet z takeonanyvaluefrom0to15withoutanyofthethreeamountsbeing negative,sincepresumablymiddle-schoolersaretooyoungtoassumedebt. Thenthetotalcapitalheldbythethreeis x + y + z =+ z + )]TJ/F21 10.9091 Tf 9.934 0 Td [(z + z =20+ z .Sotheircombined holdingscanrangeanywherefrom$20Donnaisbroketo$35Donnaisush. WewillhavemoretosayaboutthissituationinSectionTSS[50],andspecicallyTheoremCMVEI [56]. Version2.02

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SubsectionSSLE.SOLSolutions23 M70 ContributedbyRobertBeezerStatement[19] Theequation x 2 )]TJ/F21 10.9091 Tf 10.544 0 Td [(y 2 =1hasasolutionsetbyitselfthathastheshapeofahyperbolawhenplotted.The vedierentsecondequationshavesolutionsetsthatarecircleswhenplottedindividually.Wherethe hyperbolaandcircleintersectarethesolutionstothesystemoftwoequations.Asthesizeandlocationof thecirclevaries,thenumberofintersectionsvariesfromfourtononeintheordergiven.Sketchingthe relevantequationswouldbeinstructive,aswasdiscussedinExampleSTNE[9]. Theexactsolutionsetsareaccordingtothechoiceofthesecondequation, x 2 + y 2 =4: r 5 2 ; r 3 2 ; )]TJ/F27 10.9091 Tf 8.485 17.142 Td [(r 5 2 ; r 3 2 ; r 5 2 ; )]TJ/F27 10.9091 Tf 8.485 17.142 Td [(r 3 2 ; )]TJ/F27 10.9091 Tf 8.485 17.142 Td [(r 5 2 ; )]TJ/F27 10.9091 Tf 8.485 17.142 Td [(r 3 2 x 2 +2 x + y 2 =3: n ; 0 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; p 3 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 ; )]TJ 8.485 9.557 Td [(p 3 o x 2 + y 2 =1: f ; 0 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 0 g x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x + y 2 =0: f ; 0 g 4 x 2 +4 y 2 =1: fg T10 ContributedbyRobertBeezerStatement[19] Wecansaythatanintegeris odd ifwhenitisdividedby2thereisaremainderof1.So6isnotodd since6=3 2+0,while11isoddsince11=5 2+1. T20 ContributedbyRobertBeezerStatement[20] DenitionEO[11]isengineeredtomakeTheoremEOPSS[12]true.Ifweweretoallowazeroscalarto multiplyanequationthenthatequationwouldbetransformedtotheequation0=0,whichistruefor anypossiblevaluesofthevariables.Anyrestrictionsonthesolutionsetimposedbytheoriginalequation wouldbelost. However,inthethirdoperation,itisallowedtochooseazeroscalar,multiplyanequationbythis scalarandaddthetransformedequationtoasecondequationleavingtherstunchanged.Theresult? Nothing.Thesecondequationisthesameasitwasbefore.Sothetheoremistrueinthiscase,thetwo systemsareequivalent.Butinpractice,thiswouldbeasillythingtoactuallyeverdo!Westillallowit though,inordertokeepourtheoremasgeneralaspossible. NoticethelocationintheproofofTheoremEOPSS[12]wheretheexpression 1 appears|thisexplains theprohibitionon =0inthesecondequationoperation. Version2.02

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SectionRREFReducedRow-EchelonForm24 SectionRREF ReducedRow-EchelonForm Aftersolvingafewsystemsofequations,youwillrecognizethatitdoesn'tmattersomuch what wecall ourvariables,asopposedtowhatnumbersactastheircoecients.Asysteminthevariables x 1 ;x 2 ;x 3 wouldbehavethesameifwechangedthenamesofthevariablesto a;b;c andkeptalltheconstantsthe sameandinthesameplaces.Inthissection,wewillisolatethekeybitsofinformationaboutasystemof equationsintosomethingcalledamatrix,andthenusethismatrixtosystematicallysolvetheequations. Alongthewaywewillobtainoneofourmostimportantandusefulcomputationaltools. SubsectionMVNSE MatrixandVectorNotationforSystemsofEquations DenitionM Matrix An m n matrix isarectangularlayoutofnumbersfrom C having m rowsand n columns.Wewilluse upper-caseLatinlettersfromthestartofthealphabet A;B;C;::: todenotematricesandsquared-o bracketstodelimitthelayout.Manyuselargeparenthesesinsteadofbrackets|thedistinctionisnot important.Rowsofamatrixwillbereferencedstartingatthetopandworkingdowni.e.row1isatthe topandcolumnswillbereferencedstartingfromthelefti.e.column1isattheleft.Foramatrix A thenotation[ A ] ij willrefertothecomplexnumberinrow i andcolumn j of A ThisdenitioncontainsNotationM. ThisdenitioncontainsNotationMC. 4 Becarefulwiththisnotationforindividualentries,sinceitiseasytothinkthat[ A ] ij referstothe whole matrix.Itdoesnot.Itisjusta number ,butisaconvenientwaytotalkabouttheindividualentries simultaneously.ThisnotationwillgetaheavyworkoutoncewegettoChapterM[182]. ExampleAM Amatrix B = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1253 10 )]TJ/F15 10.9091 Tf 8.484 0 Td [(61 )]TJ/F15 10.9091 Tf 8.485 0 Td [(422 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 3 5 isamatrixwith m =3rowsand n =4columns.Wecansaythat[ B ] 2 ; 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(6while[ B ] 3 ; 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2. Somemathematicalsoftwareisveryparticularaboutwhichtypesofnumbersintegers,rationals, reals,complexesyouwishtoworkwith.See:ComputationR.SAGE[674].Acalculatororcomputer languagecanbeaconvenientwaytoperformcalculationswithmatrices.Butrstyouhavetoenterthe matrix.See:ComputationME.MMA[667]ComputationME.TI86[672]ComputationME.TI83[673] ComputationME.SAGE[675].Whenwedoequationoperationsonsystemofequations,thenamesof thevariablesreallyaren'tveryimportant. x 1 x 2 x 3 ,or a b c ,or x y z ,itreallydoesn'tmatter.In thissubsectionwewilldescribesomenotationthatwillmakeiteasiertodescribelinearsystems,solvethe systemsanddescribethesolutionsets.Hereisalistofdenitions,ladenwithnotation. DenitionCV ColumnVector A columnvector of size m isanorderedlistof m numbers,whichiswritteninordervertically,starting atthetopandproceedingtothebottom.Attimes,wewillrefertoacolumnvectorassimplya vector Version2.02

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SubsectionRREF.MVNSEMatrixandVectorNotationforSystemsofEquations25 Columnvectorswillbewritteninbold,usuallywithlowercaseLatinletterfromtheendofthealphabet suchas u v w x y z .Somebooksliketowritevectorswitharrows,suchas ~u .Writingbyhand,some liketoputarrowsontopofthesymbol,oratildeunderneaththesymbol,asin u .Torefertothe entry or component thatisnumber i inthelistthatisthevector v wewrite[ v ] i ThisdenitioncontainsNotationCV. ThisdenitioncontainsNotationCVC. 4 Becarefulwiththisnotation.Whilethesymbols[ v ] i mightlooksomewhatsubstantial,asanobject thisrepresentsjustonecomponentofavector,whichisjustasinglecomplexnumber. DenitionZCV ZeroColumnVector The zerovector ofsize m isthecolumnvectorofsize m whereeachentryisthenumberzero, 0 = 2 6 6 6 6 6 4 0 0 0 0 3 7 7 7 7 7 5 ordenedmuchmorecompactly,[ 0 ] i =0for1 i m ThisdenitioncontainsNotationZCV. 4 DenitionCM CoecientMatrix Forasystemoflinearequations, a 11 x 1 + a 12 x 2 + a 13 x 3 + + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + a 23 x 3 + + a 2 n x n = b 2 a 31 x 1 + a 32 x 2 + a 33 x 3 + + a 3 n x n = b 3 a m 1 x 1 + a m 2 x 2 + a m 3 x 3 + + a mn x n = b m the coecientmatrix isthe m n matrix A = 2 6 6 6 6 6 4 a 11 a 12 a 13 :::a 1 n a 21 a 22 a 23 :::a 2 n a 31 a 32 a 33 :::a 3 n a m 1 a m 2 a m 3 :::a mn 3 7 7 7 7 7 5 4 DenitionVOC VectorofConstants Forasystemoflinearequations, a 11 x 1 + a 12 x 2 + a 13 x 3 + + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + a 23 x 3 + + a 2 n x n = b 2 a 31 x 1 + a 32 x 2 + a 33 x 3 + + a 3 n x n = b 3 Version2.02

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SubsectionRREF.MVNSEMatrixandVectorNotationforSystemsofEquations26 a m 1 x 1 + a m 2 x 2 + a m 3 x 3 + + a mn x n = b m the vectorofconstants isthecolumnvectorofsize m b = 2 6 6 6 6 6 4 b 1 b 2 b 3 b m 3 7 7 7 7 7 5 4 DenitionSOLV SolutionVector Forasystemoflinearequations, a 11 x 1 + a 12 x 2 + a 13 x 3 + + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + a 23 x 3 + + a 2 n x n = b 2 a 31 x 1 + a 32 x 2 + a 33 x 3 + + a 3 n x n = b 3 a m 1 x 1 + a m 2 x 2 + a m 3 x 3 + + a mn x n = b m the solutionvector isthecolumnvectorofsize n x = 2 6 6 6 6 6 4 x 1 x 2 x 3 x n 3 7 7 7 7 7 5 4 Thesolutionvectormaydodouble-dutyonoccasion.Itmightrefertoalistofvariablequantitiesat onepoint,andsubsequentlyrefertovaluesofthosevariablesthatactuallyformaparticularsolutionto thatsystem. DenitionMRLS MatrixRepresentationofaLinearSystem If A isthecoecientmatrixofasystemoflinearequationsand b isthevectorofconstants,thenwewill write LS A; b asashorthandexpressionforthesystemoflinearequations,whichwewillrefertoasthe matrixrepresentation ofthelinearsystem. ThisdenitioncontainsNotationMRLS. 4 ExampleNSLE Notationforsystemsoflinearequations Thesystemoflinearequations 2 x 1 +4 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 +5 x 4 + x 5 =9 3 x 1 + x 2 + x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 5 =0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 1 +7 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 3 +2 x 4 +2 x 5 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 Version2.02

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SubsectionRREF.RORowOperations27 hascoecientmatrix A = 2 4 24 )]TJ/F15 10.9091 Tf 8.485 0 Td [(351 3101 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(27 )]TJ/F15 10.9091 Tf 8.485 0 Td [(522 3 5 andvectorofconstants b = 2 4 9 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 5 andsowillbereferencedas LS A; b DenitionAM AugmentedMatrix Supposewehaveasystemof m equationsin n variables,withcoecientmatrix A andvectorofconstants b .Thenthe augmentedmatrix ofthesystemofequationsisthe m n +1matrixwhoserst n columnsarethecolumnsof A andwhoselastcolumnnumber n +1isthecolumnvector b .Thismatrix willbewrittenas[ A j b ]. ThisdenitioncontainsNotationAM. 4 Theaugmentedmatrix represents alltheimportantinformationinthesystemofequations,sincethe namesofthevariableshavebeenignored,andtheonlyconnectionwiththevariablesisthelocationof theircoecientsinthematrix.Itisimportanttorealizethattheaugmentedmatrixisjustthat,amatrix, and not asystemofequations.Inparticular,theaugmentedmatrixdoesnothaveanysolutions,"though itwillbeusefulforndingsolutionstothesystemofequationsthatitisassociatedwith.Thinkabout yourobjects,andreviewTechniqueL[688].However,noticethatanaugmentedmatrixalwaysbelongs tosomesystemofequations,andviceversa,soitistemptingtotryandblurthedistinctionbetweenthe two.Here'saquickexample. ExampleAMAA AugmentedmatrixforArchetypeA ArchetypeA[702]isthefollowingsystemof3equationsin3variables. x 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 2 +2 x 3 =1 2 x 1 + x 2 + x 3 =8 x 1 + x 2 =5 Hereisitsaugmentedmatrix. 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(121 2118 1105 3 5 SubsectionRO RowOperations Anaugmentedmatrixforasystemofequationswillsaveusthetediumofcontinuallywritingdownthe namesofthevariablesaswesolvethesystem.Itwillalsoreleaseusfromanydependenceontheactual namesofthevariables.WehaveseenhowcertainoperationswecanperformonequationsDenition EO[11]willpreservetheirsolutionsTheoremEOPSS[12].Thenexttwodenitionsandthefollowing theoremcarryovertheseideastoaugmentedmatrices. Version2.02

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SubsectionRREF.RORowOperations28 DenitionRO RowOperations Thefollowingthreeoperationswilltransforman m n matrixintoadierentmatrixofthesamesize,and eachisknownasa rowoperation 1.Swapthelocationsoftworows. 2.Multiplyeachentryofasinglerowbyanonzeroquantity. 3.Multiplyeachentryofonerowbysomequantity,andaddthesevaluestotheentriesinthesame columnsofasecondrow.Leavetherstrowthesameafterthisoperation,butreplacethesecond rowbythenewvalues. Wewilluseasymbolicshorthandtodescribetheserowoperations: 1. R i $ R j :Swapthelocationofrows i and j 2. R i :Multiplyrow i bythenonzeroscalar 3. R i + R j :Multiplyrow i bythescalar andaddtorow j ThisdenitioncontainsNotationRO. 4 DenitionREM Row-EquivalentMatrices Twomatrices, A and B ,are row-equivalent ifonecanbeobtainedfromtheotherbyasequenceofrow operations. 4 ExampleTREM Tworow-equivalentmatrices Thematrices A = 2 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(134 52 )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 1106 3 5 B = 2 4 1106 30 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(134 3 5 arerow-equivalentascanbeseenfrom 2 4 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(134 52 )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 1106 3 5 R 1 $ R 3 )456()222()222()223()456(! 2 4 1106 52 )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(134 3 5 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 R 1 + R 2 )324()222()222()223()222()324(! 2 4 1106 30 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(134 3 5 Wecanalsosaythatanypairofthesethreematricesarerow-equivalent. NoticethateachofthethreerowoperationsisreversibleExerciseRREF.T10[43],sowedonothave tobecarefulaboutthedistinctionbetween A isrow-equivalentto B "and B isrow-equivalentto A ." ExerciseRREF.T11[43]Theprecedingdenitionsaredesignedtomakethefollowingtheorempossible. Itsaysthatrow-equivalentmatricesrepresentsystemsoflinearequationsthathaveidenticalsolutionsets. TheoremREMES Row-EquivalentMatricesrepresentEquivalentSystems Supposethat A and B arerow-equivalentaugmentedmatrices.Thenthesystemsoflinearequationsthat theyrepresentareequivalentsystems. Proof Ifweperformasinglerowoperationonanaugmentedmatrix,itwillhavethesameeectasif wedidtheanalogousequationoperationonthecorrespondingsystemofequations.Byexactlythesame Version2.02

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SubsectionRREF.RREFReducedRow-EchelonForm29 methodsasweusedintheproofofTheoremEOPSS[12]wecanseethateachoftheserowoperationswill preservethesetofsolutionsforthecorrespondingsystemofequations. Soatthispoint,ourstrategyistobeginwithasystemofequations,representitbyanaugmented matrix,performrowoperationswhichwillpreservesolutionsforthecorrespondingsystemstogeta simpler"augmentedmatrix,convertbacktoasimpler"systemofequationsandthensolvethatsystem, knowingthatitssolutionsarethoseoftheoriginalsystem.Here'sarehashofExampleUS[14]asan exerciseinusingournewtools. ExampleUSR Threeequations,onesolution,reprised Wesolvethefollowingsystemusingaugmentedmatricesandrowoperations.Thisisthesamesystemof equationssolvedinExampleUS[14]usingequationoperations. x 1 +2 x 2 +2 x 3 =4 x 1 +3 x 2 +3 x 3 =5 2 x 1 +6 x 2 +5 x 3 =6 Formtheaugmentedmatrix, A = 2 4 1224 1335 2656 3 5 andapplyrowoperations, )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 R 1 + R 2 )324()222()223()222()222()324(! 2 4 1224 0111 2656 3 5 )]TJ/F19 7.9701 Tf 6.587 0 Td [(2 R 1 + R 3 )324()222()223()222()222()324(! 2 4 1224 0111 021 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 5 )]TJ/F19 7.9701 Tf 6.587 0 Td [(2 R 2 + R 3 )324()222()223()222()222()324(! 2 4 1224 0111 00 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 5 )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 R 3 )275()222()276(! 2 4 1224 0111 0014 3 5 Sothematrix B = 2 4 1224 0111 0014 3 5 isrowequivalentto A andbyTheoremREMES[28]thesystemofequationsbelowhasthesamesolution setastheoriginalsystemofequations. x 1 +2 x 2 +2 x 3 =4 x 2 + x 3 =1 x 3 =4 Solvingthissimpler"systemisstraightforwardandisidenticaltotheprocessinExampleUS[14]. SubsectionRREF ReducedRow-EchelonForm Theprecedingexampleamplyillustratesthedenitionsandtheoremswehaveseensofar.Butitstill leavestwoquestionsunanswered.Exactlywhatisthissimpler"formforamatrix,andjusthowdowe getit?Here'stheanswertotherstquestion,adenitionofreducedrow-echelonform. Version2.02

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SubsectionRREF.RREFReducedRow-EchelonForm30 DenitionRREF ReducedRow-EchelonForm Amatrixisin reducedrow-echelonform ifitmeetsallofthefollowingconditions: 1.Arowwhereeveryentryiszeroliesbelowanyrowthatcontainsanonzeroentry. 2.Theleftmostnonzeroentryofarowisequalto1. 3.Theleftmostnonzeroentryofarowistheonlynonzeroentryinitscolumn. 4.Consideranytwodierentleftmostnonzeroentries,onelocatedinrow i ,column j andtheother locatedinrow s ,column t .If s>i ,then t>j Arowofonlyzeroentrieswillbecalleda zerorow andtheleftmostnonzeroentryofanonzerorowwill becalleda leading1 .Thenumberofnonzerorowswillbedenotedby r Acolumncontainingaleading1willbecalleda pivotcolumn .Thesetofcolumnindicesforallof thepivotcolumnswillbedenotedby D = f d 1 ;d 2 ;d 3 ;:::;d r g where d 1
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SubsectionRREF.RREFReducedRow-EchelonForm31 1. A and B arerow-equivalent. 2. B isinreducedrow-echelonform. Proof Supposethat A has m rowsand n columns.Wewilldescribeaprocessforconverting A into B viarowoperations.Thisprocedureisknownas Gauss{Jordanelimination .Tracingthroughthis procedurewillbeeasierifyourecognizethat i referstoarowthatisbeingconverted, j referstoacolumn thatisbeingconverted,and r keepstrackofthenumberofnonzerorows.Herewego. 1.Set j =0and r =0. 2.Increase j by1.If j nowequals n +1,thenstop. 3.Examinetheentriesof A incolumn j locatedinrows r +1through m Ifalloftheseentriesarezero,thengotoStep2. 4.Choosearowfromrows r +1through m withanonzeroentryincolumn j Let i denotetheindexforthisrow. 5.Increase r by1. 6.Usetherstrowoperationtoswaprows i and r 7.Usethesecondrowoperationtoconverttheentryinrow r andcolumn j toa1. 8.Usethethirdrowoperationwithrow r toconverteveryotherentryofcolumn j tozero. 9.GotoStep2. Theresultofthisprocedureisthatthematrix A isconvertedtoamatrixinreducedrow-echelonform, whichwewillrefertoas B .Weneedtonowprovethisclaimbyshowingthattheconvertedmatrixhasthe requisitepropertiesofDenitionRREF[30].First,thematrixisonlyconvertedthroughrowoperations Step6,Step7,Step8,so A and B arerow-equivalentDenitionREM[28]. Itisabitmoreworktobecertainthat B isinreducedrow-echelonform.Weclaimthataswebegin Step2,therst j columnsofthematrixareinreducedrow-echelonformwith r nonzerorows.Certainly thisistrueatthestartwhen j =0,sincethematrixhasnocolumnsandsovacuouslymeetstheconditions ofDenitionRREF[30]with r =0nonzerorows. InStep2weincrease j by1andbegintoworkwiththenextcolumn.Therearetwopossibleoutcomes forStep3.Supposethateveryentryofcolumn j inrows r +1through m iszero.Thenwithnochanges werecognizethattherst j columnsofthematrixhasitsrst r rowsstillinreduced-rowechelonform, withthenal m )]TJ/F21 10.9091 Tf 10.909 0 Td [(r rowsstillallzero. Supposeinsteadthattheentryinrow i ofcolumn j isnonzero.Noticethatsince r +1 i m ,we knowtherst j )]TJ/F15 10.9091 Tf 11.093 0 Td [(1entriesofthisrowareallzero.Now,inStep5weincrease r by1,andthenembark onbuildinganewnonzerorow.InStep6weswaprow r androw i .Intherst j columns,therst r )]TJ/F15 10.9091 Tf 10.916 0 Td [(1 rowsremaininreducedrow-echelonformaftertheswap.InStep7wemultiplyrow r byanonzeroscalar, creatinga1intheentryincolumn j ofrow i ,andnotchanginganyotherrows.Thisnewleading1isthe rstnonzeroentryinitsrow,andislocatedtotherightofalltheleading1'sinthepreceding r )]TJ/F15 10.9091 Tf 10.794 0 Td [(1rows. WithStep8weinsurethateveryentryinthecolumnwiththisnewleading1isnowzero,asrequiredfor reducedrow-echelonform.Also,rows r +1through m arenowallzerosintherst j columns,sowenow onlyhaveonenewnonzerorow,consistentwithourincreaseof r byone.Furthermore,sincetherst j )]TJ/F15 10.9091 Tf 10.207 0 Td [(1 entriesofrow r arezero,theemploymentofthethirdrowoperationdoesnotdestroyanyofthenecessary featuresofrows1through r )]TJ/F15 10.9091 Tf 10.909 0 Td [(1androws r +1through m ,incolumns1through j )]TJ/F15 10.9091 Tf 10.909 0 Td [(1. Version2.02

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SubsectionRREF.RREFReducedRow-EchelonForm32 Soatthisstage,therst j columnsofthematrixareinreducedrow-echelonform.WhenStep2nally increases j to n +1,thentheprocedureiscompletedandthefull n columnsofthematrixareinreduced row-echelonform,withthevalueof r correctlyrecordingthenumberofnonzerorows. TheproceduregivenintheproofofTheoremREMEF[30]canbemorepreciselydescribedusinga pseudo-codeversionofacomputerprogram,asfollows: input m n and A r 0 for j 1to n i r +1 while i m and[ A ] ij =0 i i +1 if i 6 = m +1 r r +1 swaprows i and r of A rowop1 scaleentryinrow r ,column j of A toaleading1rowop2 for k 1to m k 6 = r zerooutentryinrow k ,column j of A rowop3usingrow r output r and A Noticethatasapracticalmattertheand"usedintheconditionalstatementofthewhilestatementshould beoftheshort-circuit"varietysothatthearrayaccessthatfollowsisnotout-of-bounds. Sonowwecanputitalltogether.BeginwithasystemoflinearequationsDenitionSLE[9],and representthesystembyitsaugmentedmatrixDenitionAM[27].UserowoperationsDenitionRO [28]toconvertthismatrixintoreducedrow-echelonformDenitionRREF[30],usingtheprocedure outlinedintheproofofTheoremREMEF[30].TheoremREMEF[30]alsotellsuswecanalwaysaccomplish this,andthattheresultisrow-equivalentDenitionREM[28]totheoriginalaugmentedmatrix.Since thematrixinreduced-rowechelonformhasthesamesolutionset,wecananalyzetherow-reducedversion insteadoftheoriginalmatrix,viewingitastheaugmentedmatrixofadierentsystemofequations.The beautyofaugmentedmatricesinreducedrow-echelonformisthatthesolutionsetstotheircorresponding systemscanbeeasilydetermined,aswewillseeinthenextfewexamplesandinthenextsection. Wewillseethroughthecoursethatalmosteveryinterestingpropertyofamatrixcanbediscernedby lookingatarow-equivalentmatrixinreducedrow-echelonform.Forthisreasonitisimportanttoknow thatthematrix B guaranteedtoexistbyTheoremREMEF[30]isalsounique. Twoprooftechniquesareapplicabletotheproof.First,headoutandreadtwoprooftechniques: TechniqueCD[692]andTechniqueU[693]. TheoremRREFU ReducedRow-EchelonFormisUnique Supposethat A isan m n matrixandthat B and C are m n matricesthatarerow-equivalentto A andinreducedrow-echelonform.Then B = C Proof Weneedtobeginwithnoassumptionsaboutanyrelationshipsbetween B and C ,otherthanthey arebothinreducedrow-echelonform,andtheyarebothrow-equivalentto A If B and C arebothrow-equivalentto A ,thentheyarerow-equivalenttoeachother.Repeatedrow operationsonamatrixcombinetherowswitheachotherusingoperationsthatarelinear,andareidentical ineachcolumn.Akeyobservationforthisproofisthateachindividualrowof B islinearlyrelatedtothe rowsof C .Thisrelationshipisdierentforeachrowof B ,butoncewexarow,therelationshipisthe sameacrosscolumns.Moreprecisely,therearescalars ik ,1 i;k m suchthatforany1 i m 1 j n [ B ] ij = m X k =1 ik [ C ] kj Version2.02

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SubsectionRREF.RREFReducedRow-EchelonForm33 Youshouldreadthisassayingthatanentryofrow i of B incolumn j isalinearfunctionoftheentries ofalltherowsof C thatarealsoincolumn j ,andthescalars ik dependonwhichrowof B weare consideringthe i subscripton ik ,butarethesameforeverycolumnnodependenceon j in ik .This ideamaybecomplicatednow,butwillfeelmorefamiliaroncewediscusslinearcombinations"Denition LCCV[90]andmoresowhenwediscussrowspaces"DenitionRSM[243].Fornow,spendsometime carefullyworkingExerciseRREF.M40[42],whichisdesignedtoillustratetheoriginsofthisexpression. Thiscompletesourexploitationoftherow-equivalenceof B and C Wenowrepeatedlyexploitthefactthat B and C areinreducedrow-echelonform.Recallthatapivot columnisallzeros,exceptasingleone.Morecarefully,if R isamatrixinreducedrow-echelonform,and d ` istheindexofapivotcolumn,then[ R ] kd ` =1preciselywhen k = ` andisotherwisezero.Noticealso thatanyentryof R thatisbothbelowtheentryinrow ` and totheleftofcolumn d ` isalsozerowith belowandleftunderstoodtoincludeequality.Inotherwords,lookatexamplesofmatricesinreduced row-echelonformandchoosealeading1withaboxaroundit.Therestofthecolumnisalsozeros,and thelowerleftquadrant"ofthematrixthatbeginshereistotallyzeros. Assumingnorelationshipabouttheformof B and C ,let B have r nonzerorowsanddenotethepivot columnsas D = f d 1 ;d 2 ;d 3 ;:::;d r g .For C let r 0 denotethenumberofnonzerorowsanddenotethe pivotcolumnsas D 0 = f d 0 1 ;d 0 2 ;d 0 3 ;:::;d 0 r 0 g NotationRREFA[30].Therearefourstepsintheproof, andtherstthreeareaboutshowingthat B and C havethesamenumberofpivotcolumns,inthesame places.Inotherwords,theprimed"symbolsareanecessaryction. FirstStep.Supposethat d 1
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SubsectionRREF.RREFReducedRow-EchelonForm34 Now, 1=[ B ] p +1 ;d p +1 DenitionRREF[30] = m X k =1 p +1 ;k [ C ] kd p +1 = p X k =1 p +1 ;k [ C ] kd p +1 + m X k = p +1 p +1 ;k [ C ] kd p +1 PropertyAACN[680] = p X k =1 [ C ] kd p +1 + m X k = p +1 p +1 ;k [ C ] kd p +1 = m X k = p +1 p +1 ;k [ C ] kd p +1 = m X k = p +1 p +1 ;k d p +1
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SubsectionRREF.RREFReducedRow-EchelonForm35 = r 0 X k =1 rk [ C ] kj + m X k = r 0 +1 rk [ C ] kj PropertyCACN[680] = r 0 X k =1 rk [ C ] kj + m X k = r 0 +1 rk DenitionRREF[30] = r 0 X k =1 rk [ C ] kj = r 0 X k =1 [ C ] kj =0 Sorow r isatotallyzerorow,contradictingthatthisshouldbethebottommostnonzerorowof B .So r 0 r .Byanentirelysimilarargument,reversingtherolesof B and C ,wewouldconcludethat r 0 r andtherefore r = r 0 .Thus,combiningtherstthreestepswecansaythat D = D 0 .Inotherwords, B and C havethesamepivotcolumns,inthesamelocations. FourthStep.Inthisnalstep,wewillnotarguebycontradiction.Ourintentistodeterminethe valuesofthe ij .Noticethatwecanusethevaluesofthe d i interchangeablyfor B and C .Herewego, 1=[ B ] id i DenitionRREF[30] = m X k =1 ik [ C ] kd i = ii [ C ] id i + m X k =1 k 6 = i ik [ C ] kd i PropertyCACN[680] = ii + m X k =1 k 6 = i ik DenitionRREF[30] = ii andfor ` 6 = i 0=[ B ] id ` DenitionRREF[30] = m X k =1 ik [ C ] kd ` = i` [ C ] `d ` + m X k =1 k 6 = ` ik [ C ] kd i PropertyCACN[680] = i` + m X k =1 k 6 = ` ik DenitionRREF[30] = i` Finally,havingdeterminedthevaluesofthe ij ,wecanshowthat B = C .For1 i m ,1 j n [ B ] ij = m X k =1 ik [ C ] kj Version2.02

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SubsectionRREF.RREFReducedRow-EchelonForm36 = ii [ C ] ij + m X k =1 k 6 = i ik [ C ] kj PropertyCACN[680] =[ C ] ij + m X k =1 k 6 = i [ C ] kj =[ C ] ij So B and C haveequalvaluesineveryentry,andsoarethesamematrix. Wewillnowrunthroughsomeexamplesofusingthesedenitionsandtheoremstosolvesomesystems ofequations.Fromnowon,whenwehaveamatrixinreducedrow-echelonform,wewillmarktheleading 1'swithasmallbox.Inyourwork,youcanbox'em,circle'emorwrite'eminadierentcolor|just identify'emsomehow.Thisdevicewillproveveryusefullaterandisaverygoodhabittostartdeveloping rightnow. ExampleSAB SolutionsforArchetypeB Let'sndthesolutionstothefollowingsystemofequations, )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 x 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(33 5 x 1 +5 x 2 +7 x 3 =24 x 1 +4 x 3 =5 First,formtheaugmentedmatrix, 2 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(33 55724 1045 3 5 andworktoreducedrow-echelonform,rstwith i =1, R 1 $ R 3 )456()222()222()223()456(! 2 4 1045 55724 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(33 3 5 )]TJ/F19 7.9701 Tf 6.587 0 Td [(5 R 1 + R 2 )324()222()223()222()222()324(! 2 4 1045 05 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(33 3 5 7 R 1 + R 3 )348()222()223()222()348(! 2 4 1 045 05 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6162 3 5 Now,with i =2, 1 5 R 2 )494()222()494(! 2 4 1 045 01 )]TJ/F19 7.9701 Tf 6.586 0 Td [(13 5 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 5 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6162 3 5 6 R 2 + R 3 )348()222()223()222()348(! 2 4 1 045 0 1 )]TJ/F19 7.9701 Tf 6.587 0 Td [(13 5 )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 5 00 2 5 4 5 3 5 Andnally,with i =3, 5 2 R 3 )494()222()494(! 2 4 1 045 0 1 )]TJ/F19 7.9701 Tf 6.587 0 Td [(13 5 )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 5 0012 3 5 13 5 R 3 + R 2 )376()222()222()222()223()375(! 2 4 1 045 0 1 05 0012 3 5 )]TJ/F19 7.9701 Tf 6.587 0 Td [(4 R 3 + R 1 )324()222()223()222()222()324(! 2 4 1 00 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 0 1 05 00 1 2 3 5 Version2.02

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SubsectionRREF.RREFReducedRow-EchelonForm37 Thisisnowtheaugmentedmatrixofaverysimplesystemofequations,namely x 1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3, x 2 =5, x 3 =2, whichhasanobvioussolution.Furthermore,wecanseethatthisisthe only solutiontothissystem,sowe havedeterminedtheentiresolutionset, S = 8 < : 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 5 2 3 5 9 = ; YoumightcomparethisexamplewiththeprocedureweusedinExampleUS[14]. ArchetypesAandBaremeanttocontrasteachotherinmanyrespects.Solet'ssolveArchetypeA now. ExampleSAA SolutionsforArchetypeA Let'sndthesolutionstothefollowingsystemofequations, x 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 2 +2 x 3 =1 2 x 1 + x 2 + x 3 =8 x 1 + x 2 =5 First,formtheaugmentedmatrix, 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(121 2118 1105 3 5 andworktoreducedrow-echelonform,rstwith i =1, )]TJ/F19 7.9701 Tf 6.587 0 Td [(2 R 1 + R 2 )324()222()222()223()222()324(! 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(121 03 )]TJ/F15 10.9091 Tf 8.485 0 Td [(36 1105 3 5 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 R 1 + R 3 )324()222()223()222()222()324(! 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(121 03 )]TJ/F15 10.9091 Tf 8.485 0 Td [(36 02 )]TJ/F15 10.9091 Tf 8.485 0 Td [(24 3 5 Now,with i =2, 1 3 R 2 )494()222()494(! 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(121 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 02 )]TJ/F15 10.9091 Tf 8.485 0 Td [(24 3 5 1 R 2 + R 1 )348()222()223()222()348(! 2 4 1 013 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 02 )]TJ/F15 10.9091 Tf 8.485 0 Td [(24 3 5 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 R 2 + R 3 )324()222()223()222()222()324(! 2 4 1 013 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 0000 3 5 Thesystemofequationsrepresentedbythisaugmentedmatrixneedstobeconsideredabitdierently thanthatforArchetypeB.First,thelastrowofthematrixistheequation0=0,whichis always true,so itimposesnorestrictionsonourpossiblesolutionsandthereforewecansafelyignoreitasweanalyzethe othertwoequations.Theseequationsare, x 1 + x 3 =3 x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 3 =2 : Whilethissystemisfairlyeasytosolve,italsoappearstohaveamultitudeofsolutions.Forexample, choose x 3 =1andseethatthen x 1 =2and x 2 =3willtogetherformasolution.Orchoose x 3 =0,and thendiscoverthat x 1 =3and x 2 =2leadtoasolution.Tryityourself:pick any valueof x 3 youplease, andgureoutwhat x 1 and x 2 shouldbetomaketherstandsecondequationsrespectivelytrue.We'll Version2.02

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SubsectionRREF.RREFReducedRow-EchelonForm38 waitwhileyoudothat.Becauseofthisbehavior,wesaythat x 3 isafree"orindependent"variable.But whydowevary x 3 andnotsomeothervariable?Fornow,noticethatthethirdcolumnoftheaugmented matrixdoesnothaveanyleading1'sinitscolumn.Withthisidea,wecanrearrangethetwoequations, solvingeachforthevariablethatcorrespondstotheleading1inthatrow. x 1 =3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 3 x 2 =2+ x 3 Towritethesetofsolutionvectorsinsetnotation,wehave S = 8 < : 2 4 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 3 2+ x 3 x 3 3 5 j x 3 2 C 9 = ; We'lllearnmoreinthenextsectionaboutsystemswithinnitelymanysolutionsandhowtoexpresstheir solutionsets.Rightnow,youmightlookbackatExampleIS[15]. ExampleSAE SolutionsforArchetypeE Let'sndthesolutionstothefollowingsystemofequations, 2 x 1 + x 2 +7 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x 4 =2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x 1 +4 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 4 =3 x 1 + x 2 +4 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 4 =2 First,formtheaugmentedmatrix, 2 4 217 )]TJ/F15 10.9091 Tf 8.485 0 Td [(72 )]TJ/F15 10.9091 Tf 8.485 0 Td [(34 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(63 114 )]TJ/F15 10.9091 Tf 8.485 0 Td [(52 3 5 andworktoreducedrow-echelonform,rstwith i =1, R 1 $ R 3 )456()222()222()223()456(! 2 4 114 )]TJ/F15 10.9091 Tf 8.485 0 Td [(52 )]TJ/F15 10.9091 Tf 8.485 0 Td [(34 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(63 217 )]TJ/F15 10.9091 Tf 8.485 0 Td [(72 3 5 3 R 1 + R 2 )348()222()223()222()348(! 2 4 114 )]TJ/F15 10.9091 Tf 8.485 0 Td [(52 077 )]TJ/F15 10.9091 Tf 8.485 0 Td [(219 217 )]TJ/F15 10.9091 Tf 8.485 0 Td [(72 3 5 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 R 1 + R 3 )324()222()223()222()222()324(! 2 4 1 14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(52 077 )]TJ/F15 10.9091 Tf 8.484 0 Td [(219 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 5 Now,with i =2, R 2 $ R 3 )456()222()222()223()456(! 2 4 1 14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(52 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 077 )]TJ/F15 10.9091 Tf 8.485 0 Td [(219 3 5 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 R 2 )275()222()276(! 2 4 1 14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(52 011 )]TJ/F15 10.9091 Tf 8.485 0 Td [(32 077 )]TJ/F15 10.9091 Tf 8.485 0 Td [(219 3 5 )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 R 2 + R 1 )324()222()223()222()222()324(! 2 4 1 03 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 011 )]TJ/F15 10.9091 Tf 8.485 0 Td [(32 077 )]TJ/F15 10.9091 Tf 8.485 0 Td [(219 3 5 )]TJ/F19 7.9701 Tf 6.587 0 Td [(7 R 2 + R 3 )324()222()222()223()222()324(! 2 4 1 03 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 0 1 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(32 0000 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 5 Andnally,with i =3, )]TJ/F20 5.9776 Tf 7.782 3.258 Td [(1 5 R 3 )470()222()222()470(! 2 4 1 03 )]TJ/F15 10.9091 Tf 8.484 0 Td [(20 0 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(32 00001 3 5 )]TJ/F19 7.9701 Tf 6.587 0 Td [(2 R 3 + R 2 )324()222()223()222()222()324(! 2 4 1 03 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 0 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(30 0000 1 3 5 Version2.02

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SubsectionRREF.READReadingQuestions39 Let'sanalyzetheequationsinthesystemrepresentedbythisaugmentedmatrix.Thethirdequationwill read0=1.Thisispatentlyfalse,allthetime.Nochoiceofvaluesforourvariableswillevermakeit true.We'redone.Sincewecannotevenmakethelastequationtrue,wehavenohopeofmakingallof theequationssimultaneouslytrue.Sothissystemhasnosolutions,anditssolutionsetistheemptyset, ; = fg DenitionES[683]. Noticethatwecouldhavereachedthisconclusionsooner.Afterperformingtherowoperation )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 R 2 + R 3 ,wecanseethatthethirdequationreads0= )]TJ/F15 10.9091 Tf 8.485 0 Td [(5,afalsestatement.Sincethesystemrepresentedby thismatrixhasnosolutions,noneofthesystemsrepresentedhasanysolutions.However,forthisexample, wehavechosentobringthematrixfullytoreducedrow-echelonformforthepractice. ThesethreeexamplesExampleSAB[36],ExampleSAA[37],ExampleSAE[38]illustratethefull rangeofpossibilitiesforasystemoflinearequations|nosolutions,onesolution,orinnitelymany solutions.Inthenextsectionwe'llexaminethesethreescenariosmoreclosely. DenitionRR Row-Reducing To row-reduce thematrix A meanstoapplyrowoperationsto A andarriveatarow-equivalentmatrix B inreducedrow-echelonform. 4 Sotheterm row-reduce isusedasaverb.TheoremREMEF[30]tellsusthatthisprocesswillalways besuccessfulandTheoremRREFU[32]tellsusthattheresultwillbeunambiguous.Typically,theanalysis of A willproceedbyanalyzing B andapplyingtheoremswhosehypothesesincludetherow-equivalenceof A and B Aftersomepracticebyhand,youwillwanttouseyourfavoritecomputingdevicetodothecomputations requiredtobringamatrixtoreducedrow-echelonformExerciseRREF.C30[42].See:Computation RR.MMA[667]ComputationRR.TI86[672]ComputationRR.TI83[673]ComputationRR.SAGE [675]. SubsectionREAD ReadingQuestions 1.Isthematrixbelowinreducedrow-echelonform?Whyorwhynot? 2 4 15068 00120 00001 3 5 2.Userowoperationstoconvertthematrixbelowtoreducedrow-echelonformandreportthenal matrix. 2 4 218 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(254 3 5 3.Findallthesolutionstothesystembelowbyusinganaugmentedmatrixandrowoperations.Report yournalmatrixinreducedrow-echelonformandthesetofsolutions. 2 x 1 +3 x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 3 =0 x 1 +2 x 2 + x 3 =3 x 1 +3 x 2 +3 x 3 =7 Version2.02

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SubsectionRREF.EXCExercises40 SubsectionEXC Exercises C05 Eacharchetypebelowisasystemofequations.Formtheaugmentedmatrixofthesystemof equations,convertthematrixtoreducedrow-echelonformbyusingequationoperationsandthendescribe thesolutionsetoftheoriginalsystemofequations. ArchetypeA[702] ArchetypeB[707] ArchetypeC[712] ArchetypeD[716] ArchetypeE[720] ArchetypeF[724] ArchetypeG[729] ArchetypeH[733] ArchetypeI[737] ArchetypeJ[741] ContributedbyRobertBeezer ForproblemsC10{C19,ndallsolutionstothesystemoflinearequations.Useyourfavoritecomputing devicetorow-reducetheaugmentedmatricesforthesystems,andwritethesolutionsasaset,usingcorrect setnotation. C10 2 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 2 + x 3 +7 x 4 =14 2 x 1 +8 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 3 +5 x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 x 1 +3 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 =4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 x 1 +2 x 2 +3 x 3 +4 x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(19 ContributedbyRobertBeezerSolution[44] C11 3 x 1 +4 x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 3 +2 x 4 =6 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 +3 x 3 + x 4 =2 10 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(10 x 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 4 =1 ContributedbyRobertBeezerSolution[44] C12 2 x 1 +4 x 2 +5 x 3 +7 x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(26 x 1 +2 x 2 + x 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 2 + x 3 +11 x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 ContributedbyRobertBeezerSolution[44] C13 x 1 +2 x 2 +8 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 Version2.02

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SubsectionRREF.EXCExercises41 3 x 1 +2 x 2 +12 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 4 =6 )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 1 + x 2 + x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 ContributedbyRobertBeezerSolution[45] C14 2 x 1 + x 2 +7 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 4 =4 3 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 +11 x 4 =13 x 1 + x 2 +5 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 4 =1 ContributedbyRobertBeezerSolution[45] C15 2 x 1 +3 x 2 )]TJ/F21 10.9091 Tf 10.91 0 Td [(x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(16 x 1 +2 x 2 + x 3 =0 )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 1 +2 x 2 +3 x 3 +4 x 4 =8 ContributedbyRobertBeezerSolution[45] C16 2 x 1 +3 x 2 +19 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 4 =2 x 1 +2 x 2 +12 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 4 =1 )]TJ/F21 10.9091 Tf 8.484 0 Td [(x 1 +2 x 2 +8 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 4 =1 ContributedbyRobertBeezerSolution[46] C17 )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 1 +5 x 2 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 1 +5 x 2 +5 x 3 +2 x 4 =9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 2 +3 x 3 + x 4 =3 7 x 1 +6 x 2 +5 x 3 + x 4 =30 ContributedbyRobertBeezerSolution[46] C18 x 1 +2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 4 =32 x 1 +3 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 5 =33 x 1 +2 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 4 +3 x 5 =22 ContributedbyRobertBeezerSolution[46] Version2.02

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SubsectionRREF.EXCExercises42 C19 2 x 1 + x 2 =6 )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 x 1 +4 x 2 =4 3 x 1 +5 x 2 =2 ContributedbyRobertBeezerSolution[47] ForproblemsC30{C33,row-reducethematrixwithouttheaidofacalculator,indicatingtherow operationsyouareusingateachstepusingthenotationofDenitionRO[28]. C30 2 4 21510 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2612 3 5 ContributedbyRobertBeezerSolution[47] C31 2 4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 3 5 ContributedbyRobertBeezerSolution[47] C32 2 4 111 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 321 3 5 ContributedbyRobertBeezerSolution[48] C33 2 4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 24 )]TJ/F15 10.9091 Tf 8.485 0 Td [(14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(235 3 5 ContributedbyRobertBeezerSolution[48] M40 Considerthetwo3 4matricesbelow B = 2 4 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(22 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(58 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 5 C = 2 4 1212 1140 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(41 3 5 aRow-reduceeachmatrixanddeterminethatthereducedrow-echelonformsof B and C are identical.Fromthisarguethat B and C arerow-equivalent. bIntheproofofTheoremRREFU[32],webeginbyarguingthatentriesofrow-equivalentmatrices arerelatedbywayofcertainscalarsandsums.Inthisexample,wewouldwritethatentriesof B fromrow i thatareincolumn j arelinearlyrelatedtotheentriesof C incolumn j fromallthreerows [ B ] ij = i 1 [ C ] 1 j + i 2 [ C ] 2 j + i 3 [ C ] 3 j 1 j 4 Version2.02

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SubsectionRREF.EXCExercises43 Foreach1 i 3ndthecorrespondingthreescalarsinthisrelationship.Soyouranswerwillbenine scalars,determinedthreeatatime. ContributedbyRobertBeezerSolution[48] M50 Aparkinglothas66vehiclescars,trucks,motorcyclesandbicyclesinit.Therearefourtimes asmanycarsastrucks.Thetotalnumberoftirespercarortruck,2permotorcycleorbicycleis252. Howmanycarsarethere?Howmanybicycles? ContributedbyRobertBeezerSolution[49] T10 ProvethateachofthethreerowoperationsDenitionRO[28]isreversible.Moreprecisely,if thematrix B isobtainedfrom A byapplicationofasinglerowoperation,showthatthereisasinglerow operationthatwilltransform B backinto A ContributedbyRobertBeezerSolution[49] T11 Supposethat A B and C are m n matrices.Usethedenitionofrow-equivalenceDenition REM[28]toprovethefollowingthreefacts. 1. A isrow-equivalentto A 2.If A isrow-equivalentto B ,then B isrow-equivalentto A 3.If A isrow-equivalentto B ,and B isrow-equivalentto C ,then A isrow-equivalentto C Arelationshipthatsatisesthesethreepropertiesisknownasan equivalencerelation ,animportant ideainthestudyofvariousalgebras.Thisisaformalwayofsayingthatarelationshipbehaveslike equality,withoutrequiringtherelationshiptobeasstrictasequalityitself.We'llseeitagaininTheorem SER[433]. ContributedbyRobertBeezer T12 Supposethat B isan m n matrixinreducedrow-echelonform.Buildanew,likelysmaller, k ` matrix C asfollows.Keepanycollectionof k adjacentrows, k m .Fromtheserows,keepcolumns1 through ` ` n .Provethat C isinreducedrow-echelonform. ContributedbyRobertBeezer T13 GeneralizeExerciseRREF.T12[43]byjustkeepingany k rows,andnotrequiringtherowstobe adjacent.Provethatanysuchmatrix C isinreducedrow-echelonform. ContributedbyRobertBeezer Version2.02

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SubsectionRREF.SOLSolutions44 SubsectionSOL Solutions C10 ContributedbyRobertBeezerStatement[40] Theaugmentedmatrixrow-reducesto 2 6 6 6 4 1 0001 0 1 00 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 00 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 000 1 1 3 7 7 7 5 andweseefromthelocationsoftheleading1'sthatthesystemisconsistentTheoremRCLS[53]and that n )]TJ/F21 10.9091 Tf 11.358 0 Td [(r =4 )]TJ/F15 10.9091 Tf 11.358 0 Td [(4=0andsothesystemhasnofreevariablesTheoremCSRN[54]andhencehasa uniquesolution.Thissolutionis S = 8 > > < > > : 2 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 1 3 7 7 5 9 > > = > > ; C11 ContributedbyRobertBeezerStatement[40] Theaugmentedmatrixrow-reducesto 2 4 1 014 = 50 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 = 100 0000 1 3 5 andaleading1inthelastcolumntellsusthatthesystemisinconsistentTheoremRCLS[53].Sothe solutionsetis ; = fg C12 ContributedbyRobertBeezerStatement[40] Theaugmentedmatrixrow-reducesto 2 4 1 20 )]TJ/F15 10.9091 Tf 8.485 0 Td [(42 00 1 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 00000 3 5 TheoremRCLS[53]andTheoremCSRN[54]tellsusthesystemisconsistentandthesolutionsetcan bedescribedwith n )]TJ/F21 10.9091 Tf 10.762 0 Td [(r =4 )]TJ/F15 10.9091 Tf 10.762 0 Td [(2=2freevariables,namely x 2 and x 4 .Solvingforthedependentvariables D = f x 1 ;x 3 g therstandsecondequationsrepresentedintherow-reducedmatrixyields, x 1 =2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 +4 x 4 x 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 4 Asaset,wewritethisas 8 > > < > > : 2 6 6 4 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 +4 x 4 x 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 10.91 0 Td [(3 x 4 x 4 3 7 7 5 j x 2 ;x 4 2 C 9 > > = > > ; Version2.02

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SubsectionRREF.SOLSolutions45 C13 ContributedbyRobertBeezerStatement[40] Theaugmentedmatrixofthesystemofequationsis 2 4 128 )]TJ/F15 10.9091 Tf 8.484 0 Td [(7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3212 )]TJ/F15 10.9091 Tf 8.484 0 Td [(56 )]TJ/F15 10.9091 Tf 8.484 0 Td [(111 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 )]TJ/F15 10.9091 Tf 8.484 0 Td [(10 3 5 whichrow-reducesto 2 4 1 0210 0 1 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(40 0000 1 3 5 WithaleadingoneinthelastcolumnTheoremRCLS[53]tellsusthesystemofequationsisinconsistent, sothesolutionsetistheemptyset, ; C14 ContributedbyRobertBeezerStatement[41] Theaugmentedmatrixofthesystemofequationsis 2 4 217 )]TJ/F15 10.9091 Tf 8.485 0 Td [(24 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(201113 115 )]TJ/F15 10.9091 Tf 8.484 0 Td [(31 3 5 whichrow-reducesto 2 4 1 0213 0 1 3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 00000 3 5 Then D = f 1 ; 2 g and F = f 3 ; 4 ; 5 g ,sothesystemisconsistent 62 D andcanbedescribedbythetwofree variables x 3 and x 4 .Rearrangingtheequationsrepresentedbythetwononzerorowstogainexpressions forthedependentvariables x 1 and x 2 ,yieldsthesolutionset, S = 8 > > < > > : 2 6 6 4 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 +4 x 4 x 3 x 4 3 7 7 5 j x 3 ;x 4 2 C 9 > > = > > ; C15 ContributedbyRobertBeezerStatement[41] Theaugmentedmatrixofthesystemofequationsis 2 4 23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 )]TJ/F15 10.9091 Tf 8.484 0 Td [(16 12100 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12348 3 5 whichrow-reducesto 2 4 1 0023 0 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 00 1 47 3 5 Then D = f 1 ; 2 ; 3 g and F = f 4 ; 5 g ,sothesystemisconsistent 62 D andcanbedescribedbytheone freevariable x 4 .Rearrangingtheequationsrepresentedbythethreenonzerorowstogainexpressionsfor thedependentvariables x 1 x 2 and x 3 ,yieldsthesolutionset, S = 8 > > < > > : 2 6 6 4 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5+3 x 4 7 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 4 x 4 3 7 7 5 j x 4 2 C 9 > > = > > ; Version2.02

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SubsectionRREF.SOLSolutions46 C16 ContributedbyRobertBeezerStatement[41] Theaugmentedmatrixofthesystemofequationsis 2 4 2319 )]TJ/F15 10.9091 Tf 8.485 0 Td [(42 1212 )]TJ/F15 10.9091 Tf 8.485 0 Td [(31 )]TJ/F15 10.9091 Tf 8.485 0 Td [(128 )]TJ/F15 10.9091 Tf 8.485 0 Td [(51 3 5 whichrow-reducesto 2 4 1 0210 0 1 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 0000 1 3 5 WithaleadingoneinthelastcolumnTheoremRCLS[53]tellsusthesystemofequationsisinconsistent, sothesolutionsetistheemptyset, ; = fg C17 ContributedbyRobertBeezerStatement[41] Werow-reducetheaugmentedmatrixofthesystemofequations, 2 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1500 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.484 0 Td [(25529 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1313 765130 3 7 7 5 RREF )443()223()222()443(! 2 6 6 6 4 1 0003 0 1 00 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 00 1 02 000 1 5 3 7 7 7 5 Thereducedrow-echelonformofthematrixistheaugmentedmatrixofthesystem x 1 =3, x 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1, x 3 =2, x 4 =5,whichhasauniquesolution.Asasetofcolumnvectors,thesolutionsetis S = 8 > > < > > : 2 6 6 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 5 3 7 7 5 9 > > = > > ; C18 ContributedbyRobertBeezerStatement[41] Werow-reducetheaugmentedmatrixofthesystemofequations, 2 4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1032 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(70 )]TJ/F15 10.9091 Tf 8.485 0 Td [(133 102 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2322 3 5 RREF )443()223()222()443(! 2 4 1 02056 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(30 )]TJ/F15 10.9091 Tf 8.485 0 Td [(29 000 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 3 5 Withnoleading1inthenalcolumn,werecognizethesystemasconsistentTheoremRCLS[53].Since thesystemisconsistent,wecomputethenumberoffreevariablesas n )]TJ/F21 10.9091 Tf 11.413 0 Td [(r =5 )]TJ/F15 10.9091 Tf 11.413 0 Td [(3=2,andwesee thatcolumns3and5arenotpivotcolumns,so x 3 and x 5 arefreevariables.Weconverteachrowofthe reducedrow-echelonformofthematrixintoanequation,andsolveitforthelonedependentvariable,as inexpressioninthetwofreevariables. x 1 +2 x 3 +5 x 5 =6 x 1 =6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 5 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 5 =9 x 2 =9+3 x 3 +2 x 5 x 4 + x 5 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(8 x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 5 Theseexpressionsgiveusaconvenientwaytodescribethesolutionset, S S = 8 > > > > < > > > > : 2 6 6 6 6 4 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 5 9+3 x 3 +2 x 5 x 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 5 x 5 3 7 7 7 7 5 j x 3 ;x 5 2 C 9 > > > > = > > > > ; Version2.02

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SubsectionRREF.SOLSolutions47 C19 ContributedbyRobertBeezerStatement[42] Weformtheaugmentedmatrixofthesystem, 2 6 6 4 216 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 344 352 3 7 7 5 whichrow-reducesto 2 6 6 4 1 04 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 000 000 3 7 7 5 Withnoleading1inthenalcolumn,thissystemisconsistentTheoremRCLS[53].Thereare n =2 variablesinthesystemand r =2non-zerorowsintherow-reducedmatrix.ByTheoremFVCS[55],there are n )]TJ/F21 10.9091 Tf 11.31 0 Td [(r =2 )]TJ/F15 10.9091 Tf 11.31 0 Td [(2=0freevariablesandwethereforeknowthesolutionisunique.Formingthesystem ofequationsrepresentedbytherow-reducedmatrix,weseethat x 1 =4and x 2 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(2.Writtenassetof columnvectors, S = 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 C30 ContributedbyRobertBeezerStatement[42] 2 4 21510 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2612 3 5 R 1 $ R 2 )456()222()222()223()456(! 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 21510 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2612 3 5 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 R 1 + R 2 )324()222()223()222()222()324(! 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 07714 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2612 3 5 )]TJ/F19 7.9701 Tf 6.586 0 Td [(4 R 1 + R 3 )324()222()223()222()222()324(! 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 07714 0101020 3 5 1 7 R 2 )494()222()494(! 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0112 0101020 3 5 3 R 2 + R 1 )348()222()223()222()348(! 2 4 1024 0112 0101020 3 5 )]TJ/F19 7.9701 Tf 6.587 0 Td [(10 R 2 + R 3 )408()222()222()222()223()222()408(! 2 4 1 024 0 1 12 0000 3 5 C31 ContributedbyRobertBeezerStatement[42] 2 4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 3 5 3 R 1 + R 2 )348()222()223()222()348(! 2 4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 05 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 3 5 2 R 1 + R 3 )348()222()223()222()348(! 2 4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 05 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 05 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 3 5 1 5 R 2 )494()222()495(! 2 4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 05 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 3 5 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 R 2 + R 1 )324()222()223()222()222()324(! 2 4 102 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 05 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 3 5 )]TJ/F19 7.9701 Tf 6.586 0 Td [(5 R 2 + R 3 )324()222()223()222()222()324(! 2 4 1 02 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 000 3 5 Version2.02

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SubsectionRREF.SOLSolutions48 C32 ContributedbyRobertBeezerStatement[42] FollowingthealgorithmofTheoremREMEF[30],andworkingtocreatepivotcolumnsfromlefttoright, wehave 2 4 111 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 321 3 5 4 R 1 + R 2 )348()222()223()222()348(! 2 4 111 012 321 3 5 )]TJ/F19 7.9701 Tf 6.586 0 Td [(3 R 1 + R 3 )324()222()222()223()222()324(! 2 4 1 11 012 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 5 )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 R 2 + R 1 )324()222()223()222()222()324(! 2 4 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 012 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 5 1 R 2 + R 3 )348()222()223()222()348(! 2 4 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 1 2 000 3 5 C33 ContributedbyRobertBeezerStatement[42] FollowingthealgorithmofTheoremREMEF[30],andworkingtocreatepivotcolumnsfromlefttoright, wehave 2 4 12 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 24 )]TJ/F15 10.9091 Tf 8.484 0 Td [(14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(235 3 5 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 R 1 + R 2 )324()222()223()222()222()324(! 2 4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0016 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(235 3 5 1 R 1 + R 3 )348()222()223()222()348(! 2 4 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0016 0024 3 5 1 R 2 + R 1 )348()222()223()222()348(! 2 4 1 205 0016 0024 3 5 )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 R 2 + R 3 )324()222()223()222()222()324(! 2 4 1 205 00 1 6 000 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 3 5 )]TJ/F20 5.9776 Tf 7.782 3.258 Td [(1 8 R 3 )470()222()223()470(! 2 4 1 205 00 1 6 0001 3 5 )]TJ/F19 7.9701 Tf 6.587 0 Td [(6 R 3 + R 2 )324()222()223()222()222()324(! 2 4 1 205 00 1 0 0001 3 5 )]TJ/F19 7.9701 Tf 6.586 0 Td [(5 R 3 + R 1 )324()222()223()222()222()324(! 2 4 1 200 00 1 0 000 1 3 5 M40 ContributedbyRobertBeezerStatement[42] aLet R bethecommonreducedrow-echelonformof B and C .Asequenceofrowoperationsconverts B to R andasecondsequenceofrowoperationsconverts C to R .Ifwereverse"thesecondsequence's order,andreverseeachindividualrowoperationseeExerciseRREF.T10[43]thenwecanbeginwith B ,convertto R withtherstsequence,andthenconvertto C withthereversedsequence.Satisfying DenitionREM[28]wecansay B and C arerow-equivalentmatrices. bWewillworkthiscarefullyfortherstrowof B andjustgivethesolutionforthenexttworows. Forrow1of B take i =1andwehave [ B ] 1 j = 11 [ C ] 1 j + 12 [ C ] 2 j + 13 [ C ] 3 j 1 j 4 Ifwesubstitutethefourvaluesfor j wearriveatfourlinearequationsinthethreeunknowns 11 ; 12 ; 13 j =1[ B ] 11 = 11 [ C ] 11 + 12 [ C ] 21 + 13 [ C ] 31 1= 11 + 12 + 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 j =2[ B ] 12 = 11 [ C ] 12 + 12 [ C ] 22 + 13 [ C ] 32 3= 11 + 12 + 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 j =3[ B ] 13 = 11 [ C ] 13 + 12 [ C ] 23 + 13 [ C ] 33 )]TJ/F15 10.9091 Tf 44.751 0 Td [(2= 11 + 12 + 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 j =4[ B ] 14 = 11 [ C ] 14 + 12 [ C ] 24 + 13 [ C ] 34 2= 11 + 12 + 13 Weformtheaugmentedmatrixofthissystemandrow-reducetondthesolutions, 2 6 6 4 11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2012 3 7 7 5 RREF )443()223()222()443(! 2 6 6 4 1 002 0 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 00 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0000 3 7 7 5 Version2.02

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SubsectionRREF.SOLSolutions49 Sotheuniquesolutionis 11 =2, 12 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3, 13 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2.Entirelysimilarworkwillleadyouto 21 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 22 =1 23 =1 and 31 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 32 =8 33 =5 M50 ContributedbyRobertBeezerStatement[43] Let c;t;m;b denotethenumberofcars,trucks,motorcycles,andbicycles.Thenthestatementsfromthe problemyieldtheequations: c + t + m + b =66 c )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 t =0 4 c +4 t +2 m +2 b =252 Theaugmentedmatrixforthissystemis 2 4 111166 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4000 4422252 3 5 whichrow-reducesto 2 4 1 00048 0 1 0012 00 1 16 3 5 c =48istherstequationrepresentedintherow-reducedmatrixsothereare48cars. m + b =6isthe thirdequationrepresentedintherow-reducedmatrixsothereareanywherefrom0to6bicycles.Wecan alsosaythat b isafreevariable,butthecontextoftheproblemlimitsitto7integervaluessinceyou cannothaveanegativenumberofmotorcycles. T10 ContributedbyRobertBeezerStatement[43] Ifwecanreverseeachrowoperationindividually,thenwecanreverseasequenceofrowoperations.The operationsthatreverseeachoperationarelistedbelow,usingourshorthandnotation, R i $ R j R i $ R j R i ; 6 =0 1 R i R i + R j )]TJ/F21 10.9091 Tf 10.909 0 Td [(R i + R j Version2.02

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SectionTSSTypesofSolutionSets50 SectionTSS TypesofSolutionSets Wewillnowbemorecarefulaboutanalyzingthereducedrow-echelonformderivedfromtheaugmented matrixofasystemoflinearequations.Inparticular,wewillseehowtosystematicallyhandlethesituation whenwehaveinnitelymanysolutionstoasystem,andwewillprovethateverysystemoflinearequations haseitherzero,oneorinnitelymanysolutions.Withthesetools,wewillbeabletosolveanysystemby awell-describedmethod. SubsectionCS ConsistentSystems ThecomputerscientistDonaldKnuthsaid,Scienceiswhatweunderstandwellenoughtoexplaintoa computer.Artiseverythingelse."Inthissectionwe'llremovesolvingsystemsofequationsfromtherealm ofart,andintotherealmofscience.Webeginwithadenition. DenitionCS ConsistentSystem Asystemoflinearequationsis consistent ifithasatleastonesolution.Otherwise,thesystemiscalled inconsistent 4 Wewillwanttorstrecognizewhenasystemisinconsistentorconsistent,andinthecaseofconsistent systemswewillbeabletofurtherrenethetypesofsolutionspossible.Wewilldothisbyanalyzingthe reducedrow-echelonformofamatrix,usingthevalueof r ,andthesetsofcolumnindices, D and F ,rst denedbackinDenitionRREF[30]. Useofthenotationfortheelementsof D and F canbeabitconfusing,sincewehavesubscripted variablesthatareinturnequaltointegersusedtoindexthematrix.However,manyquestionsabout matricesandsystemsofequationscanbeansweredonceweknow r D and F .Thechoiceoftheletters D and F refertoourupcomingdenitionofdependentandfreevariablesDenitionIDV[52].Anexample willhelpusbegintogetcomfortablewiththisaspectofreducedrow-echelonform. ExampleRREFN Reducedrow-echelonformnotation Forthe5 9matrix B = 2 6 6 6 6 6 4 1 5002805 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 00 1 047020 000 1 3903 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 000000 1 42 000000000 3 7 7 7 7 7 5 inreducedrow-echelonformwehave r =4 d 1 =1 d 2 =3 d 3 =4 d 4 =7 f 1 =2 f 2 =5 f 3 =6 f 4 =8 f 5 =9 Noticethatthesets D = f d 1 ;d 2 ;d 3 ;d 4 g = f 1 ; 3 ; 4 ; 7 g F = f f 1 ;f 2 ;f 3 ;f 4 ;f 5 g = f 2 ; 5 ; 6 ; 8 ; 9 g Version2.02

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SubsectionTSS.CSConsistentSystems51 havenothingincommonandtogetheraccountforallofthecolumnsof B wesayitisa partition ofthe setofcolumnindices. Thenumber r isthesinglemostimportantpieceofinformationwecangetfromthereducedrowechelonformofamatrix.Itisdenedasthenumberofnonzerorows,butsinceeachnonzerorowhasa leading1,itisalsothenumberofleading1'spresent.Foreachleading1,wehaveapivotcolumn,so r is alsothenumberofpivotcolumns.Repeatingourselves, r isthenumberofnonzerorows,thenumberof leading1's and thenumberofpivotcolumns.Acrossdierentsituations,eachoftheseinterpretationsof themeaningof r willbeuseful. Beforeprovingsometheoremsaboutthepossibilitiesforsolutionsetstosystemsofequations,let's analyzeoneparticularsystemwithaninnitesolutionsetverycarefullyasanexample.We'llusethis techniquefrequently,andshortlywe'llreneitslightly. ArchetypesIandJarebothfairlylargefordoingcomputationsbyhandthoughnotimpossiblylarge. Theirpropertiesareverysimilar,sowewillfrequentlyanalyzethesituationinArchetypeI,andleaveyou thejoyofanalyzingArchetypeJyourself.SoworkthroughArchetypeIwiththetext,byhandand/or withacomputer,andthentackleArchetypeJyourselfandcheckyourresultswiththoselisted.Notice toothatthearchetypesdescribingsystemsofequationseachliststhevaluesof r D and F .Herewego... ExampleISSI Describinginnitesolutionsets,ArchetypeI ArchetypeI[737]isthesystemof m =4equationsin n =7variables. x 1 +4 x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 4 +7 x 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 x 7 =3 2 x 1 +8 x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 3 +3 x 4 +9 x 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(13 x 6 +7 x 7 =9 2 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 5 +12 x 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 x 7 =1 )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 2 +2 x 3 +4 x 4 +8 x 5 )]TJ/F15 10.9091 Tf 10.91 0 Td [(31 x 6 +37 x 7 =4 Thissystemhasa4 8augmentedmatrixthatisrow-equivalenttothefollowingmatrixcheckthis!,and whichisinreducedrow-echelonformtheexistenceofthismatrixisguaranteedbyTheoremREMEF[30] anditsuniquenessisguaranteedbyTheoremRREFU[32], 2 6 6 4 1 40021 )]TJ/F15 10.9091 Tf 8.484 0 Td [(34 00 1 01 )]TJ/F15 10.9091 Tf 8.484 0 Td [(352 000 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(661 00000000 3 7 7 5 Sowendthat r =3and D = f d 1 ;d 2 ;d 3 g = f 1 ; 3 ; 4 g F = f f 1 ;f 2 ;f 3 ;f 4 ;f 5 g = f 2 ; 5 ; 6 ; 7 ; 8 g Let i denoteoneofthe r =3non-zerorows,andthenweseethatwecansolvethecorrespondingequation representedbythisrowforthevariable x d i andwriteitasalinearfunctionofthevariables x f 1 ;x f 2 ;x f 3 ;x f 4 noticethat f 5 =8doesnotreferenceavariable.We'lldothisnow,butyoucanalreadyseehowthe subscriptsuponsubscriptstakessomegettingusedto. i =1 x d 1 = x 1 =4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 5 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 6 +3 x 7 i =2 x d 2 = x 3 =2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 5 +3 x 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 7 i =3 x d 3 = x 4 =1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 5 +6 x 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 7 Eachelementoftheset F = f f 1 ;f 2 ;f 3 ;f 4 ;f 5 g = f 2 ; 5 ; 6 ; 7 ; 8 g istheindexofavariable,exceptfor f 5 =8.Wereferto x f 1 = x 2 x f 2 = x 5 x f 3 = x 6 and x f 4 = x 7 asfree"orindependent"variablessince theyareallowedtoassumeanypossiblecombinationofvaluesthatwecanimagineandwecancontinueon Version2.02

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SubsectionTSS.CSConsistentSystems52 tobuildasolutiontothesystembysolvingindividualequationsforthevaluesoftheotherdependent" variables. Eachelementoftheset D = f d 1 ;d 2 ;d 3 g = f 1 ; 3 ; 4 g istheindexofavariable.Werefertothevariables x d 1 = x 1 x d 2 = x 3 and x d 3 = x 4 asdependent"variablessincethey depend onthe independent variables. Moreprecisely,foreachpossiblechoiceofvaluesfortheindependentvariablesweget exactlyone setof valuesforthedependentvariablesthatcombinetoformasolutionofthesystem. Toexpressthesolutionsasaset,wewrite 8 > > > > > > > > < > > > > > > > > : 2 6 6 6 6 6 6 6 6 4 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 5 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 6 +3 x 7 x 2 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 5 +3 x 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 7 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 5 +6 x 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 7 x 5 x 6 x 7 3 7 7 7 7 7 7 7 7 5 j x 2 ;x 5 ;x 6 ;x 7 2 C 9 > > > > > > > > = > > > > > > > > ; Theconditionthat x 2 ;x 5 ;x 6 ;x 7 2 C ishowwespecifythatthevariables x 2 ;x 5 ;x 6 ;x 7 arefree"to assumeanypossiblevalues. Thissystematicapproachtosolvingasystemofequationswillallowustocreateaprecisedescription ofthesolutionsetforanyconsistentsystemoncewehavefoundthereducedrow-echelonformofthe augmentedmatrix.Itwillworkjustaswellwhenthesetoffreevariablesisemptyandwegetjusta singlesolution.Andwecouldprogramacomputertodoit!NowhaveawhackatArchetypeJExercise TSS.T10[58],mimickingthediscussioninthisexample.We'llstillbeherewhenyougetback. Usingthereducedrow-echelonformoftheaugmentedmatrixofasystemofequationstodetermine thenatureofthesolutionsetofthesystemisaverykeyidea.Solet'slookatonemoreexamplelikethe lastone.Butrstadenition,andthentheexample.Wemixourmetaphorsabitwhenwecallvariables freeversusdependent.Maybeweshouldcalldependentvariablesenslaved"? DenitionIDV IndependentandDependentVariables Suppose A istheaugmentedmatrixofaconsistentsystemoflinearequationsand B isarow-equivalent matrixinreducedrow-echelonform.Suppose j istheindexofacolumnof B thatcontainstheleading1 forsomerowi.e.column j isapivotcolumn.Thenthevariable x j is dependent .Avariablethatisnot dependentiscalled independent or free 4 Ifyoustudiedthisdenitioncarefully,youmightwonderwhattodoifthesystemhas n variablesand column n +1isapivotcolumn?Wewillseeshortly,byTheoremRCLS[53],thatthisneverhappensfor aconsistentsystem. ExampleFDV Freeanddependentvariables Considerthesystemofveequationsinvevariables, x 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 3 + x 4 +11 x 5 =13 x 1 )]TJ/F21 10.9091 Tf 10.91 0 Td [(x 2 + x 3 + x 4 +5 x 5 =16 2 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 + x 4 +10 x 5 =21 2 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 3 +3 x 4 +20 x 5 =38 2 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 + x 3 + x 4 +8 x 5 =22 Version2.02

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SubsectionTSS.CSConsistentSystems53 whoseaugmentedmatrixrow-reducesto 2 6 6 6 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10036 00 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 000 1 49 000000 000000 3 7 7 7 7 7 5 Thereareleading1'sincolumns1,3and4,so D = f 1 ; 3 ; 4 g .Fromthisweknowthatthevariables x 1 x 3 and x 4 willbedependentvariables,andeachofthe r =3nonzerorowsoftherow-reducedmatrix willyieldanexpressionforoneofthesethreevariables.Theset F isalltheremainingcolumnindices, F = f 2 ; 5 ; 6 g .That6 2 F referstothecolumnoriginatingfromthevectorofconstants,buttheremaining indicesin F willcorrespondtofreevariables,so x 2 and x 5 theremainingvariablesareourfreevariables. Theresultingthreeequationsthatdescribeoursolutionsetarethen, x d 1 = x 1 x 1 =6+ x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 5 x d 2 = x 3 x 3 =1+2 x 5 x d 3 = x 4 x 4 =9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 5 Makesureyouunderstandwherethesethreeequationscamefrom,andnoticehowthelocationofthe leading1'sdeterminedthevariablesontheleft-handsideofeachequation.Wecancompactlydescribe thesolutionsetas, S = 8 > > > > < > > > > : 2 6 6 6 6 4 6+ x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 5 x 2 1+2 x 5 9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 5 x 5 3 7 7 7 7 5 j x 2 ;x 5 2 C 9 > > > > = > > > > ; Noticehowweexpressthefreedomfor x 2 and x 5 : x 2 ;x 5 2 C Setsareanimportantpartofalgebra,andwe'veseenafewalready.Beingcomfortablewithsetsis importantforunderstandingandwritingproofs.Ifyouhaven'talready,payavisitnowtoSectionSET [683]. Wecannowusethevaluesof m n r ,andtheindependentanddependentvariablestocategorizethe solutionsetsforlinearsystemsthroughasequenceoftheorems.Throughthefollowingsequenceofproofs, youwillwanttoconsultthreeprooftechniques.SeeTechniqueE[690].SeeTechniqueN[691].See TechniqueCP[691]. Firstwehaveanimportanttheoremthatexploresthedistinctionbetweenconsistentandinconsistent linearsystems. TheoremRCLS RecognizingConsistencyofaLinearSystem Suppose A istheaugmentedmatrixofasystemoflinearequationswith n variables.Supposealsothat B isarow-equivalentmatrixinreducedrow-echelonformwith r nonzerorows.Thenthesystemofequations isinconsistentifandonlyiftheleading1ofrow r islocatedincolumn n +1of B Proof Thersthalfoftheproofbeginswiththeassumptionthattheleading1ofrow r islocatedin column n +1of B .Thenrow r of B beginswith n consecutivezeros,nishingwiththeleading1.Thisisa representationoftheequation0=1,whichisfalse.Sincethisequationisfalseforanycollectionofvalues wemightchooseforthevariables,therearenosolutionsforthesystemofequations,anditisinconsistent. Forthesecondhalfoftheproof,wewishtoshowthatifweassumethesystemisinconsistent, thenthenalleading1islocatedinthelastcolumn.Butinsteadofprovingthisdirectly,we'llformthe logicallyequivalentstatementthatisthecontrapositive,andprovethatinsteadseeTechniqueCP[691]. Version2.02

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SubsectionTSS.CSConsistentSystems54 Turningtheimplicationaround,andnegatingeachportion,wearriveatthelogicallyequivalentstatement: Iftheleading1ofrow r isnotincolumn n +1,thenthesystemofequationsisconsistent. Iftheleading1forrow r islocatedsomewhereincolumns1through n ,then every precedingrow's leading1isalsolocatedincolumns1through n .Inotherwords,sincethelastleading1isnotinthe lastcolumn,noleading1foranyrowisinthelastcolumn,duetotheechelonlayoutoftheleading1's DenitionRREF[30].Wewillnowconstructasolutiontothesystembysettingeachdependentvariable totheentryofthenalcolumnfortherowwiththecorrespondingleading1,andsettingeachfreevariable tozero.Thatsentenceisprettyvague,solet'sbemoreprecise.Usingournotationforthesets D and F fromthereducedrow-echelonformNotationRREFA[30]: x d i =[ B ] i;n +1 ; 1 i rx f i =0 ; 1 i n )]TJ/F21 10.9091 Tf 10.909 0 Td [(r Thesevaluesforthevariablesmaketheequationsrepresentedbytherst r rowsof B alltrueconvince yourselfofthis.Rowsnumberedgreaterthan r ifanyareallzerorows,hencerepresenttheequation 0=0andarealsoalltrue.Wehavenowidentiedonesolutiontothesystemrepresentedby B ,andhence asolutiontothesystemrepresentedby A TheoremREMES[28].Sowecansaythesystemisconsistent DenitionCS[50]. Thebeautyofthistheorembeinganequivalenceisthatwecanunequivocallytesttoseeifasystem isconsistentorinconsistentbylookingatjustasingleentryofthereducedrow-echelonformmatrix.We couldprogramacomputertodoit! Noticethatforaconsistentsystemtherow-reducedaugmentedmatrixhas n +1 2 F ,sothelargest elementof F doesnotrefertoavariable.Also,foraninconsistentsystem, n +1 2 D ,anditthendoesnot makemuchsensetodiscusswhetherornotvariablesarefreeordependentsincethereisnosolution.Take alookbackatDenitionIDV[52]andseewhywedidnotneedtoconsiderthepossibilityofreferencing x n +1 asadependentvariable. WiththecharacterizationofTheoremRCLS[53],wecanexploretherelationshipsbetween r and n inlightoftheconsistencyofasystemofequations.First,asituationwherewecanquicklyconcludethe inconsistencyofasystem. TheoremISRN InconsistentSystems, r and n Suppose A istheaugmentedmatrixofasystemoflinearequationsin n variables.Supposealsothat B isa row-equivalentmatrixinreducedrow-echelonformwith r rowsthatarenotcompletelyzeros.If r = n +1, thenthesystemofequationsisinconsistent. Proof If r = n +1,then D = f 1 ; 2 ; 3 ;:::;n;n +1 g andeverycolumnof B containsaleading1andis apivotcolumn.Inparticular,theentryofcolumn n +1forrow r = n +1isaleading1.TheoremRCLS [53]thensaysthatthesystemisinconsistent. DonotconfuseTheoremISRN[54]withitsconverse!GocheckoutTechniqueCV[691]rightnow. Next,ifasystemisconsistent,wecandistinguishbetweenauniquesolutionandinnitelymany solutions,andfurthermore,werecognizethatthesearetheonlytwopossibilities. TheoremCSRN ConsistentSystems, r and n Suppose A istheaugmentedmatrixofa consistent systemoflinearequationswith n variables.Suppose alsothat B isarow-equivalentmatrixinreducedrow-echelonformwith r rowsthatarenotzerorows. Then r n .If r = n ,thenthesystemhasauniquesolution,andif r
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SubsectionTSS.FVFreeVariables55 When r = n ,wend n )]TJ/F21 10.9091 Tf 9.943 0 Td [(r =0freevariablesi.e. F = f n +1 g andanysolutionmustequaltheunique solutiongivenbytherst n entriesofcolumn n +1of B When r 0freevariables,correspondingtocolumnsof B withoutaleading1, exceptingthenalcolumn,whichalsodoesnotcontainaleading1byTheoremRCLS[53].Byvarying thevaluesofthefreevariablessuitably,wecandemonstrateinnitelymanysolutions. SubsectionFV FreeVariables Thenexttheoremsimplystatesaconclusionfromthenalparagraphofthepreviousproof,allowingus tostateexplicitlythenumberoffreevariablesforaconsistentsystem. TheoremFVCS FreeVariablesforConsistentSystems Suppose A istheaugmentedmatrixofa consistent systemoflinearequationswith n variables.Suppose alsothat B isarow-equivalentmatrixinreducedrow-echelonformwith r rowsthatarenotcompletely zeros.Thenthesolutionsetcanbedescribedwith n )]TJ/F21 10.9091 Tf 10.909 0 Td [(r freevariables. Proof SeetheproofofTheoremCSRN[54]. ExampleCFV Countingfreevariables Foreacharchetypethatisasystemofequations,thevaluesof n and r arelisted.Manyalsocontainafew samplesolutions.Wecanusethisinformationprotably,asillustratedbyfourexamples. 1.ArchetypeA[702]has n =3and r =2.Itcanbeseentobeconsistentbythesamplesolutionsgiven. Itssolutionsetthenhas n )]TJ/F21 10.9091 Tf 10.909 0 Td [(r =1freevariables,andthereforewillbeinnite. 2.ArchetypeB[707]has n =3and r =3.Itcanbeseentobeconsistentbythesinglesamplesolution given.Itssolutionsetcanthenbedescribedwith n )]TJ/F21 10.9091 Tf 11.274 0 Td [(r =0freevariables,andthereforewillhave justthesinglesolution. 3.ArchetypeH[733]has n =2and r =3.Inthiscase, r = n +1,soTheoremISRN[54]saysthe systemisinconsistent.WeshouldnottrytoapplyTheoremFVCS[55]tocountfreevariables,since thetheoremonlyappliestoconsistentsystems.Whatwouldhappenifyoudid? 4.ArchetypeE[720]has n =4and r =3.However,bylookingatthereducedrow-echelonformofthe augmentedmatrix,wendaleading1inrow3,column4.ByTheoremRCLS[53]werecognizethe systemasinconsistent.Whydoesn'tthisexamplecontradictTheoremISRN[54]? Wehaveaccomplishedalotsofar,butourmaingoalhasbeenthefollowingtheorem,whichisnow verysimpletoprove.Theproofissosimplethatweoughttocallitacorollary,buttheresultisimportant enoughthatitdeservestobecalledatheorem.SeeTechniqueLC[696].Noticethatthistheoremwas presagedrstbyExampleTTS[10]andfurtherforeshadowedbyotherexamples. TheoremPSSLS PossibleSolutionSetsforLinearSystems Asystemoflinearequationshasnosolutions,auniquesolutionorinnitelymanysolutions. Proof Byitsdenition,asystemiseitherinconsistentorconsistentDenitionCS[50].Therstcase describessystemswithnosolutions.Forconsistentsystems,wehavetheremainingtwopossibilitiesas Version2.02

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SubsectionTSS.FVFreeVariables56 guaranteedby,anddescribedin,TheoremCSRN[54]. Hereisadiagramthatconsolidatesseveralofourtheoremsfromthissection,andwhichisofpractical usewhenyouanalyzesystemsofequations. DiagramDTSLS.DecisionTreeforSolvingLinearSystems Wehaveonemoretheoremtoroundoutoursetoftoolsfordeterminingsolutionsetstosystemsoflinear equations. TheoremCMVEI Consistent,MoreVariablesthanEquations,Innitesolutions Supposeaconsistentsystemoflinearequationshas m equationsin n variables.If n>m ,thenthesystem hasinnitelymanysolutions. Proof Supposethattheaugmentedmatrixofthesystemofequationsisrow-equivalentto B ,amatrix inreducedrow-echelonformwith r nonzerorows.Because B has m rowsintotal,thenumberthatare nonzerorowsisless.Inotherwords, r m .Followthiswiththehypothesisthat n>m andwendthat thesystemhasasolutionsetdescribedbyatleastonefreevariablebecause n )]TJ/F21 10.9091 Tf 10.909 0 Td [(r n )]TJ/F21 10.9091 Tf 10.909 0 Td [(m> 0 : Aconsistentsystemwithfreevariableswillhaveaninnitenumberofsolutions,asgivenbyTheorem CSRN[54]. Noticethattousethistheoremweneedonlyknowthatthesystemisconsistent,togetherwiththe valuesof m and n .Wedonotnecessarilyhavetocomputearow-equivalentreducedrow-echelonform matrix,eventhoughwediscussedsuchamatrixintheproof.Thisisthesubstanceofthefollowing example. ExampleOSGMD Onesolutiongivesmany,ArchetypeD ArchetypeDisthesystemof m =3equationsin n =4variables, 2 x 1 + x 2 +7 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x 4 =8 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 x 1 +4 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 x 1 + x 2 +4 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 4 =4 andthesolution x 1 =0, x 2 =1, x 3 =2, x 4 =1canbecheckedeasilybysubstitution.Havingbeen handed thissolution,weknowthesystemisconsistent.This,togetherwith n>m ,allowsustoapplyTheorem CMVEI[56]andconcludethatthesystemhasinnitelymanysolutions. Thesetheoremsgiveustheproceduresandimplicationsthatallowustocompletelysolveanysystem oflinearequations.Themaincomputationaltoolisusingrowoperationstoconvertanaugmentedmatrix Version2.02

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SubsectionTSS.READReadingQuestions57 intoreducedrow-echelonform.Here'sabroadoutlineofhowwewouldinstructacomputertosolvea systemoflinearequations. 1.Representasystemoflinearequationsbyanaugmentedmatrixanarrayistheappropriatedata structureinmostcomputerlanguages. 2.Convertthematrixtoarow-equivalentmatrixinreducedrow-echelonformusingtheprocedurefrom theproofofTheoremREMEF[30]. 3.Determine r andlocatetheleading1ofrow r .Ifitisincolumn n +1,outputthestatementthatthe systemisinconsistentandhalt. 4.Withtheleading1ofrow r notincolumn n +1,therearetwopossibilities: a r = n andthesolutionisunique.Itcanbereadodirectlyfromtheentriesinrows1through n ofcolumn n +1. b r
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SubsectionTSS.EXCExercises58 SubsectionEXC Exercises C10 InthespiritofExampleISSI[51],describetheinnitesolutionsetforArchetypeJ[741]. ContributedbyRobertBeezer M45 ProvethatArchetypeJ[741]hasinnitelymanysolutions without row-reducingtheaugmented matrix. ContributedbyRobertBeezerSolution[60] ForExercisesM51{M57say asmuchaspossible abouteachsystem'ssolutionset.Besuretomake itclearwhichtheoremsyouareusingtoreachyourconclusions. M51 Aconsistentsystemof8equationsin6variables. ContributedbyRobertBeezerSolution[60] M52 Aconsistentsystemof6equationsin8variables. ContributedbyRobertBeezerSolution[60] M53 Asystemof5equationsin9variables. ContributedbyRobertBeezerSolution[60] M54 Asystemwith12equationsin35variables. ContributedbyRobertBeezerSolution[60] M56 Asystemwith6equationsin12variables. ContributedbyRobertBeezerSolution[60] M57 Asystemwith8equationsand6variables.Thereducedrow-echelonformoftheaugmentedmatrix ofthesystemhas7pivotcolumns. ContributedbyRobertBeezerSolution[60] M60 Withoutdoinganycomputations,andwithoutexamininganysolutions,sayasmuchaspossible abouttheformofthesolutionsetforeacharchetypethatisasystemofequations. ArchetypeA[702] ArchetypeB[707] ArchetypeC[712] ArchetypeD[716] ArchetypeE[720] ArchetypeF[724] ArchetypeG[729] ArchetypeH[733] ArchetypeI[737] ArchetypeJ[741] ContributedbyRobertBeezer T10 Aninconsistentsystemmayhave r>n .Ifwetryincorrectly!toapplyTheoremFVCS[55]to suchasystem,howmanyfreevariableswouldwediscover? ContributedbyRobertBeezerSolution[60] T40 Supposethatthecoecientmatrixofaconsistentsystemoflinearequationshastwocolumnsthat areidentical.Provethatthesystemhasinnitelymanysolutions. ContributedbyRobertBeezerSolution[60] T41 Considerthesystemoflinearequations LS A; b ,andsupposethateveryelementofthevectorof Version2.02

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SubsectionTSS.EXCExercises59 constants b isacommonmultipleofthecorrespondingelementofacertaincolumnof A .Moreprecisely, thereisacomplexnumber ,andacolumnindex j ,suchthat[ b ] i = [ A ] ij forall i .Provethatthesystem isconsistent. ContributedbyRobertBeezerSolution[60] Version2.02

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SubsectionTSS.SOLSolutions60 SubsectionSOL Solutions M45 ContributedbyRobertBeezerStatement[58] Demonstratethatthesystemisconsistentbyverifyinganyoneofthefoursamplesolutionsprovided.Then because n =9 > 6= m ,TheoremCMVEI[56]givesustheconclusionthatthesystemhasinnitelymany solutions. Noticethatweonlyknowthesystemwillhave atleast 9 )]TJ/F15 10.9091 Tf 11.327 0 Td [(6=3freevariables,butverywellcould havemore.Wedonotknowknowthat r =6,onlythat r 6. M51 ContributedbyRobertBeezerStatement[58] ConsistentmeansthereisatleastonesolutionDenitionCS[50].Itwillhaveeitherauniquesolution orinnitelymanysolutionsTheoremPSSLS[55]. M52 ContributedbyRobertBeezerStatement[58] With6rowsintheaugmentedmatrix,therow-reducedversionwillhave r 6.Sincethesystemis consistent,applyTheoremCSRN[54]toseethat n )]TJ/F21 10.9091 Tf 10.909 0 Td [(r 2impliesinnitelymanysolutions. M53 ContributedbyRobertBeezerStatement[58] Thesystemcouldbeinconsistent.Ifitisconsistent,thenbecauseithasmorevariablesthanequations TheoremCMVEI[56]impliesthattherewouldbeinnitelymanysolutions.So,ofallthepossibilitiesin TheoremPSSLS[55],onlythecaseofauniquesolutioncanberuledout. M54 ContributedbyRobertBeezerStatement[58] Thesystemcouldbeinconsistent.Ifitisconsistent,thenTheoremCMVEI[56]tellsusthesolutionset willbeinnite.Sowecanbecertainthatthereisnotauniquesolution. M56 ContributedbyRobertBeezerStatement[58] Thesystemcouldbeinconsistent.Ifitisconsistent,andsince12 > 6,thenTheoremCMVEI[56]says wewillhaveinnitelymanysolutions.Sotherearetwopossibilities.TheoremPSSLS[55]allowstostate equivalentlythatauniquesolutionisanimpossibility. M57 ContributedbyRobertBeezerStatement[58] 7pivotcolumnsimpliesthatthereare r =7nonzerorowssorow8isallzerosinthereducedrow-echelon form.Then n +1=6+1=7= r andTheoremISRN[54]allowstoconcludethatthesystemis inconsistent. T10 ContributedbyRobertBeezerStatement[58] TheoremFVCS[55]willindicateanegativenumberoffreevariables,butwecansayevenmore.If r>n thentheonlypossibilityisthat r = n +1,andthenwecompute n )]TJ/F21 10.9091 Tf 10.909 0 Td [(r = n )]TJ/F15 10.9091 Tf 10.909 0 Td [( n +1= )]TJ/F15 10.9091 Tf 8.484 0 Td [(1freevariables. T40 ContributedbyRobertBeezerStatement[58] Sincethesystemisconsistent,weknowthereiseitherauniquesolution,orinnitelymanysolutions TheoremPSSLS[55].IfweperformrowoperationsDenitionRO[28]ontheaugmentedmatrixofthe system,thetwoequalcolumnsofthecoecientmatrixwillsuerthesamefate,andremainequalinthe nalreducedrow-echelonform.SupposebothofthesecolumnsarepivotcolumnsDenitionRREF[30]. Thenthereissinglerowcontainingthetwoleading1'softhetwopivotcolumns,aviolationofreduced row-echelonformDenitionRREF[30].Soatleastoneofthesecolumnsisnotapivotcolumn,andthe columnindexindicatesafreevariableinthedescriptionofthesolutionsetDenitionIDV[52].Witha freevariable,wearriveataninnitesolutionsetTheoremFVCS[55]. T41 ContributedbyRobertBeezerStatement[58] Theconditionaboutthemultipleofthecolumnofconstantswillallowyoutoshowthatthefollowing Version2.02

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SubsectionTSS.SOLSolutions61 valuesformasolutionofthesystem LS A; b x 1 =0 x 2 =0 :::x j )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 =0 x j = x j +1 =0 :::x n )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 =0 x n =0 Withonesolutionofthesystemknown,wecansaythesystemisconsistentDenitionCS[50]. AmoreinvolvedproofcanbebuiltusingTheoremRCLS[53].Beginbyprovingthateachofthethree rowoperationsDenitionRO[28]willconverttheaugmentedmatrixofthesystemintoanothermatrix wherecolumn j is timestheentryofthesamerowinthelastcolumn.Inotherwords,thecolumn multipleproperty"ispreservedunderrowoperations.Theseproofswillgetsuccessivelymoreinvolvedas youworkthroughthethreeoperations. NowconstructaproofbycontradictionTechniqueCD[692],bysupposingthatthesystemisinconsistent.Thenthelastcolumnofthereducedrow-echelonformoftheaugmentedmatrixisapivotcolumn TheoremRCLS[53].Thencolumn j musthaveazerointhesamerowastheleading1ofthenal column.Butthecolumnmultipleproperty"impliesthatthereisan incolumn j inthesamerowasthe leading1.So =0.Byhypothesis,thenthevectorofconstantsisthezerovector.However,ifwebegan withanalcolumnofzeros,rowoperationswouldneverhavecreatedaleading1inthenalcolumn.This contradictsthenalcolumnbeingapivotcolumn,andthereforethesystemcannotbeinconsistent. Version2.02

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SectionHSEHomogeneousSystemsofEquations62 SectionHSE HomogeneousSystemsofEquations Inthissectionwespecializetosystemsoflinearequationswhereeveryequationhasazeroasitsconstant term.Alongtheway,wewillbegintoexpressmoreandmoreideasinthelanguageofmatricesandbegin amoveawayfromwritingoutwholesystemsofequations.Theideasinitiatedinthissectionwillcarry throughtheremainderofthecourse. SubsectionSHS SolutionsofHomogeneousSystems Asusual,webeginwithadenition. DenitionHS HomogeneousSystem Asystemoflinearequations, LS A; b is homogeneous ifthevectorofconstantsisthezerovector,in otherwords, b = 0 4 ExampleAHSAC ArchetypeCasahomogeneoussystem Foreacharchetypethatisasystemofequations,wehaveformulatedasimilar,yetdierent,homogeneous systemofequationsbyreplacingeachequation'sconstanttermwithazero.Towit,forArchetypeC[712], wecanconverttheoriginalsystemofequationsintothehomogeneoussystem, 2 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 2 + x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 4 =0 4 x 1 + x 2 +2 x 3 +9 x 4 =0 3 x 1 + x 2 + x 3 +8 x 4 =0 Canyouquicklyndasolutiontothissystemwithoutrow-reducingtheaugmentedmatrix? AsyoumighthavediscoveredbystudyingExampleAHSAC[62],settingeachvariabletozerowill always beasolutionofahomogeneoussystem.Thisisthesubstanceofthefollowingtheorem. TheoremHSC HomogeneousSystemsareConsistent Supposethatasystemoflinearequationsishomogeneous.Thenthesystemisconsistent. Proof Seteachvariableofthesystemtozero.Whensubstitutingthesevaluesintoeachequation,the left-handsideevaluatestozero,nomatterwhatthecoecientsare.Sinceahomogeneoussystemhaszero ontheright-handsideofeachequationastheconstantterm,eachequationistrue.Withonedemonstrated solution,wecancallthesystemconsistent. Sincethissolutionissoobvious,wenowdeneitasthetrivialsolution. DenitionTSHSE TrivialSolutiontoHomogeneousSystemsofEquations Supposeahomogeneoussystemoflinearequationshas n variables.Thesolution x 1 =0, x 2 =0,..., x n =0 i.e. x = 0 iscalledthe trivialsolution 4 Herearethreetypicalexamples,whichwewillreferencethroughoutthissection.Workthroughthe rowoperationsaswebringeachtoreducedrow-echelonform.Alsonoticewhatissimilarineachexample, andwhatdiers. Version2.02

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SubsectionHSE.SHSSolutionsofHomogeneousSystems63 ExampleHUSAB Homogeneous,uniquesolution,ArchetypeB ArchetypeBcanbeconvertedtothehomogeneoussystem, )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 x 1 +2 x 2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(14 x 3 =0 23 x 1 )]TJ/F15 10.9091 Tf 10.91 0 Td [(6 x 2 +33 x 3 =0 14 x 1 )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 x 2 +17 x 3 =0 whoseaugmentedmatrixrow-reducesto 2 4 1 000 0 1 00 00 1 0 3 5 ByTheoremHSC[62],thesystemisconsistent,andsothecomputation n )]TJ/F21 10.9091 Tf 11.429 0 Td [(r =3 )]TJ/F15 10.9091 Tf 11.429 0 Td [(3=0meansthe solutionsetcontainsjustasinglesolution.Then,thislonesolutionmustbethetrivialsolution. ExampleHISAA Homogeneous,innitesolutions,ArchetypeA ArchetypeA[702]canbeconvertedtothehomogeneoussystem, x 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 2 +2 x 3 =0 2 x 1 + x 2 + x 3 =0 x 1 + x 2 =0 whoseaugmentedmatrixrow-reducesto 2 4 1 010 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 0000 3 5 ByTheoremHSC[62],thesystemisconsistent,andsothecomputation n )]TJ/F21 10.9091 Tf 11.429 0 Td [(r =3 )]TJ/F15 10.9091 Tf 11.429 0 Td [(2=1meansthe solutionsetcontainsonefreevariablebyTheoremFVCS[55],andhencehasinnitelymanysolutions.We candescribethissolutionsetusingthefreevariable x 3 S = 8 < : 2 4 x 1 x 2 x 3 3 5 j x 1 = )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 3 ;x 2 = x 3 9 = ; = 8 < : 2 4 )]TJ/F21 10.9091 Tf 8.484 0 Td [(x 3 x 3 x 3 3 5 j x 3 2 C 9 = ; Geometrically,thesearepointsinthreedimensionsthatlieonalinethroughtheorigin. ExampleHISAD Homogeneous,innitesolutions,ArchetypeD ArchetypeD[716]andidentically,ArchetypeE[720]canbeconvertedtothehomogeneoussystem, 2 x 1 + x 2 +7 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x 4 =0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x 1 +4 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 4 =0 x 1 + x 2 +4 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 4 =0 whoseaugmentedmatrixrow-reducesto 2 4 1 03 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 0 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(30 00000 3 5 Version2.02

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SubsectionHSE.NSMNullSpaceofaMatrix64 ByTheoremHSC[62],thesystemisconsistent,andsothecomputation n )]TJ/F21 10.9091 Tf 11.429 0 Td [(r =4 )]TJ/F15 10.9091 Tf 11.429 0 Td [(2=2meansthe solutionsetcontainstwofreevariablesbyTheoremFVCS[55],andhencehasinnitelymanysolutions. Wecandescribethissolutionsetusingthefreevariables x 3 and x 4 S = 8 > > < > > : 2 6 6 4 x 1 x 2 x 3 x 4 3 7 7 5 j x 1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x 3 +2 x 4 ;x 2 = )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 3 +3 x 4 9 > > = > > ; = 8 > > < > > : 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x 3 +2 x 4 )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 3 +3 x 4 x 3 x 4 3 7 7 5 j x 3 ;x 4 2 C 9 > > = > > ; Afterworkingthroughtheseexamples,youmightperformthesamecomputationsfortheslightlylarger example,ArchetypeJ[741]. Noticethatwhenwedorowoperationsontheaugmentedmatrixofahomogeneoussystemoflinear equationsthelastcolumnofthematrixisallzeros.Anyoneofthethreeallowablerowoperationswill convertzerostozerosandthus,thenalcolumnofthematrixinreducedrow-echelonformwillalsobe allzeros.Sointhiscase,wemaybeaslikelytoreferenceonlythecoecientmatrixandpresumethatwe rememberthatthenalcolumnbeginswithzeros,andafteranynumberofrowoperationsisstillzero. ExampleHISAD[63]suggeststhefollowingtheorem. TheoremHMVEI Homogeneous,MoreVariablesthanEquations,Innitesolutions Supposethatahomogeneoussystemoflinearequationshas m equationsand n variableswith n>m Thenthesystemhasinnitelymanysolutions. Proof Weareassumingthesystemishomogeneous,soTheoremHSC[62]saysitisconsistent.Thenthe hypothesisthat n>m ,togetherwithTheoremCMVEI[56],givesinnitelymanysolutions. ExampleHUSAB[63]andExampleHISAA[63]areconcernedwithhomogeneoussystemswhere n = m andexposeafundamentaldistinctionbetweenthetwoexamples.Onehasauniquesolution,whilethe otherhasinnitelymany.Theseareexactlytheonlytwopossibilitiesforahomogeneoussystemand illustratethateachispossibleunlikethecasewhen n>m whereTheoremHMVEI[64]tellsusthatthere isonlyonepossibilityforahomogeneoussystem. SubsectionNSM NullSpaceofaMatrix ThesetofsolutionstoahomogeneoussystemwhichbyTheoremHSC[62]isneveremptyisofenough interesttowarrantitsownname.However,wedeneitasapropertyofthecoecientmatrix,notasa propertyofsomesystemofequations. DenitionNSM NullSpaceofaMatrix The nullspace ofamatrix A ,denoted N A ,isthesetofallthevectorsthataresolutionstothe homogeneoussystem LS A; 0 Version2.02

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SubsectionHSE.NSMNullSpaceofaMatrix65 ThisdenitioncontainsNotationNSM. 4 IntheArchetypesAppendixA[698]eachexamplethatisasystemofequationsalsohasacorrespondinghomogeneoussystemofequationslisted,andseveralsamplesolutionsaregiven.Thesesolutions willbeelementsofthenullspaceofthecoecientmatrix.We'lllookatoneexample. ExampleNSEAI NullspaceelementsofArchetypeI Thewrite-upforArchetypeI[737]listsseveralsolutionsofthecorrespondinghomogeneoussystem.Here aretwo,writtenassolutionvectors.Wecansaythattheyareinthenullspaceofthecoecientmatrix forthesystemofequationsinArchetypeI[737]. x = 2 6 6 6 6 6 6 6 6 4 3 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 0 0 1 3 7 7 7 7 7 7 7 7 5 y = 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 1 1 3 7 7 7 7 7 7 7 7 5 However,thevector z = 2 6 6 6 6 6 6 6 6 4 1 0 0 0 0 0 2 3 7 7 7 7 7 7 7 7 5 isnotinthenullspace,sinceitisnotasolutiontothehomogeneoussystem.Forexample,itfailstoeven maketherstequationtrue. Herearetwoprototypicalexamplesofthecomputationofthenullspaceofamatrix. ExampleCNS1 Computinganullspace,#1 Let'scomputethenullspaceof A = 2 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(17 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(8 10249 22 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(18 3 5 whichwewriteas N A .TranslatingDenitionNSM[64],wesimplydesiretosolvethehomogeneous system LS A; 0 .Sowerow-reducetheaugmentedmatrixtoobtain 2 4 1 02010 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3040 000 1 20 3 5 Thevariablesofthehomogeneoussystem x 3 and x 5 arefreesincecolumns1,2and4arepivotcolumns, sowearrangetheequationsrepresentedbythematrixinreducedrow-echelonformto x 1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 5 x 2 =3 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 5 x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 5 Version2.02

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SubsectionHSE.READReadingQuestions66 Sowecanwritetheinnitesolutionsetassetsusingcolumnvectors, N A = 8 > > > > < > > > > : 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 5 3 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 5 x 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 5 x 5 3 7 7 7 7 5 j x 3 ;x 5 2 C 9 > > > > = > > > > ; ExampleCNS2 Computinganullspace,#2 Let'scomputethenullspaceof C = 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(461 )]TJ/F15 10.9091 Tf 8.485 0 Td [(141 567 471 3 7 7 5 whichwewriteas N C .TranslatingDenitionNSM[64],wesimplydesiretosolvethehomogeneous system LS C; 0 .Sowerow-reducetheaugmentedmatrixtoobtain 2 6 6 4 1 000 0 1 00 00 1 0 0000 3 7 7 5 Therearenofreevariablesinthehomogeneoussystemrepresentedbytherow-reducedmatrix,sothereis onlythetrivialsolution,thezerovector, 0 .Sowecanwritethetrivialsolutionsetas N C = f 0 g = 8 < : 2 4 0 0 0 3 5 9 = ; SubsectionREAD ReadingQuestions 1.Whatis always trueofthesolutionsetforahomogeneoussystemofequations? 2.Supposeahomogeneoussystemofequationshas13variablesand8equations.Howmanysolutions willithave?Why? 3.Describeinwordsnotsymbolsthenullspaceofamatrix. Version2.02

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SubsectionHSE.EXCExercises67 SubsectionEXC Exercises C10 EachArchetypeAppendixA[698]thatisasystemofequationshasacorrespondinghomogeneous systemwiththesamecoecientmatrix.Computethesetofsolutionsforeach.Noticethatthesesolution setsarethenullspacesofthecoecientmatrices. ArchetypeA[702] ArchetypeB[707] ArchetypeC[712] ArchetypeD[716]/ArchetypeE[720] ArchetypeF[724] ArchetypeG[729]/ArchetypeH[733] ArchetypeI[737] andArchetypeJ[741] ContributedbyRobertBeezer C20 ArchetypeK[746]andArchetypeL[750]aresimply5 5matricesi.e.theyarenotsystemsof equations.Computethenullspaceofeachmatrix. ContributedbyRobertBeezer C30 Computethenullspaceofthematrix A N A A = 2 6 6 4 24138 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 240 )]TJ/F15 10.9091 Tf 8.485 0 Td [(34 24 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(74 3 7 7 5 ContributedbyRobertBeezerSolution[69] C31 Findthenullspaceofthematrix B N B B = 2 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(64 )]TJ/F15 10.9091 Tf 8.485 0 Td [(366 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(110 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(32 )]TJ/F15 10.9091 Tf 8.485 0 Td [(183 3 5 ContributedbyRobertBeezerSolution[69] M45 Withoutdoinganycomputations,andwithoutexamininganysolutions,sayasmuchaspossible abouttheformofthesolutionsetforcorrespondinghomogeneoussystemofequationsofeacharchetype thatisasystemofequations. ArchetypeA[702] ArchetypeB[707] ArchetypeC[712] ArchetypeD[716]/ArchetypeE[720] ArchetypeF[724] ArchetypeG[729]/ArchetypeH[733] ArchetypeI[737] ArchetypeJ[741] ContributedbyRobertBeezer ForExercisesM50{M52say asmuchaspossible abouteachsystem'ssolutionset.Besuretomake itclearwhichtheoremsyouareusingtoreachyourconclusions. Version2.02

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SubsectionHSE.EXCExercises68 M50 Ahomogeneoussystemof8equationsin8variables. ContributedbyRobertBeezerSolution[69] M51 Ahomogeneoussystemof8equationsin9variables. ContributedbyRobertBeezerSolution[70] M52 Ahomogeneoussystemof8equationsin7variables. ContributedbyRobertBeezerSolution[70] T10 Proveordisprove:Asystemoflinearequationsishomogeneousifandonlyifthesystemhasthe zerovectorasasolution. ContributedbyMartinJacksonSolution[70] T20 Considerthehomogeneoussystemoflinearequations LS A; 0 ,andsupposethat u = 2 6 6 6 6 6 4 u 1 u 2 u 3 u n 3 7 7 7 7 7 5 isone solutiontothesystemofequations.Provethat v = 2 6 6 6 6 6 4 4 u 1 4 u 2 4 u 3 4 u n 3 7 7 7 7 7 5 isalsoasolutionto LS A; 0 ContributedbyRobertBeezerSolution[70] Version2.02

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SubsectionHSE.SOLSolutions69 SubsectionSOL Solutions C30 ContributedbyRobertBeezerStatement[67] DenitionNSM[64]tellsusthatthenullspaceof A isthesolutionsettothehomogeneoussystem LS A; 0 Theaugmentedmatrixofthissystemis 2 6 6 4 241380 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(110 240 )]TJ/F15 10.9091 Tf 8.485 0 Td [(340 24 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(740 3 7 7 5 Tosolvethesystem,werow-reducetheaugmentedmatrixandobtain, 2 6 6 4 1 20050 00 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(80 000 1 20 000000 3 7 7 5 Thismatrixrepresentsasystemwithequationshavingthreedependentvariables x 1 x 3 ,and x 4 andtwo independentvariables x 2 and x 5 .Theseequationsrearrangeto x 1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 5 x 3 =8 x 5 x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 5 Sowecanwritethesolutionsetwhichistherequestednullspaceas N A = 8 > > > > < > > > > : 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 5 x 2 8 x 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 5 x 5 3 7 7 7 7 5 j x 2 ;x 5 2 C 9 > > > > = > > > > ; C31 ContributedbyRobertBeezerStatement[67] Weformtheaugmentedmatrixofthehomogeneoussystem LS B; 0 androw-reducethematrix, 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(64 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3660 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(110 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(32 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1830 3 5 RREF )443()223()222()443(! 2 4 1 0210 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(630 00000 3 5 WeknewaheadoftimethatthissystemwouldbeconsistentTheoremHSC[62],butwecannowsee thereare n )]TJ/F21 10.9091 Tf 10.183 0 Td [(r =4 )]TJ/F15 10.9091 Tf 10.183 0 Td [(2=2freevariables,namely x 3 and x 4 TheoremFVCS[55].Basedonthisanalysis, wecanrearrangetheequationsassociatedwitheachnonzerorowofthereducedrow-echelonformintoan expressionforthelonedependentvariableasafunctionofthefreevariables.Wearriveatthesolutionset tothehomogeneoussystem,whichisthenullspaceofthematrixbyDenitionNSM[64], N B = 8 > > < > > : 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 4 6 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 4 x 3 x 4 3 7 7 5 j x 3 ;x 4 2 C 9 > > = > > ; M50 ContributedbyRobertBeezerStatement[68] Sincethesystemishomogeneous,weknowithasthetrivialsolutionTheoremHSC[62].Wecannotsay Version2.02

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SubsectionHSE.SOLSolutions70 anymorebasedontheinformationprovided,excepttosaythatthereiseitherauniquesolutionorinnitely manysolutionsTheoremPSSLS[55].SeeArchetypeA[702]andArchetypeB[707]tounderstandthe possibilities. M51 ContributedbyRobertBeezerStatement[68] Sincetherearemorevariablesthanequations,TheoremHMVEI[64]appliesandtellsusthatthesolution setisinnite.FromtheproofofTheoremHSC[62]weknowthatthezerovectorisonesolution. M52 ContributedbyRobertBeezerStatement[68] ByTheoremHSC[62],weknowthesystemisconsistentbecausethezerovectorisalwaysasolutionofa homogeneoussystem.Thereisnomorethatwecansay,sincebothauniquesolutionandinnitelymany solutionsarepossibilities. T10 ContributedbyRobertBeezerStatement[68] Thisisatruestatement.Aproofis: Supposewehaveahomogeneoussystem LS A; 0 .Thenbysubstitutingthescalarzeroforeach variable,wearriveattruestatementsforeachequation.Sothezerovectorisasolution.Thisisthe contentofTheoremHSC[62]. Supposenowthatwehaveagenerici.e.notnecessarilyhomogeneoussystemofequations, LS A; b thathasthezerovectorasasolution.Uponsubstitutingthissolutionintothesystem,we discoverthateachcomponentof b mustalsobezero.So b = 0 T20 ContributedbyRobertBeezerStatement[68] Supposethatasingleequationfromthissystemthe i -thonehastheform, a i 1 x 1 + a i 2 x 2 + a i 3 x 3 + + a in x n =0 Evaluatetheleft-handsideofthisequationwiththecomponentsoftheproposedsolutionvector v a i 1 u 1 + a i 2 u 2 + a i 3 u 3 + + a in u n =4 a i 1 u 1 +4 a i 2 u 2 +4 a i 3 u 3 + +4 a in u n Commutativity =4 a i 1 u 1 + a i 2 u 2 + a i 3 u 3 + + a in u n Distributivity =4 u solutionto LS A; 0 =0 So v makeseachequationtrue,andsoisasolutiontothesystem. Noticethatthisresultisnottrueifwechange LS A; 0 fromahomogeneoussystemtoanonhomogeneoussystem.Canyoucreateanexampleofanon-homogeneoussystemwithasolution u suchthat v isnotasolution? Version2.02

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SectionNMNonsingularMatrices71 SectionNM NonsingularMatrices Inthissectionwespecializeandconsidermatriceswithequalnumbersofrowsandcolumns,whichwhen consideredascoecientmatricesleadtosystemswithequalnumbersofequationsandvariables.Wewill seeinthesecondhalfofthecourseChapterD[370],ChapterE[396]ChapterLT[452],ChapterR[530] thatthesematricesareespeciallyimportant. SubsectionNM NonsingularMatrices Ourtheoremswillnowestablishconnectionsbetweensystemsofequationshomogeneousorotherwise, augmentedmatricesrepresentingthosesystems,coecientmatrices,constantvectors,thereducedrowechelonformofmatricesaugmentedandcoecientandsolutionsets.Beverycarefulinyourreading, writingandspeakingaboutsystemsofequations,matricesandsetsofvectors.Asystemofequationsis notamatrix,amatrixisnotasolutionset,andasolutionsetisnotasystemofequations.Nowwouldbe agreattimetoreviewthediscussionaboutspeakingandwritingmathematicsinTechniqueL[688]. DenitionSQM SquareMatrix Amatrixwith m rowsand n columnsis square if m = n .Inthiscase,wesaythematrixhas size n .To emphasizethesituationwhenamatrixisnotsquare,wewillcallit rectangular 4 Wecannowpresentoneofthecentraldenitionsoflinearalgebra. DenitionNM NonsingularMatrix Suppose A isasquarematrix.Supposefurtherthatthesolutionsettothehomogeneouslinearsystem ofequations LS A; 0 is f 0 g ,i.e.thesystemhas only thetrivialsolution.Thenwesaythat A isa nonsingular matrix.Otherwisewesay A isa singular matrix. 4 Wecaninvestigatewhetheranysquarematrixisnonsingularornot,nomatterifthematrixisderived somehowfromasystemofequationsorifitissimplyamatrix.Thedenitionsaysthattoperformthis investigationwemustconstructaveryspecicsystemofequationshomogeneous,withthematrixas thecoecientmatrixandlookatitssolutionset.Wewillhavetheoremsinthissectionthatconnect nonsingularmatriceswithsystemsofequations,creatingmoreopportunitiesforconfusion.Convince yourselfnowoftwoobservations,wecandecidenonsingularityforanysquarematrix,andthe determinationofnonsingularityinvolvesthesolutionsetforacertainhomogeneoussystemofequations. Noticethatitmakesnosensetocallasystemofequationsnonsingularthetermdoesnotapplytoa systemofequations,nordoesitmakeanysensetocalla5 7matrixsingularthematrixisnotsquare. ExampleS Asingularmatrix,ArchetypeA ExampleHISAA[63]showsthatthecoecientmatrixderivedfromArchetypeA[702],specicallythe 3 3matrix, A = 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 211 110 3 5 Version2.02

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SubsectionNM.NMNonsingularMatrices72 isasingularmatrixsincetherearenontrivialsolutionstothehomogeneoussystem LS A; 0 ExampleNM Anonsingularmatrix,ArchetypeB ExampleHUSAB[63]showsthatthecoecientmatrixderivedfromArchetypeB[707],specicallythe 3 3matrix, B = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 557 104 3 5 isanonsingularmatrixsincethehomogeneoussystem, LS B; 0 ,hasonlythetrivialsolution. NoticethatwewillnotdiscussExampleHISAD[63]asbeingasingularornonsingularcoecient matrixsincethematrixisnotsquare. Thenexttheoremcombineswithourmaincomputationaltechniquerow-reducingamatrixtomake iteasytorecognizeanonsingularmatrix.Butrstadenition. DenitionIM IdentityMatrix The m m identitymatrix I m ,isdenedby [ I m ] ij = 1 i = j 0 i 6 = j 1 i;j m ThisdenitioncontainsNotationIM. 4 ExampleIM Anidentitymatrix The4 4identitymatrixis I 4 = 2 6 6 4 1000 0100 0010 0001 3 7 7 5 : Noticethatanidentitymatrixissquare,andinreducedrow-echelonform.Soinparticular,ifwewere toarriveattheidentitymatrixwhilebringingamatrixtoreducedrow-echelonform,thenitwouldhave allofthediagonalentriescircledasleading1's. TheoremNMRRI NonsingularMatricesRowReducetotheIdentitymatrix Supposethat A isasquarematrixand B isarow-equivalentmatrixinreducedrow-echelonform.Then A isnonsingularifandonlyif B istheidentitymatrix. Proof Suppose B istheidentitymatrix.Whentheaugmentedmatrix[ A j 0 ]isrow-reduced,the resultis[ B j 0 ]=[ I n j 0 ].Thenumberofnonzerorowsisequaltothenumberofvariablesinthelinear systemofequations LS A; 0 ,so n = r andTheoremFVCS[55]gives n )]TJ/F21 10.9091 Tf 10.884 0 Td [(r =0freevariables.Thus,the homogeneoussystem LS A; 0 hasjustonesolution,whichmustbethetrivialsolution.Thisisexactly thedenitionofanonsingularmatrix. If A isnonsingular,thenthehomogeneoussystem LS A; 0 hasauniquesolution,andhasno freevariablesinthedescriptionofthesolutionset.ThehomogeneoussystemisconsistentTheoremHSC [62]soTheoremFVCS[55]appliesandtellsusthereare n )]TJ/F21 10.9091 Tf 11.251 0 Td [(r freevariables.Thus, n )]TJ/F21 10.9091 Tf 11.25 0 Td [(r =0,andso Version2.02

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SubsectionNM.NSNMNullSpaceofaNonsingularMatrix73 n = r .So B has n pivotcolumnsamongitstotalof n columns.Thisisenoughtoforce B tobethe n n identitymatrix I n Noticethatsincethistheoremisanequivalenceitwillalwaysallowustodetermineifamatrixis eithernonsingularorsingular.Herearetwoexamplesofthis,continuingourstudyofArchetypeAand ArchetypeB. ExampleSRR Singularmatrix,row-reduced ThecoecientmatrixforArchetypeA[702]is A = 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 211 110 3 5 whichwhenrow-reducedbecomestherow-equivalentmatrix B = 2 4 1 01 0 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 000 3 5 : Sincethismatrixisnotthe3 3identitymatrix,TheoremNMRRI[72]tellsusthat A isasingularmatrix. ExampleNSR Nonsingularmatrix,row-reduced ThecoecientmatrixforArchetypeB[707]is A = 2 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 557 104 3 5 whichwhenrow-reducedbecomestherow-equivalentmatrix B = 2 4 1 00 0 1 0 00 1 3 5 : Sincethismatrixisthe3 3identitymatrix,TheoremNMRRI[72]tellsusthat A isanonsingularmatrix. SubsectionNSNM NullSpaceofaNonsingularMatrix Nonsingularmatricesandtheirnullspacesareintimatelyrelated,asthenexttwoexamplesillustrate. ExampleNSS Nullspaceofasingularmatrix GiventhecoecientmatrixfromArchetypeA[702], A = 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 211 110 3 5 Version2.02

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SubsectionNM.NSNMNullSpaceofaNonsingularMatrix74 thenullspaceisthesetofsolutionstothehomogeneoussystemofequations LS A; 0 hasasolutionset andnullspaceconstructedinExampleHISAA[63]as N A = 8 < : 2 4 )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 3 x 3 x 3 3 5 j x 3 2 C 9 = ; ExampleNSNM Nullspaceofanonsingularmatrix GiventhecoecientmatrixfromArchetypeB[707], A = 2 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(7 )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 557 104 3 5 thehomogeneoussystem LS A; 0 hasasolutionsetconstructedinExampleHUSAB[63]thatcontains onlythetrivialsolution,sothenullspacehasonlyasingleelement, N A = 8 < : 2 4 0 0 0 3 5 9 = ; Thesetwoexamplesillustratethenexttheorem,whichisanotherequivalence. TheoremNMTNS NonsingularMatriceshaveTrivialNullSpaces Supposethat A isasquarematrix.Then A isnonsingularifandonlyifthenullspaceof A N A ,contains onlythezerovector,i.e. N A = f 0 g Proof Thenullspaceofasquare matrix A ,isequaltothesetofsolutionstothehomogeneous system LS A; 0 .A matrix isnonsingularifandonlyifthesetofsolutionstothehomogeneous system LS A; 0 hasonlyatrivialsolution.Thesetwoobservationsmaybechainedtogethertoconstructthetwoproofs necessaryforeachhalfofthistheorem. Thenexttheorempullsalotofbigideastogether.TheoremNMUS[74]tellsusthatwecanlearn muchaboutsolutionstoasystemoflinearequationswithasquarecoecientmatrixbyjustexamininga similarhomogeneoussystem. TheoremNMUS NonsingularMatricesandUniqueSolutions Supposethat A isasquarematrix. A isanonsingularmatrixifandonlyifthesystem LS A; b hasa uniquesolutionforeverychoiceoftheconstantvector b Proof Thehypothesisforthishalfoftheproofisthatthesystem LS A; b hasauniquesolution for every choiceoftheconstantvector b .Wewillmakeaveryspecicchoicefor b : b = 0 .Thenweknow thatthesystem LS A; 0 hasauniquesolution.Butthisispreciselythedenitionofwhatitmeansfor A tobenonsingularDenitionNM[71].Thatalmostseemstooeasy!Noticethatwehavenotusedthe fullpowerofourhypothesis,butthereisnothingthatsayswemustuseahypothesistoitsfullest. Weassumethat A isnonsingularofsize n n ,soweknowthereisasequenceofrowoperationsthat willconvert A intotheidentitymatrix I n TheoremNMRRI[72].Formtheaugmentedmatrix A 0 =[ A j b ] andapplythissamesequenceofrowoperationsto A 0 .Theresultwillbethematrix B 0 =[ I n j c ],whichis inreducedrow-echelonformwith r = n .Thentheaugmentedmatrix B 0 representstheextremelysimple Version2.02

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SubsectionNM.READReadingQuestions75 systemofequations x i =[ c ] i ,1 i n .Thevector c isclearlyasolution,sothesystemisconsistent DenitionCS[50].Withaconsistentsystem,weuseTheoremFVCS[55]tocountfreevariables.We ndthatthereare n )]TJ/F21 10.9091 Tf 10.826 0 Td [(r = n )]TJ/F21 10.9091 Tf 10.826 0 Td [(n =0freevariables,andsowethereforeknowthatthesolutionisunique. ThishalfoftheproofwassuggestedbyAsaScherer. Thistheoremhelpstoexplainpartofourinterestinnonsingularmatrices.Ifamatrixisnonsingular, thennomatterwhatvectorofconstantswepairitwith,usingthematrixasthecoecientmatrixwill always yieldalinearsystemofequationswithasolution,andthesolutionisunique.Todetermineifa matrixhasthispropertynon-singularityitisenoughtojustsolveonelinearsystem,thehomogeneous systemwiththematrixascoecientmatrixandthezerovectorasthevectorofconstantsoranyother vectorofconstants,seeExerciseMM.T10[207]. Formulatingthenegationofthesecondpartofthistheoremisagoodexercise.Asingularmatrixhas thepropertythatfor some valueofthevector b ,thesystem LS A; b doesnothaveauniquesolution whichmeansthatithasnosolutionorinnitelymanysolutions.Wewillbeabletosaymoreaboutthis caselaterseethediscussionfollowingTheoremPSPHS[105].Squarematricesthatarenonsingularhave alonglistofinterestingproperties,whichwewillstarttocataloginthefollowing,recurring,theorem.Of course,singularmatriceswillthenhavealloftheoppositeproperties.Thefollowingtheoremisalistof equivalences.Wewanttounderstandjustwhatisinvolvedwithunderstandingandprovingatheorem thatsaysseveralconditionsareequivalent.SohavealookatTechniqueME[693]beforestudyingtherst inthisseriesoftheorems. TheoremNME1 NonsingularMatrixEquivalences,Round1 Supposethat A isasquarematrix.Thefollowingareequivalent. 1. A isnonsingular. 2. A row-reducestotheidentitymatrix. 3.Thenullspaceof A containsonlythezerovector, N A = f 0 g 4.Thelinearsystem LS A; b hasauniquesolutionforeverypossiblechoiceof b Proof That A isnonsingularisequivalenttoeachofthesubsequentstatementsby,inturn,Theorem NMRRI[72],TheoremNMTNS[74]andTheoremNMUS[74].Sothestatementofthistheoremisjusta convenientwaytoorganizealltheseresults. Finally,youmayhavewonderedwhywerefertoamatrixas nonsingular whenitcreatessystems ofequationswith single solutionsTheoremNMUS[74]!I'vewonderedthesamething.We'llhavean opportunitytoaddressthiswhenwegettoTheoremSMZD[389].Canyouwaitthatlong? SubsectionREAD ReadingQuestions 1.Whatisthedenitionofanonsingularmatrix? 2.Whatistheeasiestwaytorecognizeanonsingularmatrix? 3.Supposewehaveasystemofequationsanditscoecientmatrixisnonsingular.Whatcanyousay aboutthesolutionsetforthissystem? Version2.02

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SubsectionNM.EXCExercises76 SubsectionEXC Exercises InExercisesC30{C33determineifthematrixisnonsingularorsingular.Givereasonsforyouranswer. C30 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3128 2034 127 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(120 3 7 7 5 ContributedbyRobertBeezerSolution[78] C31 2 6 6 4 2314 1110 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1235 1213 3 7 7 5 ContributedbyRobertBeezerSolution[78] C32 2 4 9324 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(613 413 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 3 5 ContributedbyRobertBeezerSolution[78] C33 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1203 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(24 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2043 )]TJ/F15 10.9091 Tf 8.485 0 Td [(31 )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 3 7 7 5 ContributedbyRobertBeezerSolution[78] C40 Eachofthearchetypesbelowisasystemofequationswithasquarecoecientmatrix,orisitself asquarematrix.Determineifthesematricesarenonsingular,orsingular.Commentonthenullspaceof eachmatrix. ArchetypeA[702] ArchetypeB[707] ArchetypeF[724] ArchetypeK[746] ArchetypeL[750] ContributedbyRobertBeezer C50 Findthenullspaceofthematrix E below. E = 2 6 6 4 21 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 22 )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 12 )]TJ/F15 10.9091 Tf 8.484 0 Td [(80 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1212 3 7 7 5 ContributedbyRobertBeezerSolution[78] Version2.02

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SubsectionNM.EXCExercises77 M30 Let A bethecoecientmatrixofthesystemofequationsbelow.Is A nonsingularorsingular? Explainwhatyoucouldinferaboutthesolutionsetforthesystembasedonlyonwhatyouhavelearned about A beingsingularornonsingular. )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 1 +5 x 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 1 +5 x 2 +5 x 3 +2 x 4 =9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 2 +3 x 3 + x 4 =3 7 x 1 +6 x 2 +5 x 3 + x 4 =30 ContributedbyRobertBeezerSolution[79] ForExercisesM51{M52say asmuchaspossible abouteachsystem'ssolutionset.Besuretomake itclearwhichtheoremsyouareusingtoreachyourconclusions. M51 6equationsin6variables,singularcoecientmatrix. ContributedbyRobertBeezerSolution[79] M52 Asystemwithanonsingularcoecientmatrix,nothomogeneous. ContributedbyRobertBeezerSolution[79] T10 Supposethat A isasingularmatrix,and B isamatrixinreducedrow-echelonformthatisrowequivalentto A .Provethatthelastrowof B isazerorow. ContributedbyRobertBeezerSolution[79] T30 Supposethat A isanonsingularmatrixand A isrow-equivalenttothematrix B .Provethat B is nonsingular. ContributedbyRobertBeezerSolution[79] T90 ProvideanalternativeforthesecondhalfoftheproofofTheoremNMUS[74],withoutappealing topropertiesofthereducedrow-echelonformofthecoecientmatrix.Inotherwords,provethatif A is nonsingular,then LS A; b hasauniquesolutionforeverychoiceoftheconstantvector b .Constructthis proofwithoutusingTheoremREMEF[30]orTheoremRREFU[32]. ContributedbyRobertBeezerSolution[79] Version2.02

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SubsectionNM.SOLSolutions78 SubsectionSOL Solutions C30 ContributedbyRobertBeezerStatement[76] Thematrixrow-reducesto 2 6 6 6 4 1 000 0 1 00 00 1 0 000 1 3 7 7 7 5 whichisthe4 4identitymatrix.ByTheoremNMRRI[72]theoriginalmatrixmustbenonsingular. C31 ContributedbyRobertBeezerStatement[76] Row-reducingthematrixyields, 2 6 6 4 1 00 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 1 03 00 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0000 3 7 7 5 Sincethisisnotthe4 4identitymatrix,TheoremNMRRI[72]tellsusthematrixissingular. C32 ContributedbyRobertBeezerStatement[76] Thematrixisnotsquare,soneithertermisapplicable.SeeDenitionNM[71],whichisstatedforjust squarematrices. C33 ContributedbyRobertBeezerStatement[76] TheoremNMRRI[72]tellsuswecananswerthisquestionbysimplyrow-reducingthematrix.Doingthis weobtain, 2 6 6 6 4 1 000 0 1 00 00 1 0 000 1 3 7 7 7 5 Sincethereducedrow-echelonformofthematrixisthe4 4identitymatrix I 4 ,weknowthat B is nonsingular. C50 ContributedbyRobertBeezerStatement[76] Weformtheaugmentedmatrixofthehomogeneoussystem LS E; 0 androw-reducethematrix, 2 6 6 4 21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(90 22 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(60 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(800 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12120 3 7 7 5 RREF )443()223()222()443(! 2 6 6 4 1 02 )]TJ/F15 10.9091 Tf 8.485 0 Td [(60 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(530 00000 00000 3 7 7 5 WeknewaheadoftimethatthissystemwouldbeconsistentTheoremHSC[62],butwecannowsee thereare n )]TJ/F21 10.9091 Tf 11.302 0 Td [(r =4 )]TJ/F15 10.9091 Tf 11.302 0 Td [(2=2freevariables,namely x 3 and x 4 since F = f 3 ; 4 ; 5 g TheoremFVCS[55]. Basedonthisanalysis,wecanrearrangetheequationsassociatedwitheachnonzerorowofthereduced row-echelonformintoanexpressionforthelonedependentvariableasafunctionofthefreevariables.We arriveatthesolutionsettothishomogeneoussystem,whichisthenullspaceofthematrixbyDenition NSM[64], N E = 8 > > < > > : 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 3 +6 x 4 5 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 4 x 3 x 4 3 7 7 5 j x 3 ;x 4 2 C 9 > > = > > ; Version2.02

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SubsectionNM.SOLSolutions79 M30 ContributedbyRobertBeezerStatement[77] Werow-reducethecoecientmatrixofthesystemofequations, 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1500 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2552 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(131 7651 3 7 7 5 RREF )443()223()222()443(! 2 6 6 6 4 1 000 0 1 00 00 1 0 000 1 3 7 7 7 5 Sincetherow-reducedversionofthecoecientmatrixisthe4 4identitymatrix, I 4 DenitionIM[72] byTheoremNMRRI[72],weknowthecoecientmatrixisnonsingular.AccordingtoTheoremNMUS [74]weknowthatthesystemisguaranteedtohaveauniquesolution,basedonlyontheextrainformation thatthecoecientmatrixisnonsingular. M51 ContributedbyRobertBeezerStatement[77] TheoremNMRRI[72]tellsusthatthecoecientmatrixwillnotrow-reducetotheidentitymatrix.So ifweweretorow-reducetheaugmentedmatrixofthissystemofequations,wewouldnotgetaunique solution.SobyTheoremPSSLS[55]theremainingpossibilitiesarenosolutions,orinnitelymany. M52 ContributedbyRobertBeezerStatement[77] AnysystemwithanonsingularcoecientmatrixwillhaveauniquesolutionbyTheoremNMUS[74].If thesystemisnothomogeneous,thesolutioncannotbethezerovectorExerciseHSE.T10[68]. T10 ContributedbyRobertBeezerStatement[77] Let n denotethesizeofthesquarematrix A .ByTheoremNMRRI[72]thehypothesisthat A issingular impliesthat B isnottheidentitymatrix I n .If B has n pivotcolumns,thenitwouldhavetobe I n ,so B musthavefewerthan n pivotcolumns.Butthenumberofnonzerorowsin B r isequaltothenumber ofpivotcolumnsaswell.Sothe n rowsof B havefewerthan n nonzerorows,and B mustcontainatleast onezerorow.ByDenitionRREF[30],thisrowmustbeatthebottomof B T30 ContributedbyRobertBeezerStatement[77] Since A and B arerow-equivalentmatrices,considerationofthethreerowoperationsDenitionRO[28] willshowthattheaugmentedmatrices,[ A j 0 ]and[ B j 0 ],arealsorow-equivalentmatrices.Thissays thatthetwohomogeneoussystems, LS A; 0 and LS B; 0 areequivalentsystems. LS A; 0 hasonly thezerovectorasasolutionDenitionNM[71],thus LS B; 0 hasonlythezerovectorasasolution. Finally,byDenitionNM[71],weseethat B isnonsingular. Formasimilartheoremreplacingnonsingular"bysingular"inboththehypothesisandtheconclusion.Provethisnewtheoremwithanapproachjustliketheoneabove,and/oremploytheresultabout nonsingularmatricesinaproofbycontradiction. T90 ContributedbyRobertBeezerStatement[77] Weassume A isnonsingular,andtrytosolvethesystem LS A; b withoutmakinganyassumptionsabout b .Todothiswewillbeginbyconstructinganewhomogeneouslinearsystemofequationsthatlooksvery muchliketheoriginal.Suppose A hassize n whymustitbesquare?andwritetheoriginalsystemas, a 11 x 1 + a 12 x 2 + a 13 x 3 + + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + a 23 x 3 + + a 2 n x n = b 2 a 31 x 1 + a 32 x 2 + a 33 x 3 + + a 3 n x n = b 3 a n 1 x 1 + a n 2 x 2 + a n 3 x 3 + + a nn x n = b n formthenew,homogeneoussystemin n equationswith n +1variables,byaddinganewvariable y ,whose coecientsarethenegativesoftheconstantterms, a 11 x 1 + a 12 x 2 + a 13 x 3 + + a 1 n x n )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 1 y =0 Version2.02

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SubsectionNM.SOLSolutions80 a 21 x 1 + a 22 x 2 + a 23 x 3 + + a 2 n x n )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 2 y =0 a 31 x 1 + a 32 x 2 + a 33 x 3 + + a 3 n x n )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 3 y =0 a n 1 x 1 + a n 2 x 2 + a n 3 x 3 + + a nn x n )]TJ/F21 10.9091 Tf 10.909 0 Td [(b n y =0 Sincethisisahomogeneoussystemwithmorevariablesthanequations m = n +1 >n ,TheoremHMVEI [64]saysthatthesystemhasinnitelymanysolutions.Wewillchooseoneofthesesolutions, any oneof thesesolutions,solongasitis not thetrivialsolution.Writethissolutionas x 1 = c 1 x 2 = c 2 x 3 = c 3 :::x n = c n y = c n +1 Weknowthatatleastonevalueofthe c i isnonzero,butwewillnowshowthatinparticular c n +1 6 =0. WedothisusingaproofbycontradictionTechniqueCD[692].Sosupposethe c i formasolutionas described,andinadditionthat c n +1 =0.Thenwecanwritethe i -thequationofsystem as, a i 1 c 1 + a i 2 c 2 + a i 3 c 3 + + a in c n )]TJ/F21 10.9091 Tf 10.909 0 Td [(b i =0 whichbecomes a i 1 c 1 + a i 2 c 2 + a i 3 c 3 + + a in c n =0 Sincethisistrueforeach i ,wehavethat x 1 = c 1 ;x 2 = c 2 ;x 3 = c 3 ;:::;x n = c n isasolutiontothe homogeneoussystem LS A; 0 formedwithanonsingularcoecientmatrix.Thismeansthattheonly possiblesolutionisthetrivialsolution,so c 1 =0 ;c 2 =0 ;c 3 =0 ;:::;c n =0.So,assumingsimplythat c n +1 =0,weconcludethat all ofthe c i arezero.Butthiscontradictsourchoiceofthe c i asnotbeingthe trivialsolutiontothesystem .So c n +1 6 =0. Wenowproposeandverifyasolutiontotheoriginalsystem .Set x 1 = c 1 c n +1 x 2 = c 2 c n +1 x 3 = c 3 c n +1 :::x n = c n c n +1 Noticehowitwasnecessarythatweknowthat c n +1 6 =0forthissteptosucceed.Now,evaluatethe i -th equationofsystem withthisproposedsolution,andrecognizeinthethirdlinethat c 1 through c n +1 appearasiftheyweresubstitutedintotheleft-handsideofthe i -thequationofsystem a i 1 c 1 c n +1 + a i 2 c 2 c n +1 + a i 3 c 3 c n +1 + + a in c n c n +1 = 1 c n +1 a i 1 c 1 + a i 2 c 2 + a i 3 c 3 + + a in c n = 1 c n +1 a i 1 c 1 + a i 2 c 2 + a i 3 c 3 + + a in c n )]TJ/F21 10.9091 Tf 10.909 0 Td [(b i c n +1 + b i = 1 c n +1 + b i = b i Sincethisequationistrueforevery i ,wehavefoundasolutiontosystem .Tonish,westillneedto establishthatthissolutionis unique Withonesolutioninhand,wewillentertainthepossibilityofasecondsolution.Soassumesystem hastwosolutions, x 1 = d 1 x 2 = d 2 x 3 = d 3 :::x n = d n Version2.02

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SubsectionNM.SOLSolutions81 x 1 = e 1 x 2 = e 2 x 3 = e 3 :::x n = e n Then, a i 1 d 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(e 1 + a i 2 d 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(e 2 + a i 3 d 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(e 3 + + a in d n )]TJ/F21 10.9091 Tf 10.909 0 Td [(e n = a i 1 d 1 + a i 2 d 2 + a i 3 d 3 + + a in d n )]TJ/F15 10.9091 Tf 10.909 0 Td [( a i 1 e 1 + a i 2 e 2 + a i 3 e 3 + + a in e n = b i )]TJ/F21 10.9091 Tf 10.909 0 Td [(b i =0 Thisisthe i -thequationofthehomogeneoussystem LS A; 0 evaluatedwith x j = d j )]TJ/F21 10.9091 Tf 11.278 0 Td [(e j ,1 j n Since A isnonsingular,wemustconcludethatthissolutionisthetrivialsolution,andso0= d j )]TJ/F21 10.9091 Tf 11.529 0 Td [(e j 1 j n .Thatis, d j = e j forall j andthetwosolutionsareidentical,meaninganysolutionto is unique. Noticethattheproposedsolution x i = c i c n +1 appearedinthisproofwithnomotivationwhatsoever. Thisisjustneinaproof.Aproofshould convince youthatatheoremis true .Itisyourjobto read the proofandbeconvincedofeveryassertion.QuestionslikeWheredidthatcomefrom?"orHowwouldI thinkofthat?"havenobearingonthe validity oftheproof. Version2.02

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AnnotatedAcronymsNM.SLESystemsofLinearEquations82 AnnotatedAcronymsSLE SystemsofLinearEquations Attheconclusionofeachchapteryouwillndasectionlikethis,reviewingselecteddenitionsand theorems.Therearemanyreasonsforwhyadenitionortheoremmightbeplacedhere.Itmight representakeyconcept,itmightbeusedfrequentlyforcomputations,providethecriticalstepinmany proofs,oritmaydeservespecialcomment. Theselistsarenotmeanttobeexhaustive,butshouldstillbeusefulaspartofreviewingeachchapter. Wewillmentionafewofthesethatyoumighteventuallyrecognizeonsightasbeingworthmemorization. Bythatwemeanthatyoucanassociatetheacronymwitharoughstatementofthetheorem|notthat theexactdetailsofthetheoremneedtobememorized.Anditiscertainlynotourintentthateverything ontheselistsisimportantenoughtomemorize. TheoremRCLS[53] Wewillrepeatedlyappealtothistheoremtodetermineifasystemoflinearequations,does,ordoesn't, haveasolution.Thisonewewillseeoftenenoughthatitisworthmemorizing. TheoremHMVEI[64] Thistheoremisthetheoreticalbasisofseveralofourmostimportanttheorems.Sokeepaneyeoutfor it,anditsdescendants,asyoustudyotherproofs.Forexample,TheoremHMVEI[64]iscriticaltothe proofofTheoremSSLD[341],TheoremSSLD[341]iscriticaltotheproofofTheoremG[355],Theorem G[355]iscriticaltotheproofsofthepairofsimilartheorems,TheoremILTD[486]andTheoremSLTD [502],whilenallyTheoremILTD[486]andTheoremSLTD[502]arecriticaltotheproofofanimportant result,TheoremIVSED[516].Thischainofimplicationsmightnotmakemuchsenseonarstreading, butcomebacklatertoseehowsomeveryimportanttheoremsbuildontheseeminglysimpleresultthatis TheoremHMVEI[64].Usingthend"featureinwhateversoftwareyouusetoreadtheelectronicversion ofthetextcanbeafunwaytoexploretheserelationships. TheoremNMRRI[72] Thistheoremgivesusoneofsimplestways,computationally,torecognizeifamatrixisnonsingular,or singular.Wewillseethisoneoften,incomputationalexercisesespecially. TheoremNMUS[74] NonsingularmatriceswillbeanimportanttopicgoingforwardwitnesstheNMExseriesoftheorems. Thisisourrstresultalongtheselines,ausefultheoremforotherproofs,andalsoillustratesamore generalconceptfromChapterLT[452]. Version2.02

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ChapterV Vectors Wehaveworkedextensivelyinthelastchapterwithmatrices,andsomewithvectors.Inthischapterwewill developthepropertiesofvectors,whilepreparingtostudyvectorspacesChapterVS[279].Initiallywe willdepartfromourstudyofsystemsoflinearequations,butinSectionLC[90]wewillforgeaconnection betweenlinearcombinationsandsystemsoflinearequationsinTheoremSLSLC[93].Thisconnectionwill allowustounderstandsystemsoflinearequationsatahigherlevel,whileconsequentlydiscussingthem lessfrequently. SectionVO VectorOperations Inthissectionwedenesomenewoperationsinvolvingvectors,andcollectsomebasicpropertiesofthese operations.Beginbyrecallingourdenitionofacolumnvectorasanorderedlistofcomplexnumbers, writtenverticallyDenitionCV[24].Thecollectionofallpossiblevectorsofaxedsizeisacommonly usedset,sowestartwithitsdenition. DenitionVSCV VectorSpaceofColumnVectors Thevectorspace C m isthesetofallcolumnvectorsDenitionCV[24]ofsize m withentriesfromthe setofcomplexnumbers, C ThisdenitioncontainsNotationVSCV. 4 Whenasetsimilartothisisdenedusingonlycolumnvectorswherealltheentriesarefromthereal numbers,itiswrittenas R m andisknownas Euclidean m -space Thetermvector"isusedinavarietyofdierentways.Wehavedeneditasanorderedlistwritten vertically.Itcouldsimplybeanorderedlistofnumbers,andwrittenas ; 3 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 6.Oritcouldbe interpretedasapointin m dimensions,suchas ; 4 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2representingapointinthreedimensionsrelative to x y and z axes.Withaninterpretationasapoint,wecanconstructanarrowfromtheorigintothe pointwhichisconsistentwiththenotionthatavectorhasdirectionandmagnitude. Alloftheseideascanbeshowntoberelatedandequivalent,sokeepthatinmindasyouconnectthe ideasofthiscoursewithideasfromotherdisciplines.Fornow,we'llstickwiththeideathatavectorisa justalistofnumbers,insomeparticularorder. 83

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SubsectionVO.VEASMVectorEquality,Addition,ScalarMultiplication84 SubsectionVEASM VectorEquality,Addition,ScalarMultiplication Westartourstudyofthissetbyrstdeningwhatitmeansfortwovectorstobethesame. DenitionCVE ColumnVectorEquality Supposethat u ; v 2 C m .Then u and v are equal ,written u = v if [ u ] i =[ v ] i 1 i m ThisdenitioncontainsNotationCVE. 4 Nowthismayseemlikeasillyorevenstupidthingtosaysocarefully.Ofcoursetwovectorsare equaliftheyareequalforeachcorrespondingentry!Well,thisisnotassillyasitappears.Wewillseea fewoccasionslaterwheretheobviousdenitionis not therightone.Andbesides,indoingmathematics weneedtobeverycarefulaboutmakingallthenecessarydenitionsandmakingthemunambiguous.And we'vedonethathere. Noticenowthatthesymbol`='isnowdoingtriple-duty.Weknowfromourearliereducationwhatit meansfortwonumbersrealorcomplextobeequal,andwetakethisforgranted.InDenitionSE[684] wedenedwhatitmeantfortwosetstobeequal.Nowwehavedenedwhatitmeansfortwovectors tobeequal,andthatdenitionbuildsonourdenitionforwhentwonumbersareequalwhenweusethe condition u i = v i forall1 i m .Sothinkcarefullyaboutyourobjectswhenyouseeanequalsignand thinkaboutjustwhichnotionofequalityyouhaveencountered.Thiswillbeespeciallyimportantwhen youareaskedtoconstructproofswhoseconclusionstatesthattwoobjectsareequal. OK,let'sdoanexampleofvectorequalitythatbeginstohintattheutilityofthisdenition. ExampleVESE Vectorequalityforasystemofequations ConsiderthesystemoflinearequationsinArchetypeB[707], )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 x 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(33 5 x 1 +5 x 2 +7 x 3 =24 x 1 +4 x 3 =5 Notetheuseofthreeequalssigns|eachindicatesanequalityofnumbersthelinearexpressionsare numberswhenweevaluatethemwithxedvaluesofthevariablequantities.Nowwritethevector equality, 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 x 3 5 x 1 +5 x 2 +7 x 3 x 1 +4 x 3 3 5 = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(33 24 5 3 5 : ByDenitionCVE[84],this single equalityoftwocolumnvectorstranslatesinto three simultaneous equalitiesofnumbersthatformthesystemofequations.Sowiththisnewnotionofvectorequalitywe canbecomelessreliantonreferringto systems of simultaneous equations.There'smoretovectorequality thanjustthis,butthisisagoodexampleforstartersandwewilldevelopitfurther. Wewillnowdenetwooperationsontheset C m .Bythiswemeanwell-denedproceduresthat somehowconvertvectorsintoothervectors.Herearetwoofthemostbasicdenitionsoftheentirecourse. DenitionCVA ColumnVectorAddition Supposethat u ; v 2 C m .The sum of u and v isthevector u + v denedby [ u + v ] i =[ u ] i +[ v ] i 1 i m Version2.02

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SubsectionVO.VEASMVectorEquality,Addition,ScalarMultiplication85 ThisdenitioncontainsNotationCVA. 4 Sovectoradditiontakestwovectorsofthesamesizeandcombinestheminanaturalway!tocreatea newvectorofthesamesize.Noticethatthisdenitionisrequired,evenifweagreethatthisistheobvious, right,naturalorcorrectwaytodoit.Noticetoothatthesymbol`+'isbeingrecycled.Weallknowhow toadd numbers ,butnowwehavethesamesymbolextendedtodouble-dutyandweuseittoindicatehow toaddtwonewobjects,vectors.Andthisdenitionofournewmeaningisbuiltonourpreviousmeaning ofadditionviatheexpressions u i + v i .Thinkaboutyourobjects,especiallywhendoingproofs.Vector additioniseasy,here'sanexamplefrom C 4 ExampleVA Additionoftwovectorsin C 4 If u = 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 4 2 3 7 7 5 v = 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 5 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 3 7 7 5 then u + v = 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 4 2 3 7 7 5 + 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 5 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 3 7 7 5 = 2 6 6 4 2+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3+5 4+2 2+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 3 7 7 5 = 2 6 6 4 1 2 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 7 7 5 : Oursecondoperationtakestwoobjectsofdierenttypes,specicallyanumberandavector,and combinesthemtocreateanothervector.Inthiscontextwecallanumbera scalar inordertoemphasize thatitisnotavector. DenitionCVSM ColumnVectorScalarMultiplication Suppose u 2 C m and 2 C ,thenthe scalarmultiple of u by isthevector u denedby [ u ] i = [ u ] i 1 i m ThisdenitioncontainsNotationCVSM. 4 Noticethatwearedoingakindofmultiplicationhere,butweare dening anewtype,perhapsinwhat appearstobeanaturalway.Weusejuxtapositionsmashingtwosymbolstogetherside-by-sidetodenote thisoperationratherthanusingasymbollikewedidwithvectoraddition.Sothiscanbeanothersource ofconfusion.Whentwosymbolsarenexttoeachother,arewedoingregularoldmultiplication,thekind we'vedoneforyears,orarewedoingscalarvectormultiplication,theoperationwejustdened?Think aboutyourobjects|iftherstobjectisascalar,andthesecondisavector,thenit must bethatweare doingournewoperation,andthe result ofthisoperationwillbeanothervector. Noticehowconsistencyinnotationcanbeanaidhere.IfwewritescalarsaslowercaseGreekletters fromthestartofthealphabetsuchas ,...andwritevectorsinboldLatinlettersfromtheend ofthealphabet u v ,...,thenwehavesomehintsaboutwhattypeofobjectsweareworkingwith. Thiscanbeablessing and acurse,sincewhenwegoreadanotherbookaboutlinearalgebra,orreadan applicationinanotherdisciplinephysics,economics,...thetypesofnotationemployedmaybevery dierentandhenceunfamiliar. Again,computationally,vectorscalarmultiplicationisveryeasy. Version2.02

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SubsectionVO.VSPVectorSpaceProperties86 ExampleCVSM Scalarmultiplicationin C 5 If u = 2 6 6 6 6 4 3 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 7 7 5 and =6,then u =6 2 6 6 6 6 4 3 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 7 7 5 = 2 6 6 6 6 4 6 6 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 6 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 7 7 5 = 2 6 6 6 6 4 18 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 24 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 3 7 7 7 7 5 : Vectoradditionandscalarmultiplicationarethemostnaturalandbasicoperationstoperformon vectors,soitshouldbeeasytohaveyourcomputationaldeviceformalinearcombination.See:ComputationVLC.MMA[668]ComputationVLC.TI86[672]ComputationVLC.TI83[674]Computation VLC.SAGE[677]. SubsectionVSP VectorSpaceProperties Withdenitionsofvectoradditionandscalarmultiplicationwecanstate,andprove,severalpropertiesof eachoperation,andsomepropertiesthatinvolvetheirinterplay.Wenowcollecttenofthemhereforlater reference. TheoremVSPCV VectorSpacePropertiesofColumnVectors Supposethat C m isthesetofcolumnvectorsofsize m DenitionVSCV[83]withadditionandscalar multiplicationasdenedinDenitionCVA[84]andDenitionCVSM[85].Then ACCAdditiveClosure,ColumnVectors If u ; v 2 C m ,then u + v 2 C m SCCScalarClosure,ColumnVectors If 2 C and u 2 C m ,then u 2 C m CCCommutativity,ColumnVectors If u ; v 2 C m ,then u + v = v + u AACAdditiveAssociativity,ColumnVectors If u ; v ; w 2 C m ,then u + v + w = u + v + w ZCZeroVector,ColumnVectors Thereisavector, 0 ,calledthe zerovector ,suchthat u + 0 = u forall u 2 C m AICAdditiveInverses,ColumnVectors If u 2 C m ,thenthereexistsavector )]TJ/F36 10.9091 Tf 8.485 0 Td [(u 2 C m sothat u + )]TJ/F36 10.9091 Tf 8.484 0 Td [(u = 0 SMACScalarMultiplicationAssociativity,ColumnVectors If ; 2 C and u 2 C m ,then u = u Version2.02

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SubsectionVO.READReadingQuestions87 DVACDistributivityacrossVectorAddition,ColumnVectors If 2 C and u ; v 2 C m ,then u + v = u + v DSACDistributivityacrossScalarAddition,ColumnVectors If ; 2 C and u 2 C m ,then + u = u + u OCOne,ColumnVectors If u 2 C m ,then1 u = u Proof Whilesomeofthesepropertiesseemveryobvious,theyallrequireproof.However,theproofsare notveryinteresting,andborderontedious.We'llproveoneversionofdistributivityverycarefully,and youcantestyourproof-buildingskillsonsomeoftheothers.Weneedtoestablishanequality,sowewill dosobybeginningwithonesideoftheequality,applyvariousdenitionsandtheoremslistedtotheright ofeachsteptomassagetheexpressionfromtheleftintotheexpressionontheright.Herewegowitha proofofPropertyDSAC[87].For1 i m [ + u ] i = + [ u ] i DenitionCVSM[85] = [ u ] i + [ u ] i Distributivityin C =[ u ] i +[ u ] i DenitionCVSM[85] =[ u + u ] i DenitionCVA[84] Sincetheindividualcomponentsofthevectors + u and u + u areequalfor all i ,1 i m DenitionCVE[84]tellsusthevectorsareequal. Manyoftheconclusionsofourtheoremscanbecharacterizedasidentities,"especiallywhenweare establishingbasicpropertiesofoperationssuchasthoseinthissection.Mostofthepropertieslistedin TheoremVSPCV[86]areexamples.Sosomeadviceaboutthestyleweuseforprovingidentitiesis appropriaterightnow.HavealookatTechniquePI[693]. Becarefulwiththenotionofthevector )]TJ/F36 10.9091 Tf 8.485 0 Td [(u .Thisisavectorthatweaddto u sothattheresultisthe particularvector 0 .Thisisbasicallyapropertyofvectoraddition.Ithappensthatwecancompute )]TJ/F36 10.9091 Tf 8.485 0 Td [(u usingthe other operation,scalarmultiplication.Wecanprovethisdirectlybywritingthat [ )]TJ/F36 10.9091 Tf 8.485 0 Td [(u ] i = )]TJ/F15 10.9091 Tf 10.303 0 Td [([ u ] i = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1[ u ] i =[ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 u ] i Wewillseelaterhowtoderivethispropertyasa consequence ofseveralofthetenpropertieslistedin TheoremVSPCV[86]. SubsectionREAD ReadingQuestions 1.Wherehaveyouseenvectorsusedbeforeinothercourses?Howweretheydierent? 2.Inwords,whenaretwovectorsequal? 3.Performthefollowingcomputationwithvectoroperations 2 2 4 1 5 0 3 5 + )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 2 4 7 6 5 3 5 Version2.02

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SubsectionVO.EXCExercises88 SubsectionEXC Exercises C10 Compute 4 2 6 6 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 4 1 0 3 7 7 7 7 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 6 6 6 6 4 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 2 4 3 7 7 7 7 5 + 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 0 1 2 3 7 7 7 7 5 ContributedbyRobertBeezerSolution[89] T13 ProvePropertyCC[86]ofTheoremVSPCV[86].Writeyourproofinthestyleoftheproofof PropertyDSAC[87]giveninthissection. ContributedbyRobertBeezerSolution[89] T17 ProvePropertySMAC[86]ofTheoremVSPCV[86].Writeyourproofinthestyleoftheproofof PropertyDSAC[87]giveninthissection. ContributedbyRobertBeezer T18 ProvePropertyDVAC[87]ofTheoremVSPCV[86].Writeyourproofinthestyleoftheproofof PropertyDSAC[87]giveninthissection. ContributedbyRobertBeezer Version2.02

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SubsectionVO.SOLSolutions89 SubsectionSOL Solutions C10 ContributedbyRobertBeezerStatement[88] 2 6 6 6 6 4 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 26 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 3 7 7 7 7 5 T13 ContributedbyRobertBeezerStatement[88] Forall1 i m [ u + v ] i =[ u ] i +[ v ] i DenitionCVA[84] =[ v ] i +[ u ] i Commutativityin C =[ v + u ] i DenitionCVA[84] Withequalityofeachcomponentofthevectors u + v and v + u beingequalDenitionCVE[84]tellsus thetwovectorsareequal. Version2.02

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SectionLCLinearCombinations90 SectionLC LinearCombinations InSectionVO[83]wedenedvectoradditionandscalarmultiplication.Thesetwooperationscombine nicelytogiveusaconstructionknownasalinearcombination,aconstructthatwewillworkwiththroughoutthiscourse. SubsectionLC LinearCombinations DenitionLCCV LinearCombinationofColumnVectors Given n vectors u 1 ; u 2 ; u 3 ;:::; u n from C m and n scalars 1 ; 2 ; 3 ;:::; n ,their linearcombination isthevector 1 u 1 + 2 u 2 + 3 u 3 + + n u n 4 Sothisdenitiontakesanequalnumberofscalarsandvectors,combinesthemusingourtwonew operationsscalarmultiplicationandvectoradditionandcreatesasinglebrand-newvector,ofthesame sizeastheoriginalvectors.Whenadenitionortheorememploysalinearcombination,thinkaboutthe natureoftheobjectsthatgointoitscreationlistsofscalarsandvectors,andthetypeofobjectthat resultsasinglevector.Computationally,alinearcombinationisprettyeasy. ExampleTLC Twolinearcombinationsin C 6 Supposethat 1 =1 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 =2 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 and u 1 = 2 6 6 6 6 6 6 4 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 2 9 3 7 7 7 7 7 7 5 u 2 = 2 6 6 6 6 6 6 4 6 3 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 4 3 7 7 7 7 7 7 5 u 3 = 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 2 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 3 7 7 7 7 7 7 5 u 4 = 2 6 6 6 6 6 6 4 3 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 7 1 3 3 7 7 7 7 7 7 5 thentheirlinearcombinationis 1 u 1 + 2 u 2 + 3 u 3 + 4 u 4 = 2 6 6 6 6 6 6 4 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 2 9 3 7 7 7 7 7 7 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 2 6 6 6 6 6 6 4 6 3 0 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 1 4 3 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 2 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 3 7 7 7 7 7 7 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 6 6 6 6 6 6 4 3 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 7 1 3 3 7 7 7 7 7 7 5 Version2.02

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SubsectionLC.LCLinearCombinations91 = 2 6 6 6 6 6 6 4 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 2 9 3 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(24 )]TJ/F15 10.9091 Tf 8.484 0 Td [(12 0 8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(16 3 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 4 2 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 0 3 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(35 )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 4 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 3 7 7 7 7 7 7 5 : Adierentlinearcombination,ofthesamesetofvectors,canbeformedwithdierentscalars.Take 1 =3 2 =0 3 =5 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 andformthelinearcombination 1 u 1 + 2 u 2 + 3 u 3 + 4 u 4 = 2 6 6 6 6 6 6 4 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 2 9 3 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 4 6 3 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 4 3 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 2 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 3 7 7 7 7 7 7 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 6 6 6 6 6 6 4 3 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 7 1 3 3 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 4 6 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 3 6 27 3 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 4 0 0 0 0 0 0 3 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(25 10 5 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 0 3 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(22 20 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 24 3 7 7 7 7 7 7 5 : Noticehowwecouldkeepoursetofvectorsxed,andusedierentsetsofscalarstoconstructdierent vectors.Youmightbuildafewnewlinearcombinationsof u 1 ; u 2 ; u 3 ; u 4 rightnow.We'llberighthere whenyougetback.Whatvectorswereyouabletocreate?Doyouthinkyoucouldcreatethevector w = 2 6 6 6 6 6 6 4 13 15 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(17 2 25 3 7 7 7 7 7 7 5 withasuitable"choiceoffourscalars?Doyouthinkyoucouldcreate any possiblevectorfrom C 6 by choosingtheproperscalars?Theselasttwoquestionsareveryfundamental,andtimespentconsidering them now willprovebeneciallater. Ournexttwoexamplesarekeyones,andadiscussionaboutdecompositionsistimely.Havealookat TechniqueDC[694]beforestudyingthenexttwoexamples. ExampleABLC ArchetypeBasalinearcombination InthisexamplewewillrewriteArchetypeB[707]inthelanguageofvectors,vectorequalityandlinear combinations.InExampleVESE[84]wewrotethesystemof m =3equationsasthevectorequality 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 x 3 5 x 1 +5 x 2 +7 x 3 x 1 +4 x 3 3 5 = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(33 24 5 3 5 : Version2.02

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SubsectionLC.LCLinearCombinations92 Nowwewillbustupthelinearexpressionsontheleft,rstusingvectoraddition, 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 x 1 5 x 1 x 1 3 5 + 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 x 2 5 x 2 0 x 2 3 5 + 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 x 3 7 x 3 4 x 3 3 5 = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(33 24 5 3 5 : Nowwecanrewriteeachofthese n =3vectorsasascalarmultipleofaxedvector,wherethescalaris oneoftheunknownvariables,convertingtheleft-handsideintoalinearcombination x 1 2 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(7 5 1 3 5 + x 2 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 5 0 3 5 + x 3 2 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(12 7 4 3 5 = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(33 24 5 3 5 : Wecannowinterprettheproblemofsolvingthesystemofequationsasdeterminingvaluesforthescalar multiplesthatmakethevectorequationtrue.IntheanalysisofArchetypeB[707],wewereableto determinethatithadonlyonesolution.Aquickwaytoseethisistorow-reducethecoecientmatrix tothe3 3identitymatrixandapplyTheoremNMRRI[72]todeterminethatthecoecientmatrixis nonsingular.ThenTheoremNMUS[74]tellsusthatthesystemofequationshasauniquesolution.This solutionis x 1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x 2 =5 x 3 =2 : So,inthecontextofthisexample,wecanexpressthefactthatthesevaluesofthevariablesareasolution bywritingthelinearcombination, )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 5 1 3 5 + 2 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 5 0 3 5 + 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 7 4 3 5 = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(33 24 5 3 5 : Furthermore,thesearetheonlythreescalarsthatwillaccomplishthisequality,sincetheycomefroma uniquesolution. Noticehowthethreevectorsinthisexamplearethecolumnsofthecoecientmatrixofthesystemof equations.Thisisourrsthintoftheimportantinterplaybetweenthevectorsthatformthecolumnsof amatrix,andthematrixitself. WithanydiscussionofArchetypeA[702]orArchetypeB[707]weshouldbesuretocontrastwiththe other. ExampleAALC ArchetypeAasalinearcombination Asavectorequality,ArchetypeA[702]canbewrittenas 2 4 x 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 2 +2 x 3 2 x 1 + x 2 + x 3 x 1 + x 2 3 5 = 2 4 1 8 5 3 5 : Nowbustupthelinearexpressionsontheleft,rstusingvectoraddition, 2 4 x 1 2 x 1 x 1 3 5 + 2 4 )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 2 x 2 x 2 3 5 + 2 4 2 x 3 x 3 0 x 3 3 5 = 2 4 1 8 5 3 5 : Rewriteeachofthese n =3vectorsasascalarmultipleofaxedvector,wherethescalarisoneofthe unknownvariables,convertingtheleft-handsideintoalinearcombination x 1 2 4 1 2 1 3 5 + x 2 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 1 3 5 + x 3 2 4 2 1 0 3 5 = 2 4 1 8 5 3 5 : Version2.02

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SubsectionLC.LCLinearCombinations93 Row-reducingtheaugmentedmatrixforArchetypeA[702]leadstotheconclusionthatthesystemis consistentandhasfreevariables,henceinnitelymanysolutions.Soforexample,thetwosolutions x 1 =2 x 2 =3 x 3 =1 x 1 =3 x 2 =2 x 3 =0 canbeusedtogethertosaythat, 2 4 1 2 1 3 5 + 2 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 1 1 3 5 + 2 4 2 1 0 3 5 = 2 4 1 8 5 3 5 = 2 4 1 2 1 3 5 + 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 1 3 5 + 2 4 2 1 0 3 5 Ignorethemiddleofthisequation,andmoveallthetermstotheleft-handside, 2 4 1 2 1 3 5 + 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 1 3 5 + 2 4 2 1 0 3 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 2 4 1 2 1 3 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 1 3 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 2 4 2 1 0 3 5 = 2 4 0 0 0 3 5 : Regroupinggives )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 4 1 2 1 3 5 + 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 1 3 5 + 2 4 2 1 0 3 5 = 2 4 0 0 0 3 5 : Noticethatthesethreevectorsarethecolumnsofthecoecientmatrixforthesystemofequationsin ArchetypeA[702].Thisequalitysaysthereisalinearcombinationofthosecolumnsthatequalsthevector ofallzeros.Giveitsomethought,butthissaysthat x 1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 x 2 =1 x 3 =1 isanontrivialsolutiontothehomogeneoussystemofequationswiththecoecientmatrixfortheoriginal systeminArchetypeA[702].Inparticular,thisdemonstratesthatthiscoecientmatrixissingular. There'salotgoingoninthelasttwoexamples.Comebacktotheminawhileandmakesome connectionswiththeinterveningmaterial.Fornow,wewillsummarizeandexplainsomeofthisbehavior withatheorem. TheoremSLSLC SolutionstoLinearSystemsareLinearCombinations Denotethecolumnsofthe m n matrix A asthevectors A 1 ; A 2 ; A 3 ;:::; A n .Then x isasolutionto thelinearsystemofequations LS A; b ifandonlyif [ x ] 1 A 1 +[ x ] 2 A 2 +[ x ] 3 A 3 + +[ x ] n A n = b Proof Theproofofthistheoremisasmuchaboutachangeinnotationasitisaboutmakinglogical deductions.Writethesystemofequations LS A; b as a 11 x 1 + a 12 x 2 + a 13 x 3 + + a 1 n x n = b 1 a 21 x 1 + a 22 x 2 + a 23 x 3 + + a 2 n x n = b 2 a 31 x 1 + a 32 x 2 + a 33 x 3 + + a 3 n x n = b 3 a m 1 x 1 + a m 2 x 2 + a m 3 x 3 + + a mn x n = b m : Version2.02

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SubsectionLC.VFSSVectorFormofSolutionSets94 Noticethenthattheentryofthecoecientmatrix A inrow i andcolumn j hastwonames: a ij asthe coecientof x j inequation i ofthesystemand[ A j ] i asthe i -thentryofthecolumnvectorincolumn j ofthecoecientmatrix A .Likewise,entry i of b hastwonames: b i fromthelinearsystemand[ b ] i asanentryofavector.OurtheoremisanequivalenceTechniqueE[690]soweneedtoproveboth directions." Supposewehavethevectorequalitybetween b andthelinearcombinationofthecolumnsof A Thenfor1 i n b i =[ b ] i Notation =[[ x ] 1 A 1 +[ x ] 2 A 2 +[ x ] 3 A 3 + +[ x ] n A n ] i Hypothesis =[[ x ] 1 A 1 ] i +[[ x ] 2 A 2 ] i +[[ x ] 3 A 3 ] i + +[[ x ] n A n ] i DenitionCVA[84] =[ x ] 1 [ A 1 ] i +[ x ] 2 [ A 2 ] i +[ x ] 3 [ A 3 ] i + +[ x ] n [ A n ] i DenitionCVSM[85] =[ x ] 1 a i 1 +[ x ] 2 a i 2 +[ x ] 3 a i 3 + +[ x ] n a in Notation = a i 1 [ x ] 1 + a i 2 [ x ] 2 + a i 3 [ x ] 3 + + a in [ x ] n Commutativityin C Thissaysthattheentriesof x formasolutiontoequation i of LS A; b forall1 i n ,inotherwords, x isasolutionto LS A; b Supposenowthat x isasolutiontothelinearsystem LS A; b .Thenforall1 i n [ b ] i = b i Notation = a i 1 [ x ] 1 + a i 2 [ x ] 2 + a i 3 [ x ] 3 + + a in [ x ] n Hypothesis =[ x ] 1 a i 1 +[ x ] 2 a i 2 +[ x ] 3 a i 3 + +[ x ] n a in Commutativityin C =[ x ] 1 [ A 1 ] i +[ x ] 2 [ A 2 ] i +[ x ] 3 [ A 3 ] i + +[ x ] n [ A n ] i Notation =[[ x ] 1 A 1 ] i +[[ x ] 2 A 2 ] i +[[ x ] 3 A 3 ] i + +[[ x ] n A n ] i DenitionCVSM[85] =[[ x ] 1 A 1 +[ x ] 2 A 2 +[ x ] 3 A 3 + +[ x ] n A n ] i DenitionCVA[84] Sincethecomponentsof b andthelinearcombinationofthecolumnsof A agreeforall1 i n ,Denition CVE[84]tellsusthatthevectorsareequal. Inotherwords,thistheoremtellsusthatsolutionstosystemsofequationsarelinearcombinationsof thecolumnvectorsofthecoecientmatrix A i whichyieldtheconstantvector b .Orsaidanotherway, asolutiontoasystemofequations LS A; b isananswertothequestionHowcanIformthevector b asalinearcombinationofthecolumnsof A ?"Lookthroughthearchetypesthataresystemsofequations andexamineafewoftheadvertisedsolutions.Ineachcaseusethesolutiontoformalinearcombination ofthecolumnsofthecoecientmatrixandverifythattheresultequalstheconstantvectorseeExercise LC.C21[108]. SubsectionVFSS VectorFormofSolutionSets Wehavewrittensolutionstosystemsofequationsascolumnvectors.ForexampleArchetypeB[707]has thesolution x 1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ;x 2 =5 ;x 3 =2whichwenowwriteas x = 2 4 x 1 x 2 x 3 3 5 = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 5 2 3 5 : Now,wewillusecolumnvectorsandlinearcombinationstoexpress all ofthesolutionstoalinearsystem ofequationsinacompactandunderstandableway.First,here'stwoexamplesthatwillmotivateournext Version2.02

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SubsectionLC.VFSSVectorFormofSolutionSets95 theorem.Thisisavaluabletechnique,almosttheequalofrow-reducingamatrix,sobesureyouget comfortablewithitoverthecourseofthissection. ExampleVFSAD VectorformofsolutionsforArchetypeD ArchetypeD[716]isalinearsystemof3equationsin4variables.Row-reducingtheaugmentedmatrix yields 2 4 1 03 )]TJ/F15 10.9091 Tf 8.485 0 Td [(24 0 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(30 00000 3 5 andwesee r =2nonzerorows.Also, D = f 1 ; 2 g sothedependentvariablesarethen x 1 and x 2 F = f 3 ; 4 ; 5 g sothetwofreevariablesare x 3 and x 4 .Wewillexpressagenericsolutionforthesystemby twoslightlydierentmethods,thoughbotharriveatthesameconclusion. First,wewilldecomposeTechniqueDC[694]asolutionvector.Rearrangingeachequationrepresented intherow-reducedformoftheaugmentedmatrixbysolvingforthedependentvariableineachrowyields thevectorequality, 2 6 6 4 x 1 x 2 x 3 x 4 3 7 7 5 = 2 6 6 4 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 +2 x 4 )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 3 +3 x 4 x 3 x 4 3 7 7 5 Nowwewillusethedenitionsofcolumnvectoradditionandscalarmultiplicationtoexpressthisvector asalinearcombination, = 2 6 6 4 4 0 0 0 3 7 7 5 + 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x 3 )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 3 x 3 0 3 7 7 5 + 2 6 6 4 2 x 4 3 x 4 0 x 4 3 7 7 5 DenitionCVA[84] = 2 6 6 4 4 0 0 0 3 7 7 5 + x 3 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 0 3 7 7 5 + x 4 2 6 6 4 2 3 0 1 3 7 7 5 DenitionCVSM[85] Wewilldevelopthesamelinearcombinationabitquicker,usingthreesteps.Whilethemethodaboveis instructive,themethodbelowwillbeourpreferredapproach. Step1.Writethevectorofvariablesasaxedvector,plusalinearcombinationof n )]TJ/F21 10.9091 Tf 10.326 0 Td [(r vectors,using thefreevariablesasthescalars. x = 2 6 6 4 x 1 x 2 x 3 x 4 3 7 7 5 = 2 6 6 4 3 7 7 5 + x 3 2 6 6 4 3 7 7 5 + x 4 2 6 6 4 3 7 7 5 Step2.Use0'sand1'stoensureequalityfortheentriesofthethevectorswithindicesin F corresponding tothefreevariables. x = 2 6 6 4 x 1 x 2 x 3 x 4 3 7 7 5 = 2 6 6 4 0 0 3 7 7 5 + x 3 2 6 6 4 1 0 3 7 7 5 + x 4 2 6 6 4 0 1 3 7 7 5 Version2.02

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SubsectionLC.VFSSVectorFormofSolutionSets96 Step3.Foreachdependentvariable,usetheaugmentedmatrixtoformulateanequationexpressingthe dependentvariableasaconstantplusmultiplesofthefreevariables.Convertthisequationintoentriesof thevectorsthatensureequalityforeachdependentvariable,oneatatime. x 1 =4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 +2 x 4 x = 2 6 6 4 x 1 x 2 x 3 x 4 3 7 7 5 = 2 6 6 4 4 0 0 3 7 7 5 + x 3 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 0 3 7 7 5 + x 4 2 6 6 4 2 0 1 3 7 7 5 x 2 =0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 3 +3 x 4 x = 2 6 6 4 x 1 x 2 x 3 x 4 3 7 7 5 = 2 6 6 4 4 0 0 0 3 7 7 5 + x 3 2 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 1 0 3 7 7 5 + x 4 2 6 6 4 2 3 0 1 3 7 7 5 Thisnal form ofatypicalsolutionisespeciallypleasinganduseful.Forexample,wecanbuildsolutions quicklybychoosingvaluesforourfreevariables,andthencomputealinearcombination.Suchas x 3 =2 ;x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 x = 2 6 6 4 x 1 x 2 x 3 x 4 3 7 7 5 = 2 6 6 4 4 0 0 0 3 7 7 5 + 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 0 3 7 7 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 2 6 6 4 2 3 0 1 3 7 7 5 = 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(17 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 3 7 7 5 or, x 3 =1 ;x 4 =3 x = 2 6 6 4 x 1 x 2 x 3 x 4 3 7 7 5 = 2 6 6 4 4 0 0 0 3 7 7 5 + 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 0 3 7 7 5 + 2 6 6 4 2 3 0 1 3 7 7 5 = 2 6 6 4 7 8 1 3 3 7 7 5 : You'llndthesecondsolutionlistedinthewrite-upforArchetypeD[716],andyoumightchecktherst solutionbysubstitutingitbackintotheoriginalequations. Whilethisformisusefulforquicklycreatingsolutions,itsevenbetterbecauseittellsus exactly what everysolutionlookslike.Weknowthesolutionsetisinnite,whichisprettybig,butnowwecansaythat asolutionissomemultipleof 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 0 3 7 7 5 plusamultipleof 2 6 6 4 2 3 0 1 3 7 7 5 plusthexedvector 2 6 6 4 4 0 0 0 3 7 7 5 .Period.Soitonly takesus three vectorstodescribetheentireinnitesolutionset,providedwealsoagreeonhowtocombine thethreevectorsintoalinearcombination. Thisissuchanimportantandfundamentaltechnique,we'lldoanotherexample. ExampleVFS Vectorformofsolutions Consideralinearsystemof m =5equationsin n =7variables,havingtheaugmentedmatrix A A = 2 6 6 6 6 4 21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(221521 11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(31112 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8511 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 33 )]TJ/F15 10.9091 Tf 8.485 0 Td [(93652 )]TJ/F15 10.9091 Tf 8.485 0 Td [(24 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11211 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(30 3 7 7 7 7 5 Version2.02

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SubsectionLC.VFSSVectorFormofSolutionSets97 Row-reducingweobtainthematrix B = 2 6 6 6 6 6 4 1 02 )]TJ/F15 10.9091 Tf 8.485 0 Td [(300915 0 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5400 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 0000 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(611 00000 1 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 00000000 3 7 7 7 7 7 5 andwesee r =4nonzerorows.Also, D = f 1 ; 2 ; 5 ; 6 g sothedependentvariablesarethen x 1 ;x 2 ;x 5 ; and x 6 F = f 3 ; 4 ; 7 ; 8 g sothe n )]TJ/F21 10.9091 Tf 11.075 0 Td [(r =3freevariablesare x 3 ;x 4 and x 7 .Wewillexpressagenericsolution forthesystembytwodierentmethods:bothadecompositionandaconstruction. First,wewilldecomposeTechniqueDC[694]asolutionvector.Rearrangingeachequationrepresented intherow-reducedformoftheaugmentedmatrixbysolvingforthedependentvariableineachrowyields thevectorequality, 2 6 6 6 6 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 x 6 x 7 3 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 4 15 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 3 +3 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 x 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10+5 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 4 +8 x 7 x 3 x 4 11+6 x 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 )]TJ/F15 10.9091 Tf 10.91 0 Td [(7 x 7 x 7 3 7 7 7 7 7 7 7 7 5 Nowwewillusethedenitionsofcolumnvectoradditionandscalarmultiplicationtodecomposethis genericsolutionvectorasalinearcombination, = 2 6 6 6 6 6 6 6 6 4 15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 0 0 11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 0 3 7 7 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 3 5 x 3 x 3 0 0 0 0 3 7 7 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 6 6 4 3 x 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 x 4 0 x 4 0 0 0 3 7 7 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 x 7 8 x 7 0 0 6 x 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 x 7 x 7 3 7 7 7 7 7 7 7 7 5 DenitionCVA[84] = 2 6 6 6 6 6 6 6 6 4 15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 0 0 11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 0 3 7 7 7 7 7 7 7 7 5 + x 3 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 5 1 0 0 0 0 3 7 7 7 7 7 7 7 7 5 + x 4 2 6 6 6 6 6 6 6 6 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 0 1 0 0 0 3 7 7 7 7 7 7 7 7 5 + x 7 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 8 0 0 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 1 3 7 7 7 7 7 7 7 7 5 DenitionCVSM[85] Wewillnowdevelopthesamelinearcombinationabitquicker,usingthreesteps.Whilethemethodabove isinstructive,themethodbelowwillbeourpreferredapproach. Step1.Writethevectorofvariablesasaxedvector,plusalinearcombinationof n )]TJ/F21 10.9091 Tf 10.325 0 Td [(r vectors,using thefreevariablesasthescalars. x = 2 6 6 6 6 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 x 6 x 7 3 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 + x 3 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 + x 4 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 + x 7 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 Version2.02

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SubsectionLC.VFSSVectorFormofSolutionSets98 Step2.Use0'sand1'stoensureequalityfortheentriesofthethevectorswithindicesin F corresponding tothefreevariables. x = 2 6 6 6 6 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 x 6 x 7 3 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 4 0 0 0 3 7 7 7 7 7 7 7 7 5 + x 3 2 6 6 6 6 6 6 6 6 4 1 0 0 3 7 7 7 7 7 7 7 7 5 + x 4 2 6 6 6 6 6 6 6 6 4 0 1 0 3 7 7 7 7 7 7 7 7 5 + x 7 2 6 6 6 6 6 6 6 6 4 0 0 1 3 7 7 7 7 7 7 7 7 5 Step3.Foreachdependentvariable,usetheaugmentedmatrixtoformulateanequationexpressingthe dependentvariableasaconstantplusmultiplesofthefreevariables.Convertthisequationintoentriesof thevectorsthatensureequalityforeachdependentvariable,oneatatime. x 1 =15 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 3 +3 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 x 7 x = 2 6 6 6 6 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 x 6 x 7 3 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 4 15 0 0 0 3 7 7 7 7 7 7 7 7 5 + x 3 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 0 0 3 7 7 7 7 7 7 7 7 5 + x 4 2 6 6 6 6 6 6 6 6 4 3 0 1 0 3 7 7 7 7 7 7 7 7 5 + x 7 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 0 0 1 3 7 7 7 7 7 7 7 7 5 x 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(10+5 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 4 +8 x 7 x = 2 6 6 6 6 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 x 6 x 7 3 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 4 15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 0 0 0 3 7 7 7 7 7 7 7 7 5 + x 3 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 5 1 0 0 3 7 7 7 7 7 7 7 7 5 + x 4 2 6 6 6 6 6 6 6 6 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 0 1 0 3 7 7 7 7 7 7 7 7 5 + x 7 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 8 0 0 1 3 7 7 7 7 7 7 7 7 5 x 5 =11+6 x 7 x = 2 6 6 6 6 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 x 6 x 7 3 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 4 15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 0 0 11 0 3 7 7 7 7 7 7 7 7 5 + x 3 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 5 1 0 0 0 3 7 7 7 7 7 7 7 7 5 + x 4 2 6 6 6 6 6 6 6 6 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 0 1 0 0 3 7 7 7 7 7 7 7 7 5 + x 7 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 8 0 0 6 1 3 7 7 7 7 7 7 7 7 5 x 6 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x 7 x = 2 6 6 6 6 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 x 6 x 7 3 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 4 15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 0 0 11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 0 3 7 7 7 7 7 7 7 7 5 + x 3 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 5 1 0 0 0 0 3 7 7 7 7 7 7 7 7 5 + x 4 2 6 6 6 6 6 6 6 6 4 3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 0 1 0 0 0 3 7 7 7 7 7 7 7 7 5 + x 7 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 8 0 0 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 1 3 7 7 7 7 7 7 7 7 5 Thisnal form ofatypicalsolutionisespeciallypleasinganduseful.Forexample,wecanbuildsolutions quicklybychoosingvaluesforourfreevariables,andthencomputealinearcombination.Forexample x 3 =2 ;x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 ;x 7 =3 Version2.02

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SubsectionLC.VFSSVectorFormofSolutionSets99 x = 2 6 6 6 6 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 x 6 x 7 3 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 4 15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 0 0 11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 0 3 7 7 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 5 1 0 0 0 0 3 7 7 7 7 7 7 7 7 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 2 6 6 6 6 6 6 6 6 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 0 1 0 0 0 3 7 7 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 8 0 0 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 1 3 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(28 40 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 29 )]TJ/F15 10.9091 Tf 8.485 0 Td [(42 3 3 7 7 7 7 7 7 7 7 5 orperhaps, x 3 =5 ;x 4 =2 ;x 7 =1 x = 2 6 6 6 6 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 x 6 x 7 3 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 4 15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 0 0 11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 0 3 7 7 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 5 1 0 0 0 0 3 7 7 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 6 6 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 0 1 0 0 0 3 7 7 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 8 0 0 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 1 3 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 4 2 15 5 2 17 )]TJ/F15 10.9091 Tf 8.485 0 Td [(28 1 3 7 7 7 7 7 7 7 7 5 oreven, x 3 =0 ;x 4 =0 ;x 7 =0 x = 2 6 6 6 6 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 x 6 x 7 3 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 4 15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 0 0 11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 0 3 7 7 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 5 1 0 0 0 0 3 7 7 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 6 6 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 0 1 0 0 0 3 7 7 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 8 0 0 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 1 3 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 4 15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 0 0 11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 0 3 7 7 7 7 7 7 7 7 5 Sowecancompactlyexpress all ofthesolutionstothislinearsystemwithjust4xedvectors,provided weagreehowtocombinetheminalinearcombinationstocreatesolutionvectors. Supposeyouweretoldthatthevector w belowwasasolutiontothissystemofequations.Couldyou turntheproblemaroundandwrite w asalinearcombinationofthefourvectors c u 1 u 2 u 3 ?See ExerciseLC.M11[109]. w = 2 6 6 6 6 6 6 6 6 4 100 )]TJ/F15 10.9091 Tf 8.485 0 Td [(75 7 9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(37 35 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 3 7 7 7 7 7 7 7 7 5 c = 2 6 6 6 6 6 6 6 6 4 15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 0 0 11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 0 3 7 7 7 7 7 7 7 7 5 u 1 = 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 5 1 0 0 0 0 3 7 7 7 7 7 7 7 7 5 u 2 = 2 6 6 6 6 6 6 6 6 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 0 1 0 0 0 3 7 7 7 7 7 7 7 7 5 u 3 = 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 8 0 0 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 1 3 7 7 7 7 7 7 7 7 5 Didyouthinkafewweeksagothatyoucouldsoquicklyandeasilylist all thesolutionstoalinear systemof5equationsin7variables? We'llnowformalizethelasttwoimportantexamplesasatheorem. TheoremVFSLS VectorFormofSolutionstoLinearSystems Supposethat[ A j b ]istheaugmentedmatrixforaconsistentlinearsystem LS A; b of m equationsin Version2.02

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SubsectionLC.VFSSVectorFormofSolutionSets100 n variables.Let B bearow-equivalent m n +1matrixinreducedrow-echelonform.Supposethat B has r nonzerorows,columnswithoutleading1'swithindices F = f f 1 ;f 2 ;f 3 ;:::;f n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r ;n +1 g ,and columnswithleading1'spivotcolumnshavingindices D = f d 1 ;d 2 ;d 3 ;:::;d r g .Denevectors c u j 1 j n )]TJ/F21 10.9091 Tf 10.909 0 Td [(r ofsize n by [ c ] i = 0if i 2 F [ B ] k;n +1 if i 2 D i = d k [ u j ] i = 8 > < > : 1if i 2 F i = f j 0if i 2 F i 6 = f j )]TJ/F15 10.9091 Tf 10.303 0 Td [([ B ] k;f j if i 2 D i = d k : Thenthesetofsolutionstothesystemofequations LS A; b is S = f c + 1 u 1 + 2 u 2 + 3 u 3 + + n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r u n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r j 1 ; 2 ; 3 ;:::; n )]TJ/F22 7.9701 Tf 6.586 0 Td [(r 2 C g Proof First, LS A; b isequivalenttothelinearsystemofequationsthathasthematrix B asits augmentedmatrixTheoremREMES[28],soweneedonlyshowthat S isthesolutionsetforthesystem with B asitsaugmentedmatrix.Theconclusionofthistheoremisthatthesolutionsetisequaltotheset S ,sowewillapplyDenitionSE[684]. Webeginbyshowingthateveryelementof S isindeedasolutiontothesystem.Let 1 ; 2 ; 3 ;:::; n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r beonechoiceofthescalarsusedtodescribeelementsof S .Soanarbitraryelementof S ,whichwewill considerasaproposedsolutionis x = c + 1 u 1 + 2 u 2 + 3 u 3 + + n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r u n )]TJ/F22 7.9701 Tf 6.586 0 Td [(r When r +1 ` m ,row ` ofthematrix B isazerorow,sotheequationrepresentedbythatrowis alwaystrue,nomatterwhichsolutionvectorwepropose.Soconcentrateonrowsrepresentingequations 1 ` r .Weevaluateequation ` ofthesystemrepresentedby B withtheproposedsolutionvector x andrefertothevalueoftheleft-handsideoftheequationas ` ` =[ B ] ` 1 [ x ] 1 +[ B ] ` 2 [ x ] 2 +[ B ] ` 3 [ x ] 3 + +[ B ] `n [ x ] n Since[ B ] `d i =0forall1 i r ,exceptthat[ B ] `d ` =1,weseethat ` simpliesto ` =[ x ] d ` +[ B ] `f 1 [ x ] f 1 +[ B ] `f 2 [ x ] f 2 +[ B ] `f 3 [ x ] f 3 + +[ B ] `f n )]TJ/F23 5.9776 Tf 5.756 0 Td [(r [ x ] f n )]TJ/F23 5.9776 Tf 5.756 0 Td [(r Noticethatfor1 i n )]TJ/F21 10.9091 Tf 10.909 0 Td [(r [ x ] f i =[ c ] f i + 1 [ u 1 ] f i + 2 [ u 2 ] f i + 3 [ u 3 ] f i + + i [ u i ] f i + + n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r [ u n )]TJ/F40 7.9701 Tf 6.586 0 Td [(r ] f i =0+ 1 + 2 + 3 + + i + + n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r = i So ` simpliesfurther,andweexpandtherstterm ` =[ x ] d ` +[ B ] `f 1 1 +[ B ] `f 2 2 +[ B ] `f 3 3 + +[ B ] `f n )]TJ/F23 5.9776 Tf 5.757 0 Td [(r n )]TJ/F22 7.9701 Tf 6.586 0 Td [(r =[ c + 1 u 1 + 2 u 2 + 3 u 3 + + n )]TJ/F22 7.9701 Tf 6.586 0 Td [(r u n )]TJ/F22 7.9701 Tf 6.586 0 Td [(r ] d ` + [ B ] `f 1 1 +[ B ] `f 2 2 +[ B ] `f 3 3 + +[ B ] `f n )]TJ/F23 5.9776 Tf 5.757 0 Td [(r n )]TJ/F22 7.9701 Tf 6.586 0 Td [(r =[ c ] d ` + 1 [ u 1 ] d ` + 2 [ u 2 ] d ` + 3 [ u 3 ] d ` + + n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r [ u n )]TJ/F40 7.9701 Tf 6.586 0 Td [(r ] d ` + [ B ] `f 1 1 +[ B ] `f 2 2 +[ B ] `f 3 3 + +[ B ] `f n )]TJ/F23 5.9776 Tf 5.756 0 Td [(r n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r Version2.02

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SubsectionLC.VFSSVectorFormofSolutionSets101 =[ B ] `;n +1 + 1 )]TJ/F15 10.9091 Tf 10.303 0 Td [([ B ] `;f 1 + 2 )]TJ/F15 10.9091 Tf 10.303 0 Td [([ B ] `;f 2 + 3 )]TJ/F15 10.9091 Tf 10.303 0 Td [([ B ] `;f 3 + + n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r )]TJ/F15 10.9091 Tf 10.303 0 Td [([ B ] `;f n )]TJ/F23 5.9776 Tf 5.756 0 Td [(r + [ B ] `f 1 1 +[ B ] `f 2 2 +[ B ] `f 3 3 + +[ B ] `f n )]TJ/F23 5.9776 Tf 5.757 0 Td [(r n )]TJ/F22 7.9701 Tf 6.586 0 Td [(r =[ B ] `;n +1 So ` beganastheleft-handsideofequation ` ofthesystemrepresentedby B andwenowknowitequals [ B ] `;n +1 ,theconstanttermforequation ` ofthissystem.Sothearbitrarilychosenvectorfrom S makes everyequationofthesystemtrue,andthereforeisasolutiontothesystem.Soalltheelementsof S are solutionstothesystem. Forthesecondhalfoftheproof,assumethat x isasolutionvectorforthesystemhaving B asits augmentedmatrix.Forconvenienceandclarity,denotetheentriesof x by x i ,inotherwords, x i =[ x ] i Wedesiretoshowthatthissolutionvectorisalsoanelementoftheset S .Beginwiththeobservation thatasolutionvector'sentriesmakesequation ` ofthesystemtrueforall1 ` m [ B ] `; 1 x 1 +[ B ] `; 2 x 2 +[ B ] `; 3 x 3 + +[ B ] `;n x n =[ B ] `;n +1 When ` r ,thepivotcolumnsof B havezeroentriesinrow ` withtheexceptionofcolumn d ` ,whichwill containa1.Sofor1 ` r ,equation ` simpliesto 1 x d ` +[ B ] `;f 1 x f 1 +[ B ] `;f 2 x f 2 +[ B ] `;f 3 x f 3 + +[ B ] `;f n )]TJ/F23 5.9776 Tf 5.756 0 Td [(r x f n )]TJ/F23 5.9776 Tf 5.756 0 Td [(r =[ B ] `;n +1 Thisallowsustowrite, [ x ] d ` = x d ` =[ B ] `;n +1 )]TJ/F15 10.9091 Tf 10.909 0 Td [([ B ] `;f 1 x f 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [([ B ] `;f 2 x f 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [([ B ] `;f 3 x f 3 )-222()]TJ/F15 10.9091 Tf 36.969 0 Td [([ B ] `;f n )]TJ/F23 5.9776 Tf 5.757 0 Td [(r x f n )]TJ/F23 5.9776 Tf 5.756 0 Td [(r =[ c ] d ` + x f 1 [ u 1 ] d ` + x f 2 [ u 2 ] d ` + x f 3 [ u 3 ] d ` + + x f n )]TJ/F23 5.9776 Tf 5.756 0 Td [(r [ u n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r ] d ` = c + x f 1 u 1 + x f 2 u 2 + x f 3 u 3 + + x f n )]TJ/F23 5.9776 Tf 5.756 0 Td [(r u n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r d ` Thistellsusthattheentriesofthesolutionvector x correspondingtodependentvariablesindicesin D areequaltothoseofavectorintheset S .Westillneedtochecktheotherentriesofthesolutionvector x correspondingtothefreevariablesindicesin F toseeiftheyareequaltotheentriesofthesamevector intheset S .Tothisend,suppose i 2 F and i = f j .Then [ x ] i = x i = x f j =0+0 x f 1 +0 x f 2 +0 x f 3 + +0 x f j )]TJ/F20 5.9776 Tf 5.756 0 Td [(1 +1 x f j +0 x f j +1 + +0 x f n )]TJ/F23 5.9776 Tf 5.756 0 Td [(r =[ c ] i + x f 1 [ u 1 ] i + x f 2 [ u 2 ] i + x f 3 [ u 3 ] i + + x f j [ u j ] i + + x f n )]TJ/F23 5.9776 Tf 5.756 0 Td [(r [ u n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r ] i = c + x f 1 u 1 + x f 2 u 2 + + x f n )]TJ/F23 5.9776 Tf 5.756 0 Td [(r u n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r i Soentriesof x and c + x f 1 u 1 + x f 2 u 2 + + x f n )]TJ/F23 5.9776 Tf 5.756 0 Td [(r u n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r areequalandthereforebyDenitionCVE[84]they areequalvectors.Since x f 1 ;x f 2 ;x f 3 ;:::;x f n )]TJ/F23 5.9776 Tf 5.756 0 Td [(r arescalars,thisshowsusthat x qualiesformembership in S .Sotheset S containsallofthesolutionstothesystem. NotethatbothhalvesoftheproofofTheoremVFSLS[99]indicatethat i =[ x ] f i .Inotherwords, thearbitraryscalars, i ,inthedescriptionoftheset S actuallyhavemoremeaning|theyarethevalues ofthefreevariables[ x ] f i ,1 i n )]TJ/F21 10.9091 Tf 11.226 0 Td [(r .Sowewilloftenexploitthisobservationinourdescriptionsof solutionsets. TheoremVFSLS[99]formalizeswhathappenedinthethreestepsofExampleVFSAD[95].The theoremwillbeusefulinprovingothertheorems,andititisusefulsinceittellsusanexactprocedurefor simplydescribinganinnitesolutionset.Wecouldprogramacomputertoimplementit,oncewehave theaugmentedmatrixrow-reducedandhavecheckedthatthesystemisconsistent.ByKnuth'sdenition, thiscompletesourconversionoflinearequationsolvingfromartintoscience.Noticethatitevenapplies Version2.02

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SubsectionLC.VFSSVectorFormofSolutionSets102 butisoverkillinthecaseofauniquesolution.However,asapracticalmatter,Ipreferthethree-step processofExampleVFSAD[95]whenIneedtodescribeaninnitesolutionset.Solet'spracticesome more,butwithabiggerexample. ExampleVFSAI VectorformofsolutionsforArchetypeI ArchetypeI[737]isalinearsystemof m =4equationsin n =7variables.Row-reducingtheaugmented matrixyields 2 6 6 4 1 40021 )]TJ/F15 10.9091 Tf 8.484 0 Td [(34 00 1 01 )]TJ/F15 10.9091 Tf 8.484 0 Td [(352 000 1 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(661 00000000 3 7 7 5 andwesee r =3nonzerorows.Thecolumnswithleading1'sare D = f 1 ; 3 ; 4 g sothe r dependent variablesare x 1 ;x 3 ;x 4 .Thecolumnswithoutleading1'sare F = f 2 ; 5 ; 6 ; 7 ; 8 g ,sothe n )]TJ/F21 10.9091 Tf 11.365 0 Td [(r =4free variablesare x 2 ;x 5 ;x 6 ;x 7 Step1.Writethevectorofvariables x asaxedvector c ,plusalinearcombinationof n )]TJ/F21 10.9091 Tf 11.072 0 Td [(r =4 vectors u 1 ; u 2 ; u 3 ; u 4 ,usingthefreevariablesasthescalars. x = 2 6 6 6 6 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 x 6 x 7 3 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 + x 2 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 + x 5 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 + x 6 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 + x 7 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 Step2.Foreachfreevariable,use0'sand1'stoensureequalityforthecorrespondingentryofthethe vectors.Takenoteofthepatternof0'sand1'satthisstage,becausethisisthebestlookyou'llhaveatit. We'llstateanimportanttheoreminthenextsectionandtheproofwillessentiallyrelyonthisobservation. x = 2 6 6 6 6 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 x 6 x 7 3 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 4 0 0 0 0 3 7 7 7 7 7 7 7 7 5 + x 2 2 6 6 6 6 6 6 6 6 4 1 0 0 0 3 7 7 7 7 7 7 7 7 5 + x 5 2 6 6 6 6 6 6 6 6 4 0 1 0 0 3 7 7 7 7 7 7 7 7 5 + x 6 2 6 6 6 6 6 6 6 6 4 0 0 1 0 3 7 7 7 7 7 7 7 7 5 + x 7 2 6 6 6 6 6 6 6 6 4 0 0 0 1 3 7 7 7 7 7 7 7 7 5 Step3.Foreachdependentvariable,usetheaugmentedmatrixtoformulateanequationexpressingthe dependentvariableasaconstantplusmultiplesofthefreevariables.Convertthisequationintoentriesof thevectorsthatensureequalityforeachdependentvariable,oneatatime. x 1 =4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 6 +3 x 7 x = 2 6 6 6 6 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 x 6 x 7 3 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 4 4 0 0 0 0 3 7 7 7 7 7 7 7 7 5 + x 2 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 1 0 0 0 3 7 7 7 7 7 7 7 7 5 + x 5 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 1 0 0 3 7 7 7 7 7 7 7 7 5 + x 6 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 0 1 0 3 7 7 7 7 7 7 7 7 5 + x 7 2 6 6 6 6 6 6 6 6 4 3 0 0 0 1 3 7 7 7 7 7 7 7 7 5 x 3 =2+0 x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 5 +3 x 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 7 Version2.02

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SubsectionLC.VFSSVectorFormofSolutionSets103 x = 2 6 6 6 6 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 x 6 x 7 3 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 4 4 0 2 0 0 0 3 7 7 7 7 7 7 7 7 5 + x 2 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 1 0 0 0 0 3 7 7 7 7 7 7 7 7 5 + x 5 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 0 0 3 7 7 7 7 7 7 7 7 5 + x 6 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 3 0 1 0 3 7 7 7 7 7 7 7 7 5 + x 7 2 6 6 6 6 6 6 6 6 4 3 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 0 0 1 3 7 7 7 7 7 7 7 7 5 x 4 =1+0 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 5 +6 x 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 7 x = 2 6 6 6 6 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 x 6 x 7 3 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 4 4 0 2 1 0 0 0 3 7 7 7 7 7 7 7 7 5 + x 2 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 1 0 0 0 0 0 3 7 7 7 7 7 7 7 7 5 + x 5 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 0 0 3 7 7 7 7 7 7 7 7 5 + x 6 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 3 6 0 1 0 3 7 7 7 7 7 7 7 7 5 + x 7 2 6 6 6 6 6 6 6 6 4 3 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 0 0 1 3 7 7 7 7 7 7 7 7 5 Wecannowusethisnalexpressiontoquicklybuildsolutionstothesystem.Youmighttrytorecreate eachofthesolutionslistedinthewrite-upforArchetypeI[737].Hint:lookatthevaluesofthefree variablesineachsolution,andnoticethatthevector c has0'sintheselocations. Evenbetter,wehaveadescriptionoftheinnitesolutionset,basedonjust5vectors,whichwecombine inlinearcombinationstoproducesolutions. WheneverwediscussArchetypeI[737]youknowthat'syourcuetogoworkthroughArchetypeJ[741] byyourself.Remembertotakenoteofthe0/1patternattheconclusionofStep2.Havefun|wewon't goanywherewhileyou'reaway. Thistechniqueissoimportant,thatwe'lldoonemoreexample.However,animportantdistinction willbethatthissystemishomogeneous. ExampleVFSAL VectorformofsolutionsforArchetypeL ArchetypeL[750]ispresentedsimplyasthe5 5matrix L = 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(44 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(46 107710 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(910 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(66 3 7 7 7 7 5 We'llinterpretithereasthecoecientmatrixofahomogeneoussystemandreferencethismatrixas L .Sowearesolvingthehomogeneoussystem LS L; 0 having m =5equationsin n =5variables.If webuilttheaugmentedmatrix,wewouldaddasixthcolumnto L containingallzeros.Aswedidrow operations,thissixthcolumnwouldremainallzeros.Soinsteadwewillrow-reducethecoecientmatrix, andmentallyrememberthemissingsixthcolumnofzeros.Thisrow-reducedmatrixis 2 6 6 6 6 6 4 1 001 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(22 00 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 00000 00000 3 7 7 7 7 7 5 Version2.02

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SubsectionLC.VFSSVectorFormofSolutionSets104 andwesee r =3nonzerorows.Thecolumnswithleading1'sare D = f 1 ; 2 ; 3 g sothe r dependent variablesare x 1 ;x 2 ;x 3 .Thecolumnswithoutleading1'sare F = f 4 ; 5 g ,sothe n )]TJ/F21 10.9091 Tf 11.101 0 Td [(r =2freevariables are x 4 ;x 5 .Noticethatifwehadincludedtheall-zerovectorofconstantstoformtheaugmentedmatrix forthesystem,thentheindex6wouldhaveappearedintheset F ,andsubsequentlywouldhavebeen ignoredwhenlistingthefreevariables. Step1.Writethevectorofvariables x asaxedvector c ,plusalinearcombinationof n )]TJ/F21 10.9091 Tf 11.072 0 Td [(r =2 vectors u 1 ; u 2 ,usingthefreevariablesasthescalars. x = 2 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 3 7 7 7 7 5 = 2 6 6 6 6 4 3 7 7 7 7 5 + x 4 2 6 6 6 6 4 3 7 7 7 7 5 + x 5 2 6 6 6 6 4 3 7 7 7 7 5 Step2.Foreachfreevariable,use0'sand1'stoensureequalityforthecorrespondingentryofthethe vectors.Takenoteofthepatternof0'sand1'satthisstage,evenifitisnotasilluminatingasinother examples. x = 2 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 3 7 7 7 7 5 = 2 6 6 6 6 4 0 0 3 7 7 7 7 5 + x 4 2 6 6 6 6 4 1 0 3 7 7 7 7 5 + x 5 2 6 6 6 6 4 0 1 3 7 7 7 7 5 Step3.Foreachdependentvariable,usetheaugmentedmatrixtoformulateanequationexpressingthe dependentvariableasaconstantplusmultiplesofthefreevariables.Don'tforgetaboutthemissing" sixthcolumnbeingfullofzeros.Convertthisequationintoentriesofthevectorsthatensureequalityfor eachdependentvariable,oneatatime. x 1 =0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 4 +2 x 5 x = 2 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 3 7 7 7 7 5 = 2 6 6 6 6 4 0 0 0 3 7 7 7 7 5 + x 4 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 0 3 7 7 7 7 5 + x 5 2 6 6 6 6 4 2 0 1 3 7 7 7 7 5 x 2 =0+2 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 5 x = 2 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 3 7 7 7 7 5 = 2 6 6 6 6 4 0 0 0 0 3 7 7 7 7 5 + x 4 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 2 1 0 3 7 7 7 7 5 + x 5 2 6 6 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 1 3 7 7 7 7 5 x 3 =0 )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 x 4 +1 x 5 x = 2 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 3 7 7 7 7 5 = 2 6 6 6 6 4 0 0 0 0 0 3 7 7 7 7 5 + x 4 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 1 0 3 7 7 7 7 5 + x 5 2 6 6 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 0 1 3 7 7 7 7 5 Thevector c willalwayshave0'sintheentriescorrespondingtofreevariables.However,sinceweare solvingahomogeneoussystem,therow-reducedaugmentedmatrixhaszerosincolumn n +1=6,and hence all theentriesof c arezero.Sowecanwrite x = 2 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 3 7 7 7 7 5 = 0 + x 4 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 0 3 7 7 7 7 5 + x 5 2 6 6 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 0 1 3 7 7 7 7 5 = x 4 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 0 3 7 7 7 7 5 + x 5 2 6 6 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 0 1 3 7 7 7 7 5 Version2.02

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SubsectionLC.PSHSParticularSolutions,HomogeneousSolutions105 Itwillalwayshappenthatthesolutionstoahomogeneoussystemhas c = 0 eveninthecaseofaunique solution?.Soourexpressionforthesolutionsisabitmorepleasing.Inthisexampleitsaysthatthe solutionsare allpossible linearcombinationsofthetwovectors u 1 = 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 0 3 7 7 7 7 5 and u 2 = 2 6 6 6 6 4 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 1 0 1 3 7 7 7 7 5 ,withno mentionofanyxedvectorenteringintothelinearcombination. Thisobservationwillmotivateournextsectionandthemaindenitionofthatsection,andafterthat wewillconcludethesectionbyformalizingthissituation. SubsectionPSHS ParticularSolutions,HomogeneousSolutions Thenexttheoremtellsusthatinordertondallofthesolutionstoalinearsystemofequations,itis sucienttondjustonesolution,andthenndallofthesolutionstothecorrespondinghomogeneous system.Thisexplainspartofourinterestinthenullspace,thesetofallsolutionstoahomogeneous system. TheoremPSPHS ParticularSolutionPlusHomogeneousSolutions Supposethat w isonesolutiontothelinearsystemofequations LS A;b .Then y isasolutionto LS A;b ifandonlyif y = w + z forsomevector z 2N A Proof Let A 1 ; A 2 ; A 3 ;:::; A n bethecolumnsofthecoecientmatrix A Suppose y = w + z and z 2N A .Then b =[ w ] 1 A 1 +[ w ] 2 A 2 +[ w ] 3 A 3 + +[ w ] n A n TheoremSLSLC[93] =[ w ] 1 A 1 +[ w ] 2 A 2 +[ w ] 3 A 3 + +[ w ] n A n + 0 PropertyZC[86] =[ w ] 1 A 1 +[ w ] 2 A 2 +[ w ] 3 A 3 + +[ w ] n A n TheoremSLSLC[93] +[ z ] 1 A 1 +[ z ] 2 A 2 +[ z ] 3 A 3 + +[ z ] n A n =[ w ] 1 +[ z ] 1 A 1 +[ w ] 2 +[ z ] 2 A 2 + +[ w ] n +[ z ] n A n TheoremVSPCV[86] =[ w + z ] 1 A 1 +[ w + z ] 2 A 2 +[ w + z ] 3 A 3 + +[ w + z ] n A n DenitionCVA[84] =[ y ] 1 A 1 +[ y ] 2 A 2 +[ y ] 3 A 3 + +[ y ] n A n Denitionof y ApplyingTheoremSLSLC[93]weseethatthevector y isasolutionto LS A; b Suppose y isasolutionto LS A;b .Then 0 = b )]TJ/F36 10.9091 Tf 10.909 0 Td [(b =[ y ] 1 A 1 +[ y ] 2 A 2 +[ y ] 3 A 3 + +[ y ] n A n TheoremSLSLC[93] )]TJ/F15 10.9091 Tf 10.909 0 Td [([ w ] 1 A 1 +[ w ] 2 A 2 +[ w ] 3 A 3 + +[ w ] n A n =[ y ] 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [([ w ] 1 A 1 +[ y ] 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [([ w ] 2 A 2 + +[ y ] n )]TJ/F15 10.9091 Tf 10.909 0 Td [([ w ] n A n TheoremVSPCV[86] =[ y )]TJ/F36 10.9091 Tf 10.909 0 Td [(w ] 1 A 1 +[ y )]TJ/F36 10.9091 Tf 10.909 0 Td [(w ] 2 A 2 +[ y )]TJ/F36 10.9091 Tf 10.909 0 Td [(w ] 3 A 3 + +[ y )]TJ/F36 10.9091 Tf 10.909 0 Td [(w ] n A n DenitionCVA[84] ByTheoremSLSLC[93]weseethatthevector y )]TJ/F36 10.9091 Tf 10.913 0 Td [(w isasolutiontothehomogeneoussystem LS A; 0 andbyDenitionNSM[64], y )]TJ/F36 10.9091 Tf 11.667 0 Td [(w 2N A .Inotherwords, y )]TJ/F36 10.9091 Tf 11.667 0 Td [(w = z forsomevector z 2N A Rewritten,thisis y = w + z ,asdesired. AfterprovingTheoremNMUS[74]wecommentedinsucientlyonthenegationofonehalfofthetheorem.Nonsingularcoecientmatricesleadtouniquesolutionsforeverychoiceofthevectorofconstants. Version2.02

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SubsectionLC.PSHSParticularSolutions,HomogeneousSolutions106 Whatdoesthissayaboutsingularmatrices?Asingularmatrix A hasanontrivialnullspaceTheorem NMTNS[74].Foragivenvectorofconstants, b ,thesystem LS A;b couldbeinconsistent,meaning therearenosolutions.Butifthereisatleastonesolution w ,thenTheoremPSPHS[105]tellsusthere willbeinnitelymanysolutionsbecauseoftheroleoftheinnitenullspaceforasingularmatrix.Soa systemofequationswithasingularcoecientmatrix never hasauniquesolution.Eitherthereareno solutions,orinnitelymanysolutions,dependingonthechoiceofthevectorofconstants b ExamplePSHS Particularsolutions,homogeneoussolutions,ArchetypeD ArchetypeD[716]isaconsistentsystemofequationswithanontrivialnullspace.Let A denotethe coecientmatrixofthissystem.Thewrite-upforthissystembeginswiththreesolutions, y 1 = 2 6 6 4 0 1 2 1 3 7 7 5 y 2 = 2 6 6 4 4 0 0 0 3 7 7 5 y 3 = 2 6 6 4 7 8 1 3 3 7 7 5 Wewillchoosetohave y 1 playtheroleof w inthestatementofTheoremPSPHS[105],anyoneofthe threevectorslistedhereorotherscouldhavebeenchosen.Toillustratethetheorem,weshouldbeable towriteeachofthesethreesolutionsasthevector w plusasolutiontothecorrespondinghomogeneous systemofequations.Since 0 isalwaysasolutiontoahomogeneoussystemwecaneasilywrite y 1 = w = w + 0 : Thevectors y 2 and y 3 willrequireabitmoreeort.Solutionstothehomogeneoussystem LS A; 0 are exactlytheelementsofthenullspaceofthecoecientmatrix,whichbyanapplicationofTheoremVFSLS [99]is N A = 8 > > < > > : x 3 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 0 3 7 7 5 + x 4 2 6 6 4 2 3 0 1 3 7 7 5 j x 3 ;x 4 2 C 9 > > = > > ; Then y 2 = 2 6 6 4 4 0 0 0 3 7 7 5 = 2 6 6 4 0 1 2 1 3 7 7 5 + 2 6 6 4 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 5 = 2 6 6 4 0 1 2 1 3 7 7 5 + 0 B B @ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 1 0 3 7 7 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 6 6 4 2 3 0 1 3 7 7 5 1 C C A = w + z 2 where z 2 = 2 6 6 4 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 5 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 0 3 7 7 5 + )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 2 6 6 4 2 3 0 1 3 7 7 5 isobviouslyasolutionofthehomogeneoussystemsinceitiswrittenasalinearcombinationofthevectors describingthenullspaceofthecoecientmatrixorasacheck,youcouldjustevaluatetheequationsin thehomogeneoussystemwith z 2 Again y 3 = 2 6 6 4 7 8 1 3 3 7 7 5 = 2 6 6 4 0 1 2 1 3 7 7 5 + 2 6 6 4 7 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 7 7 5 = 2 6 6 4 0 1 2 1 3 7 7 5 + 0 B B @ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 1 0 3 7 7 5 +2 2 6 6 4 2 3 0 1 3 7 7 5 1 C C A = w + z 3 Version2.02

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SubsectionLC.READReadingQuestions107 where z 3 = 2 6 6 4 7 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 7 7 5 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 0 3 7 7 5 +2 2 6 6 4 2 3 0 1 3 7 7 5 isobviouslyasolutionofthehomogeneoussystemsinceitiswrittenasalinearcombinationofthevectors describingthenullspaceofthecoecientmatrixorasacheck,youcouldjustevaluatetheequationsin thehomogeneoussystemwith z 2 Here'sanotherviewofthistheorem,inthecontextofthisexample.Grabtwonewsolutionsofthe originalsystemofequations,say y 4 = 2 6 6 4 11 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 5 y 5 = 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 2 4 2 3 7 7 5 andformtheirdierence, u = 2 6 6 4 11 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 5 )]TJ/F27 10.9091 Tf 10.909 28.473 Td [(2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 2 4 2 3 7 7 5 = 2 6 6 4 15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 7 7 5 : Itisnoaccidentthat u isasolutiontothehomogeneoussystemcheckthis!.Inotherwords,thedierence betweenanytwosolutionstoalinearsystemofequationsisanelementofthenullspaceofthecoecient matrix.ThisisanequivalentwaytostateTheoremPSPHS[105].SeeExerciseMM.T50[207]. TheideasofthissubsectionwillbeappearagaininChapterLT[452]whenwediscusspre-imagesof lineartransformationsDenitionPI[465]. SubsectionREAD ReadingQuestions 1.Earlier,areadingquestionaskedyoutosolvethesystemofequations 2 x 1 +3 x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 3 =0 x 1 +2 x 2 + x 3 =3 x 1 +3 x 2 +3 x 3 =7 Usealinearcombinationtorewritethissystemofequationsasavectorequality. 2.Findalinearcombinationofthevectors S = 8 < : 2 4 1 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 5 ; 2 4 2 0 4 3 5 ; 2 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 3 5 9 = ; thatequalsthevector 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 11 3 5 Version2.02

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SubsectionLC.READReadingQuestions108 3.Thematrixbelowistheaugmentedmatrixofasystemofequations,row-reducedtoreducedrowechelonform.Writethevectorformofthesolutionstothesystem. 2 4 1 30609 00 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 0000 1 3 3 5 Version2.02

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SubsectionLC.EXCExercises109 SubsectionEXC Exercises C21 Considereacharchetypethatisasystemofequations.Forindividualsolutionslistedbothforthe originalsystemandthecorrespondinghomogeneoussystemexpressthevectorofconstantsasalinear combinationofthecolumnsofthecoecientmatrix,asguaranteedbyTheoremSLSLC[93].Verifythis equalitybycomputingthelinearcombination.Forsystemswithnosolutions,recognizethatitisthen impossibletowritethevectorofconstantsasalinearcombinationofthecolumnsofthecoecientmatrix. Notetoo,forhomogeneoussystems,thatthesolutionsgiverisetolinearcombinationsthatequalthezero vector. ArchetypeA[702] ArchetypeB[707] ArchetypeC[712] ArchetypeD[716] ArchetypeE[720] ArchetypeF[724] ArchetypeG[729] ArchetypeH[733] ArchetypeI[737] ArchetypeJ[741] ContributedbyRobertBeezerSolution[110] C22 Considereacharchetypethatisasystemofequations.Writeelementsofthesolutionsetinvector form,asguaranteedbyTheoremVFSLS[99]. ArchetypeA[702] ArchetypeB[707] ArchetypeC[712] ArchetypeD[716] ArchetypeE[720] ArchetypeF[724] ArchetypeG[729] ArchetypeH[733] ArchetypeI[737] ArchetypeJ[741] ContributedbyRobertBeezerSolution[110] C40 Findthevectorformofthesolutionstothesystemofequationsbelow. 2 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 2 +3 x 3 + x 5 =6 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 3 +14 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 5 =15 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 + x 3 +2 x 4 + x 5 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 1 +4 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 x 4 + x 5 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 ContributedbyRobertBeezerSolution[110] C41 Findthevectorformofthesolutionstothesystemofequationsbelow. )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 x 3 +8 x 4 +4 x 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 x 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 7 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 8 )]TJ/F15 10.9091 Tf 10.909 0 Td [(18 x 9 =3 Version2.02

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SubsectionLC.EXCExercises110 3 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 +5 x 3 +2 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 6 +1 x 7 +2 x 8 +15 x 9 =10 4 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 +8 x 3 +2 x 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(14 x 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 8 +2 x 9 =36 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 x 1 +2 x 2 +1 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 4 +7 x 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 7 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 9 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 3 x 1 +2 x 2 +13 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(14 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 5 +5 x 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 8 +12 x 9 =15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 1 +2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 4 +1 x 5 +6 x 6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 7 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 8 )]TJ/F15 10.9091 Tf 10.91 0 Td [(15 x 9 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 ContributedbyRobertBeezerSolution[110] M10 ExampleTLC[90]asksifthevector w = 2 6 6 6 6 6 6 4 13 15 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(17 2 25 3 7 7 7 7 7 7 5 canbewrittenasalinearcombinationofthefourvectors u 1 = 2 6 6 6 6 6 6 4 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 2 9 3 7 7 7 7 7 7 5 u 2 = 2 6 6 6 6 6 6 4 6 3 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 4 3 7 7 7 7 7 7 5 u 3 = 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 2 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 3 7 7 7 7 7 7 5 u 4 = 2 6 6 6 6 6 6 4 3 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 7 1 3 3 7 7 7 7 7 7 5 Canit?Cananyvectorin C 6 bewrittenasalinearcombinationofthefourvectors u 1 ; u 2 ; u 3 ; u 4 ? ContributedbyRobertBeezerSolution[111] M11 AttheendofExampleVFS[96],thevector w isclaimedtobeasolutiontothelinearsystem underdiscussion.Verifythat w reallyisasolution.Thendeterminethefourscalarsthatexpress w asa linearcombinationof c u 1 u 2 u 3 ContributedbyRobertBeezerSolution[111] Version2.02

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SubsectionLC.SOLSolutions111 SubsectionSOL Solutions C21 ContributedbyRobertBeezerStatement[108] SolutionsforArchetypeA[702]andArchetypeB[707]aredescribedcarefullyinExampleAALC[92]and ExampleABLC[91]. C22 ContributedbyRobertBeezerStatement[108] SolutionsforArchetypeD[716]andArchetypeI[737]aredescribedcarefullyinExampleVFSAD[95]and ExampleVFSAI[102].Thetechniquedescribedintheseexamplesisprobablymoreusefulthancarefully decipheringthenotationofTheoremVFSLS[99].Thesolutionforeacharchetypeiscontainedinits description.Sonowyoucancheck-otheboxforthatitem. C40 ContributedbyRobertBeezerStatement[108] Row-reducetheaugmentedmatrixrepresentingthissystem,tond 2 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20601 00 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(403 0000 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 000000 3 7 7 5 Thesystemisconsistentnoleadingoneincolumn6,TheoremRCLS[53]. x 2 and x 4 arethefreevariables. NowapplyTheoremVFSLS[99]directly,orfollowthethree-stepprocessofExampleVFS[96],Example VFSAD[95],ExampleVFSAI[102],orExampleVFSAL[103]toobtain 2 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 3 7 7 7 7 5 = 2 6 6 6 6 4 1 0 3 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 7 7 7 7 5 + x 2 2 6 6 6 6 4 2 1 0 0 0 3 7 7 7 7 5 + x 4 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 0 4 1 0 3 7 7 7 7 5 C41 ContributedbyRobertBeezerStatement[108] Row-reducetheaugmentedmatrixrepresentingthissystem,tond 2 6 6 6 6 6 6 6 4 1 03 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10036 0 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(403002 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0000 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(200 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 000000 1 040 0000000 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0000000000 3 7 7 7 7 7 7 7 5 Thesystemisconsistentnoleadingoneincolumn10,TheoremRCLS[53]. F = f 3 ; 4 ; 6 ; 9 ; 10 g ,sothe freevariablesare x 3 ;x 4 ;x 6 and x 9 .NowapplyTheoremVFSLS[99]directly,orfollowthethree-step processofExampleVFS[96],ExampleVFSAD[95],ExampleVFSAI[102],orExampleVFSAL[103]to Version2.02

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SubsectionLC.SOLSolutions112 obtainthesolutionset S = 8 > > > > > > > > > > > > < > > > > > > > > > > > > : 2 6 6 6 6 6 6 6 6 6 6 6 6 4 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 0 3 0 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 3 7 7 7 7 7 7 7 7 7 7 7 7 5 + x 3 2 6 6 6 6 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 0 0 0 0 0 0 3 7 7 7 7 7 7 7 7 7 7 7 7 5 + x 4 2 6 6 6 6 6 6 6 6 6 6 6 6 4 2 4 0 1 0 0 0 0 0 3 7 7 7 7 7 7 7 7 7 7 7 7 5 + x 6 2 6 6 6 6 6 6 6 6 6 6 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 0 2 1 0 0 0 3 7 7 7 7 7 7 7 7 7 7 7 7 5 + x 9 2 6 6 6 6 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 0 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 3 7 7 7 7 7 7 7 7 7 7 7 7 5 j x 3 ;x 4 ;x 6 ;x 9 2 C 9 > > > > > > > > > > > > = > > > > > > > > > > > > ; M10 ContributedbyRobertBeezerStatement[109] No,itisnotpossibletocreate w asalinearcombinationofthefourvectors u 1 ; u 2 ; u 3 ; u 4 .Bycreatingthe desiredlinearcombinationwithunknownsasscalars,TheoremSLSLC[93]providesasystemofequations thathasnosolution.Thisonecomputationisenoughtoshowusthatitisnotpossibletocreateallthe vectorsof C 6 throughlinearcombinationsofthefourvectors u 1 ; u 2 ; u 3 ; u 4 M11 ContributedbyRobertBeezerStatement[109] Thecoecientof c is1.Thecoecientsof u 1 u 2 u 3 lieinthethird,fourthandseventhentriesof w Canyouseewhy?Hint: F = f 3 ; 4 ; 7 ; 8 g ,sothefreevariablesare x 3 ;x 4 and x 7 Version2.02

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SectionSSSpanningSets113 SectionSS SpanningSets Inthissectionwewilldescribeacompactwaytoindicatetheelementsofaninnitesetofvectors,making useoflinearcombinations.Thiswillgiveusaconvenientwaytodescribetheelementsofasetofsolutions toalinearsystem,ortheelementsofthenullspaceofamatrix,ormanyothersetsofvectors. SubsectionSSV SpanofaSetofVectors InExampleVFSAL[103]wesawthesolutionsetofahomogeneoussystemdescribedasallpossiblelinear combinationsoftwoparticularvectors.Thishappenstobeausefulwaytoconstructordescribeinnite setsofvectors,soweencapsulatethisideainadenition. DenitionSSCV SpanofaSetofColumnVectors Givenasetofvectors S = f u 1 ; u 2 ; u 3 ;:::; u p g ,their span h S i ,isthesetofallpossiblelinearcombinationsof u 1 ; u 2 ; u 3 ;:::; u p .Symbolically, h S i = f 1 u 1 + 2 u 2 + 3 u 3 + + p u p j i 2 C ; 1 i p g = p X i =1 i u i j i 2 C ; 1 i p ThisdenitioncontainsNotationSSV. 4 Thespanisjustasetofvectors,thoughinallbutonesituationitisaninniteset.Justwhenisit notinnite?Sowestartwithanitecollectionofvectors S p ofthemtobeprecise,andusethisnite settodescribeaninnitesetofvectors, h S i .Confusingthe nite set S withthe innite set h S i isoneof themostpervasiveproblemsinunderstandingintroductorylinearalgebra.Wewillseethisconstruction repeatedly,solet'sworkthroughsomeexamplestogetcomfortablewithit.Themostobviousquestion aboutasetisifaparticularitemofthecorrecttypeisintheset,ornot. ExampleABS Abasicspan Considerthesetof5vectors, S ,from C 4 S = 8 > > < > > : 2 6 6 4 1 1 3 1 3 7 7 5 ; 2 6 6 4 2 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 5 ; 2 6 6 4 7 3 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 7 7 5 ; 2 6 6 4 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 7 7 5 ; 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 9 0 3 7 7 5 9 > > = > > ; andconsidertheinnitesetofvectors h S i formedfromallpossiblelinearcombinationsoftheelementsof S .Herearefourvectorswedenitelyknowareelementsof h S i ,sincewewillconstructtheminaccordance withDenitionSSCV[112], w = 2 6 6 4 1 1 3 1 3 7 7 5 + 2 6 6 4 2 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 6 6 4 7 3 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 7 7 5 + 2 6 6 4 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 7 7 5 + 2 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 0 9 0 3 7 7 5 = 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 2 28 10 3 7 7 5 Version2.02

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SubsectionSS.SSVSpanofaSetofVectors114 x = 2 6 6 4 1 1 3 1 3 7 7 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 2 6 6 4 2 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 2 6 6 4 7 3 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 7 7 5 + 2 6 6 4 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 7 7 5 + 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 9 0 3 7 7 5 = 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(26 )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 2 34 3 7 7 5 y = 2 6 6 4 1 1 3 1 3 7 7 5 + 2 6 6 4 2 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 5 + 2 6 6 4 7 3 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 7 7 5 + 2 6 6 4 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 7 7 5 + 2 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 0 9 0 3 7 7 5 = 2 6 6 4 7 4 17 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 7 7 5 z = 2 6 6 4 1 1 3 1 3 7 7 5 + 2 6 6 4 2 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 5 + 2 6 6 4 7 3 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 7 7 5 + 2 6 6 4 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 7 7 5 + 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 9 0 3 7 7 5 = 2 6 6 4 0 0 0 0 3 7 7 5 Thepurposeofasetistocollectobjectswithsomecommonproperty,andtoexcludeobjectswithoutthat property.Sothemostfundamentalquestionaboutasetisifagivenobjectisanelementofthesetornot. Let'slearnmoreabout h S i byinvestigatingwhichvectorsareelementsoftheset,andwhicharenot. First,is u = 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 19 5 3 7 7 5 anelementof h S i ?Weareaskingiftherearescalars 1 ; 2 ; 3 ; 4 ; 5 suchthat 1 2 6 6 4 1 1 3 1 3 7 7 5 + 2 2 6 6 4 2 1 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 3 7 7 5 + 3 2 6 6 4 7 3 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 7 7 5 + 4 2 6 6 4 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 7 7 5 + 5 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 9 0 3 7 7 5 = u = 2 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 19 5 3 7 7 5 ApplyingTheoremSLSLC[93]werecognizethesearchforthesescalarsasasolutiontoalinearsystemof equationswithaugmentedmatrix 2 6 6 4 1271 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 11310 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 325 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1919 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5205 3 7 7 5 whichrow-reducesto 2 6 6 4 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10310 0 1 40 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 000 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 000000 3 7 7 5 Atthispoint,weseethatthesystemisconsistentTheoremRCLS[53],soweknowthere is asolution forthevescalars 1 ; 2 ; 3 ; 4 ; 5 .Thisisenoughevidenceforustosaythat u 2h S i .Ifwewished furtherevidence,wecouldcomputeanactualsolution,say 1 =2 2 =1 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 5 =2 Thisparticularsolutionallowsustowrite 2 6 6 4 1 1 3 1 3 7 7 5 + 2 6 6 4 2 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 6 6 4 7 3 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 7 7 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 2 6 6 4 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 7 7 5 + 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 9 0 3 7 7 5 = u = 2 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 19 5 3 7 7 5 Version2.02

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SubsectionSS.SSVSpanofaSetofVectors115 makingitevenmoreobviousthat u 2h S i Letsdoitagain.Is v = 2 6 6 4 3 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 5 anelementof h S i ?Weareaskingiftherearescalars 1 ; 2 ; 3 ; 4 ; 5 suchthat 1 2 6 6 4 1 1 3 1 3 7 7 5 + 2 2 6 6 4 2 1 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 3 7 7 5 + 3 2 6 6 4 7 3 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 7 7 5 + 4 2 6 6 4 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 7 7 5 + 5 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 9 0 3 7 7 5 = v = 2 6 6 4 3 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 5 ApplyingTheoremSLSLC[93]werecognizethesearchforthesescalarsasasolutiontoalinearsystemof equationswithaugmentedmatrix 2 6 6 4 1271 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 113101 325 )]TJ/F15 10.9091 Tf 8.485 0 Td [(192 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(520 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 5 whichrow-reducesto 2 6 6 6 4 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1030 0 1 40 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 000 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 00000 1 3 7 7 7 5 Atthispoint,weseethatthesystemisinconsistentbyTheoremRCLS[53],soweknowthere isnot a solutionforthevescalars 1 ; 2 ; 3 ; 4 ; 5 .Thisisenoughevidenceforustosaythat v 62h S i .Endof story. ExampleSCAA SpanofthecolumnsofArchetypeA Beginwiththenitesetofthreevectorsofsize3 S = f u 1 ; u 2 ; u 3 g = 8 < : 2 4 1 2 1 3 5 ; 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 1 3 5 ; 2 4 2 1 0 3 5 9 = ; andconsidertheinniteset h S i .Thevectorsof S couldhavebeenchosentobeanything,butforreasons thatwillbecomeclearlater,wehavechosenthethreecolumnsofthecoecientmatrixinArchetypeA [702].First,asanexample,notethat v = 2 4 1 2 1 3 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 1 3 5 + 2 4 2 1 0 3 5 = 2 4 22 14 2 3 5 isin h S i ,sinceitisalinearcombinationof u 1 ; u 2 ; u 3 .Wewritethissuccinctlyas v 2h S i .Thereis nothingmagicalaboutthescalars 1 =5 ; 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 3 =7,theycouldhavebeenchosentobeanything. Sorepeatthispartoftheexampleyourself,usingdierentvaluesof 1 ; 2 ; 3 .Whathappensifyou chooseallthreescalarstobezero? Soweknowhowtoquicklyconstructsampleelementsoftheset h S i .Aslightlydierentquestionarises whenyouarehandedavectorofthecorrectsizeandaskedifitisanelementof h S i .Forexample,is w = 2 4 1 8 5 3 5 in h S i ?Moresuccinctly, w 2h S i ? Version2.02

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SubsectionSS.SSVSpanofaSetofVectors116 Toanswerthisquestion,wewilllookforscalars 1 ; 2 ; 3 sothat 1 u 1 + 2 u 2 + 3 u 3 = w : ByTheoremSLSLC[93]solutionstothisvectorequationaresolutionstothesystemofequations 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [( 2 +2 3 =1 2 1 + 2 + 3 =8 1 + 2 =5 : Buildingtheaugmentedmatrixforthislinearsystem,androw-reducing,gives 2 4 1 013 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 0000 3 5 : Thissystemhasinnitelymanysolutionsthere'safreevariablein x 3 ,butallweneedisonesolution vector.Thesolution, 1 =2 2 =3 3 =1 tellsusthat u 1 + u 2 + u 3 = w soweareconvincedthat w reallyisin h S i .Noticethatthereareaninnitenumberofwaystoanswer thisquestionarmatively.Wecouldchooseadierentsolution,thistimechoosingthefreevariabletobe zero, 1 =3 2 =2 3 =0 showsusthat u 1 + u 2 + u 3 = w Verifyingthearithmeticinthissecondsolutionmaybemakesitseemobviousthat w isinthisspan?And ofcourse,wenowrealizethatthereareaninnitenumberofwaystorealize w aselementof h S i .Let's askthesametypeofquestionagain,butthistimewith y = 2 4 2 4 3 3 5 ,i.e.is y 2h S i ? Sowe'lllookforscalars 1 ; 2 ; 3 sothat 1 u 1 + 2 u 2 + 3 u 3 = y : ByTheoremSLSLC[93]solutionstothisvectorequationarethesolutionstothesystemofequations 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [( 2 +2 3 =2 2 1 + 2 + 3 =4 1 + 2 =3 : Buildingtheaugmentedmatrixforthislinearsystem,androw-reducing,gives 2 4 1 010 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 000 1 3 5 Version2.02

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SubsectionSS.SSVSpanofaSetofVectors117 Thissystemisinconsistentthere'saleading1inthelastcolumn,TheoremRCLS[53],sothereareno scalars 1 ; 2 ; 3 thatwillcreatealinearcombinationof u 1 ; u 2 ; u 3 thatequals y .Moreprecisely, y 62h S i Therearethreethingstoobserveinthisexample.Itiseasytoconstructvectorsin h S i .Itis possiblethatsomevectorsarein h S i e.g. w ,whileothersarenote.g. y .Decidingifagivenvector isin h S i leadstosolvingalinearsystemofequationsandaskingifthesystemisconsistent. Withacomputerprograminhandtosolvesystemsoflinearequations,couldyoucreateaprogramto decideifavectorwas,orwasn't,inthespanofagivensetofvectors?Isthisartorscience? ThisexamplewasbuiltonvectorsfromthecolumnsofthecoecientmatrixofArchetypeA[702]. Studythedeterminationthat v 2h S i andseeifyoucanconnectitwithsomeoftheotherpropertiesof ArchetypeA[702]. HavinganalyzedArchetypeA[702]inExampleSCAA[114],wewillofcoursesubjectArchetypeB [707]toasimilarinvestigation. ExampleSCAB SpanofthecolumnsofArchetypeB Beginwiththenitesetofthreevectorsofsize3thatarethecolumnsofthecoecientmatrixinArchetype B[707], R = f v 1 ; v 2 ; v 3 g = 8 < : 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 5 1 3 5 ; 2 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 5 0 3 5 ; 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 7 4 3 5 9 = ; andconsidertheinniteset V = h R i .First,asanexample,notethat x = 2 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(7 5 1 3 5 + 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 5 0 3 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 7 4 3 5 = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 3 5 isin h R i ,sinceitisalinearcombinationof v 1 ; v 2 ; v 3 .Inotherwords, x 2h R i .Trysomedierentvalues of 1 ; 2 ; 3 yourself,andseewhatvectorsyoucancreateaselementsof h R i Nowaskifagivenvectorisanelementof h R i .Forexample,is z = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(33 24 5 3 5 in h R i ?Is z 2h R i ? Toanswerthisquestion,wewilllookforscalars 1 ; 2 ; 3 sothat 1 v 1 + 2 v 2 + 3 v 3 = z : ByTheoremSLSLC[93]solutionstothisvectorequationarethesolutionstothesystemofequations )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(33 5 1 +5 2 +7 3 =24 1 +4 3 =5 : Buildingtheaugmentedmatrixforthislinearsystem,androw-reducing,gives 2 4 1 00 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 1 05 00 1 2 3 5 : Thissystemhasauniquesolution, 1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 2 =5 3 =2 Version2.02

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SubsectionSS.SSNSSpanningSetsofNullSpaces118 tellingusthat )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 v 1 + v 2 + v 3 = z soweareconvincedthat z reallyisin h R i .Noticethatinthiscasewehaveonlyonewaytoanswerthe questionarmativelysincethesolutionisunique. Let'saskaboutanothervector,sayis x = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 5 in h R i ?Is x 2h R i ? Wedesirescalars 1 ; 2 ; 3 sothat 1 v 1 + 2 v 2 + 3 v 3 = x : ByTheoremSLSLC[93]solutionstothisvectorequationarethesolutionstothesystemofequations )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 1 )]TJ/F15 10.9091 Tf 10.91 0 Td [(6 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 3 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(7 5 1 +5 2 +7 3 =8 1 +4 3 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 : Buildingtheaugmentedmatrixforthislinearsystem,androw-reducing,gives 2 4 1 001 0 1 02 00 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 5 Thissystemhasauniquesolution, 1 =1 2 =2 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 tellingusthat v 1 + v 2 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 v 3 = x soweareconvincedthat x reallyisin h R i .Noticethatinthiscaseweagainhaveonlyonewaytoanswer thequestionarmativelysincethesolutionisagainunique. Wecouldcontinuetotestothervectorsformembershipin h R i ,butthereisnopoint.Aquestion aboutmembershipin h R i inevitablyleadstoasystemofthreeequationsinthethreevariables 1 ; 2 ; 3 withacoecientmatrixwhosecolumnsarethevectors v 1 ; v 2 ; v 3 .Thisparticularcoecientmatrixis nonsingular,sobyTheoremNMUS[74],thesystemisguaranteedtohaveasolution.Thissolutionis unique,butthat'snotcriticalhere.So nomatter whichvectorwemighthavechosenfor z ,wewouldhave been certain todiscoverthatitwasanelementof h R i .Stateddierently,everyvectorofsize3isin h R i or h R i = C 3 ComparethisexamplewithExampleSCAA[114],andseeifyoucanconnect z withsomeaspectsof thewrite-upforArchetypeB[707]. SubsectionSSNS SpanningSetsofNullSpaces WesawinExampleVFSAL[103]thatwhenasystemofequationsishomogeneousthesolutionsetcan beexpressedintheformdescribedbyTheoremVFSLS[99]wherethevector c isthezerovector.Wecan essentiallyignorethisvector,sothattheremainderofthetypicalexpressionforasolutionlookslikeanarbitrarylinearcombination,wherethescalarsarethefreevariablesandthevectorsare u 1 ; u 2 ; u 3 ;:::; u n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r Whichsoundsalotlikeaspan.Thisisthesubstanceofthenexttheorem. Version2.02

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SubsectionSS.SSNSSpanningSetsofNullSpaces119 TheoremSSNS SpanningSetsforNullSpaces Supposethat A isan m n matrix,and B isarow-equivalentmatrixinreducedrow-echelonformwith r nonzerorows.Let D = f d 1 ;d 2 ;d 3 ;:::;d r g bethecolumnindiceswhere B hasleading1'spivotcolumns and F = f f 1 ;f 2 ;f 3 ;:::;f n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r g bethesetofcolumnindiceswhere B doesnothaveleading1's.Construct the n )]TJ/F21 10.9091 Tf 10.909 0 Td [(r vectors z j ,1 j n )]TJ/F21 10.9091 Tf 10.909 0 Td [(r ofsize n as [ z j ] i = 8 > < > : 1if i 2 F i = f j 0if i 2 F i 6 = f j )]TJ/F15 10.9091 Tf 10.303 0 Td [([ B ] k;f j if i 2 D i = d k Thenthenullspaceof A isgivenby N A = hf z 1 ; z 2 ; z 3 ;:::; z n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r gi : Proof Considerthehomogeneoussystemwith A asacoecientmatrix, LS A; 0 .Itssetofsolutions, S ,isbyDenitionNSM[64],thenullspaceof A N A .Let B 0 denotetheresultofrow-reducingthe augmentedmatrixofthishomogeneoussystem.Sincethesystemishomogeneous,thenalcolumnof theaugmentedmatrixwillbeallzeros,andafteranynumberofrowoperationsDenitionRO[28],the columnwillstillbeallzeros.So B 0 hasanalcolumnthatistotallyzeros. NowapplyTheoremVFSLS[99]to B 0 ,afternotingthatourhomogeneoussystemmustbeconsistent TheoremHSC[62].Thevector c haszerosforeachentrythatcorrespondstoanindexin F .Forentries thatcorrespondtoanindexin D ,thevalueis )]TJ/F15 10.9091 Tf 10.303 0 Td [([ B 0 ] k;n +1 ,butfor B 0 anyentryinthenalcolumnindex n +1iszero.So c = 0 .Thevectors z j ,1 j n )]TJ/F21 10.9091 Tf 11.315 0 Td [(r areidenticaltothevectors u j ,1 j n )]TJ/F21 10.9091 Tf 11.315 0 Td [(r describedinTheoremVFSLS[99].PuttingitalltogetherandapplyingDenitionSSCV[112]inthenal step, N A = S = f c + 1 u 1 + 2 u 2 + 3 u 3 + + n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r u n )]TJ/F22 7.9701 Tf 6.586 0 Td [(r j 1 ; 2 ; 3 ;:::; n )]TJ/F22 7.9701 Tf 6.586 0 Td [(r 2 C g = f 1 u 1 + 2 u 2 + 3 u 3 + + n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r u n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r j 1 ; 2 ; 3 ;:::; n )]TJ/F22 7.9701 Tf 6.586 0 Td [(r 2 C g = hf z 1 ; z 2 ; z 3 ;:::; z n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r gi ExampleSSNS Spanningsetofanullspace Findasetofvectors, S ,sothatthenullspaceofthematrix A belowisthespanof S ,thatis, h S i = N A A = 2 6 6 4 133 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 25711 11515 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(204 3 7 7 5 Thenullspaceof A isthesetofallsolutionstothehomogeneoussystem LS A; 0 .Ifwendthevector formofthesolutionstothishomogeneoussystemTheoremVFSLS[99]thenthevectors u j ,1 j n )]TJ/F21 10.9091 Tf 9.565 0 Td [(r inthelinearcombinationareexactlythevectors z j ,1 j n )]TJ/F21 10.9091 Tf 11.027 0 Td [(r describedinTheoremSSNS[118].So wecanmimicExampleVFSAL[103]toarriveatthesevectorsratherthanbeingaslavetotheformulas inthestatementofthetheorem. Version2.02

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SubsectionSS.SSNSSpanningSetsofNullSpaces120 Beginbyrow-reducing A .Theresultis 2 6 6 4 1 0604 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 000 1 3 00000 3 7 7 5 With D = f 1 ; 2 ; 4 g and F = f 3 ; 5 g werecognizethat x 3 and x 5 arefreevariablesandwecanexpresseach nonzerorowasanexpressionforthedependentvariables x 1 x 2 x 4 respectivelyinthefreevariables x 3 and x 5 .Withthiswecanwritethevectorformofasolutionvectoras 2 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 3 7 7 7 7 5 = 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 5 x 3 +2 x 5 x 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x 5 x 5 3 7 7 7 7 5 = x 3 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 1 1 0 0 3 7 7 7 7 5 + x 5 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 2 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 3 7 7 7 7 5 TheninthenotationofTheoremSSNS[118], z 1 = 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 1 1 0 0 3 7 7 7 7 5 z 2 = 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 2 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 3 7 7 7 7 5 and N A = hf z 1 ; z 2 gi = 8 > > > > < > > > > : 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 1 1 0 0 3 7 7 7 7 5 ; 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 2 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 3 7 7 7 7 5 9 > > > > = > > > > ; + ExampleNSDS Nullspacedirectlyasaspan Let'sexpressthenullspaceof A asthespanofasetofvectors,applyingTheoremSSNS[118]aseconomicallyaspossible,withoutreferencetotheunderlyinghomogeneoussystemofequationsincontrastto ExampleSSNS[118]. A = 2 6 6 6 6 4 215151 11316 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(104 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(32 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(70 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15223 3 7 7 7 7 5 TheoremSSNS[118]createsvectorsforthespanbyrstrow-reducingthematrixinquestion.Therowreducedversionof A is B = 2 6 6 6 6 6 4 1 020 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 0 1 103 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 000 1 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 000000 000000 3 7 7 7 7 7 5 WewillmechanicallyfollowtheprescriptionofTheoremSSNS[118].Herewego,intwobigsteps. Version2.02

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SubsectionSS.SSNSSpanningSetsofNullSpaces121 First,theindicesofthenon-pivotcolumnshaveindices F = f 3 ; 5 ; 6 g ,sowewillconstructthe n )]TJ/F21 10.9091 Tf 10.401 0 Td [(r = 6 )]TJ/F15 10.9091 Tf 9.916 0 Td [(3=3vectorswithapatternofzerosandonescorrespondingtotheindicesin F .Thisistherealization ofthersttwolinesofthethree-casedenitionofthevectors z j ,1 j n )]TJ/F21 10.9091 Tf 10.909 0 Td [(r z 1 = 2 6 6 6 6 6 6 4 1 0 0 3 7 7 7 7 7 7 5 z 2 = 2 6 6 6 6 6 6 4 0 1 0 3 7 7 7 7 7 7 5 z 3 = 2 6 6 6 6 6 6 4 0 0 1 3 7 7 7 7 7 7 5 Eachofthesevectorsarisesduetothepresenceofacolumnthatisnotapivotcolumn.Theremaining entriesofeachvectoraretheentriesofthecorrespondingnon-pivotcolumn,negated,anddistributedinto theemptyslotsinordertheseslotshaveindicesintheset D andcorrespondtopivotcolumns.Thisis therealizationofthethirdlineofthethree-casedenitionofthevectors z j ,1 j n )]TJ/F21 10.9091 Tf 10.909 0 Td [(r z 1 = 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 0 0 0 3 7 7 7 7 7 7 5 z 2 = 2 6 6 6 6 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 1 0 3 7 7 7 7 7 7 5 z 3 = 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 1 0 2 0 1 3 7 7 7 7 7 7 5 So,byTheoremSSNS[118],wehave N A = hf z 1 ; z 2 ; z 3 gi = 8 > > > > > > < > > > > > > : 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 0 0 0 3 7 7 7 7 7 7 5 ; 2 6 6 6 6 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 1 0 3 7 7 7 7 7 7 5 ; 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 1 0 2 0 1 3 7 7 7 7 7 7 5 9 > > > > > > = > > > > > > ; + Weknowthatthenullspaceof A isthesolutionsetofthehomogeneoussystem LS A; 0 ,butnowhere inthisapplicationofTheoremSSNS[118]havewefoundoccasiontoreferencethevariablesorequations ofthissystem.ThesedetailsareallburiedintheproofofTheoremSSNS[118]. Moreadvancedcomputationaldeviceswillcomputethenullspaceofamatrix.See:Computation NS.MMA[669].Here'sanexamplethatwillsimultaneouslyexercisethespanconstructionandTheorem SSNS[118],whilealsopointingthewaytothenextsection. ExampleSCAD SpanofthecolumnsofArchetypeD Beginwiththesetoffourvectorsofsize3 T = f w 1 ; w 2 ; w 3 ; w 4 g = 8 < : 2 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 3 5 ; 2 4 1 4 1 3 5 ; 2 4 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 4 3 5 ; 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 5 9 = ; andconsidertheinniteset W = h T i .Thevectorsof T havebeenchosenasthefourcolumnsofthe coecientmatrixinArchetypeD[716].Checkthatthevector z 2 = 2 6 6 4 2 3 0 1 3 7 7 5 Version2.02

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SubsectionSS.SSNSSpanningSetsofNullSpaces122 isasolutiontothehomogeneoussystem LS D; 0 itisthevector z 2 providedbythedescriptionofthe nullspaceofthecoecientmatrix D fromTheoremSSNS[118].ApplyingTheoremSLSLC[93],wecan writethelinearcombination, 2 w 1 +3 w 2 +0 w 3 +1 w 4 = 0 whichwecansolvefor w 4 w 4 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 w 1 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 w 2 : Thisequationsaysthatwheneverweencounterthevector w 4 ,wecanreplaceitwithaspeciclinear combinationofthevectors w 1 and w 2 .Sousing w 4 intheset T ,alongwith w 1 and w 2 ,isexcessive.An exampleofwhatwemeanherecanbeillustratedbythecomputation, 5 w 1 + )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 w 2 +6 w 3 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 w 4 =5 w 1 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 w 2 +6 w 3 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 w 1 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 w 2 =5 w 1 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 w 2 +6 w 3 + w 1 +9 w 2 =11 w 1 +5 w 2 +6 w 3 : Sowhatbeganasalinearcombinationofthevectors w 1 ; w 2 ; w 3 ; w 4 hasbeenreducedtoalinearcombinationofthevectors w 1 ; w 2 ; w 3 .AcarefulproofusingourdenitionofsetequalityDenitionSE[684] wouldnowallowustoconcludethatthisreductionispossibleforanyvectorin W ,so W = hf w 1 ; w 2 ; w 3 gi : Sothespanofoursetofvectors, W ,hasnotchanged,butwehave described itbythespanofasetof three vectors,ratherthan four .Furthermore,wecanachieveyetanother,similar,reduction. Checkthatthevector z 1 = 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 0 3 7 7 5 isasolutiontothehomogeneoussystem LS D; 0 itisthevector z 1 providedbythedescriptionofthe nullspaceofthecoecientmatrix D fromTheoremSSNS[118].ApplyingTheoremSLSLC[93],wecan writethelinearcombination, )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 w 1 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 w 2 +1 w 3 = 0 whichwecansolvefor w 3 w 3 =3 w 1 +1 w 2 : Thisequationsaysthatwheneverweencounterthevector w 3 ,wecanreplaceitwithaspeciclinear combinationofthevectors w 1 and w 2 .So,asbefore,thevector w 3 isnotneededinthedescriptionof W providedwehave w 1 and w 2 available.Inparticular,acarefulproofsuchasisdoneinExampleRSC5 [153]wouldshowthat W = hf w 1 ; w 2 gi : So W beganlifeasthespanofasetoffourvectors,andwehavenowshownutilizingsolutionstoa homogeneoussystemthat W canalsobedescribedasthespanofasetofjusttwovectors.Convince yourselfthatwecannotgoanyfurther.Inotherwords,itisnotpossibletodismisseither w 1 or w 2 ina similarfashionandwinnowthesetdowntojustonevector. Whatwasitabouttheoriginalsetoffourvectorsthatallowedustodeclarecertainvectorsassurplus? Andjustwhichvectorswereweabletodismiss?Andwhydidwehavetostoponcewehadtwovectors remaining?Theanswerstothesequestionsmotivatelinearindependence,"ournextsectionandnext denition,andsoareworthconsideringcarefully now Itispossibletohaveyourcomputationaldevicecrankoutthevectorformofthesolutionsettoalinear systemofequations.See:ComputationVFSS.MMA[669]. Version2.02

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SubsectionSS.READReadingQuestions123 SubsectionREAD ReadingQuestions 1.LetSbethesetofthreevectorsbelow. S = 8 < : 2 4 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 5 ; 2 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 2 3 5 ; 2 4 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 3 5 9 = ; Let W = h S i bethespanofS.Isthevector 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 5 in W ?Giveanexplanationofthereasonforyour answer. 2.Use S and W fromthepreviousquestion.Isthevector 2 4 6 5 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 3 5 in W ?Giveanexplanationofthe reasonforyouranswer. 3.Forthematrix A below,ndaset S sothat h S i = N A ,where N A isthenullspaceof A .See TheoremSSNS[118]. A = 2 4 1319 21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(38 11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 3 5 Version2.02

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SubsectionSS.EXCExercises124 SubsectionEXC Exercises C22 Foreacharchetypethatisasystemofequations,considerthecorrespondinghomogeneoussystemof equations.Writeelementsofthesolutionsettothesehomogeneoussystemsinvectorform,asguaranteed byTheoremVFSLS[99].Thenwritethenullspaceofthecoecientmatrixofeachsystemasthespanof asetofvectors,asdescribedinTheoremSSNS[118]. ArchetypeA[702] ArchetypeB[707] ArchetypeC[712] ArchetypeD[716]/ArchetypeE[720] ArchetypeF[724] ArchetypeG[729]/ArchetypeH[733] ArchetypeI[737] ArchetypeJ[741] ContributedbyRobertBeezerSolution[126] C23 ArchetypeK[746]andArchetypeL[750]aredenedasmatrices.UseTheoremSSNS[118]directly tondaset S sothat h S i isthenullspaceofthematrix.Donotmakeanyreferencetotheassociated homogeneoussystemofequationsinyoursolution. ContributedbyRobertBeezerSolution[126] C40 Supposethat S = 8 > > < > > : 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 4 3 7 7 5 ; 2 6 6 4 3 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 3 7 7 5 9 > > = > > ; .Let W = h S i andlet x = 2 6 6 4 5 8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 7 7 5 .Is x 2 W ?Ifso,provide anexplicitlinearcombinationthatdemonstratesthis. ContributedbyRobertBeezerSolution[126] C41 Supposethat S = 8 > > < > > : 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 4 3 7 7 5 ; 2 6 6 4 3 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 3 7 7 5 9 > > = > > ; .Let W = h S i andlet y = 2 6 6 4 5 1 3 5 3 7 7 5 .Is y 2 W ?Ifso,providean explicitlinearcombinationthatdemonstratesthis. ContributedbyRobertBeezerSolution[126] C42 Suppose R = 8 > > > > < > > > > : 2 6 6 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 4 0 3 7 7 7 7 5 ; 2 6 6 6 6 4 1 1 2 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 7 7 5 ; 2 6 6 6 6 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 7 7 5 9 > > > > = > > > > ; .Is y = 2 6 6 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 7 7 7 7 5 in h R i ? ContributedbyRobertBeezerSolution[127] C43 Suppose R = 8 > > > > < > > > > : 2 6 6 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 4 0 3 7 7 7 7 5 ; 2 6 6 6 6 4 1 1 2 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 7 7 5 ; 2 6 6 6 6 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 7 7 5 9 > > > > = > > > > ; .Is z = 2 6 6 6 6 4 1 1 5 3 1 3 7 7 7 7 5 in h R i ? ContributedbyRobertBeezerSolution[127] Version2.02

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SubsectionSS.EXCExercises125 C44 Supposethat S = 8 < : 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 1 3 5 ; 2 4 3 1 2 3 5 ; 2 4 1 5 4 3 5 ; 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 5 1 3 5 9 = ; .Let W = h S i andlet y = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 0 3 5 .Is y 2 W ?If so,provideanexplicitlinearcombinationthatdemonstratesthis. ContributedbyRobertBeezerSolution[128] C45 Supposethat S = 8 < : 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 1 3 5 ; 2 4 3 1 2 3 5 ; 2 4 1 5 4 3 5 ; 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 5 1 3 5 9 = ; .Let W = h S i andlet w = 2 4 2 1 3 3 5 .Is w 2 W ?Ifso, provideanexplicitlinearcombinationthatdemonstratesthis. ContributedbyRobertBeezerSolution[128] C50 Let A bethematrixbelow. aFindaset S sothat N A = h S i bIf z = 2 6 6 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 1 2 3 7 7 5 ,thenshowdirectlythat z 2N A cWrite z asalinearcombinationofthevectorsin S A = 2 4 2314 1213 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1011 3 5 ContributedbyRobertBeezerSolution[129] C60 Forthematrix A below,ndasetofvectors S sothatthespanof S equalsthenullspaceof A h S i = N A A = 2 4 116 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(201 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(67 3 5 ContributedbyRobertBeezerSolution[130] M20 InExampleSCAD[120]webeganwiththefourcolumnsofthecoecientmatrixofArchetypeD [716],andusedthesecolumnsinaspanconstruction.Thenwemethodicallyarguedthatwecouldremove thelastcolumn,thenthethirdcolumn,andcreatethesamesetbyjustdoingaspanconstructionwiththe rsttwocolumns.Weclaimedwecouldnotgoanyfurther,andhadremovedasmanyvectorsaspossible. Provideaconvincingargumentforwhyathirdvectorcannotberemoved. ContributedbyRobertBeezer M21 InthespiritofExampleSCAD[120],beginwiththefourcolumnsofthecoecientmatrixof ArchetypeC[712],andusethesecolumnsinaspanconstructiontobuildtheset S .Arguethat S canbe expressedasthespanofjustthreeofthecolumnsofthecoecientmatrixsayingexactlywhichthree andinthespiritofExerciseSS.M20[124]arguethatnooneofthesethreevectorscanberemovedand stillhaveaspanconstructioncreate S ContributedbyRobertBeezerSolution[130] T10 Supposethat v 1 ; v 2 2 C m .Provethat hf v 1 ; v 2 gi = hf v 1 ; v 2 ; 5 v 1 +3 v 2 gi ContributedbyRobertBeezerSolution[130] Version2.02

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SubsectionSS.EXCExercises126 T20 Supposethat S isasetofvectorsfrom C m .Provethatthezerovector, 0 ,isanelementof h S i ContributedbyRobertBeezerSolution[131] T21 Supposethat S isasetofvectorsfrom C m and x ; y 2h S i .Provethat x + y 2h S i ContributedbyRobertBeezer T22 Supposethat S isasetofvectorsfrom C m 2 C ,and x 2h S i .Provethat x 2h S i ContributedbyRobertBeezer Version2.02

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SubsectionSS.SOLSolutions127 SubsectionSOL Solutions C22 ContributedbyRobertBeezerStatement[123] Thevectorformofthesolutionsobtainedinthismannerwillinvolvepreciselythevectorsdescribedin TheoremSSNS[118]asprovidingthenullspaceofthecoecientmatrixofthesystemasaspan.These vectorsoccurineacharchetypeinadescriptionofthenullspace.StudyingExampleVFSAL[103]may beofsomehelp. C23 ContributedbyRobertBeezerStatement[123] StudyExampleNSDS[119]tounderstandthecorrectapproachtothisquestion.Thesolutionforeachis listedintheArchetypesAppendixA[698]themselves. C40 ContributedbyRobertBeezerStatement[123] Rephrasingthequestion,wewanttoknowiftherearescalars 1 and 2 suchthat 1 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 4 3 7 7 5 + 2 2 6 6 4 3 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 3 7 7 5 = 2 6 6 4 5 8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 3 7 7 5 TheoremSLSLC[93]allowsustorephrasethequestionagainasaquestforsolutionstothesystemoffour equationsintwounknownswithanaugmentedmatrixgivenby 2 6 6 4 235 )]TJ/F15 10.9091 Tf 8.485 0 Td [(128 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 41 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 7 7 5 Thismatrixrow-reducesto 2 6 6 4 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 1 3 000 000 3 7 7 5 Fromtheformofthismatrix,wecanseethat 1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2and 2 =3isanarmativeanswertoourquestion. Moreconvincingly, )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 4 3 7 7 5 + 2 6 6 4 3 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 3 7 7 5 = 2 6 6 4 5 8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 3 7 7 5 C41 ContributedbyRobertBeezerStatement[123] Rephrasingthequestion,wewanttoknowiftherearescalars 1 and 2 suchthat 1 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 3 4 3 7 7 5 + 2 2 6 6 4 3 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 3 7 7 5 = 2 6 6 4 5 1 3 5 3 7 7 5 TheoremSLSLC[93]allowsustorephrasethequestionagainasaquestforsolutionstothesystemoffour Version2.02

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SubsectionSS.SOLSolutions128 equationsintwounknownswithanaugmentedmatrixgivenby 2 6 6 4 235 )]TJ/F15 10.9091 Tf 8.485 0 Td [(121 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 415 3 7 7 5 Thismatrixrow-reducesto 2 6 6 4 1 00 0 1 0 00 1 000 3 7 7 5 Withaleading1inthelastcolumnofthismatrixTheoremRCLS[53]wecanseethatthesystemof equationshasnosolution,sotherearenovaluesfor 1 and 2 thatwillallowustoconcludethat y isin W .So y 62 W C42 ContributedbyRobertBeezerStatement[123] Formalinearcombination,withunknownscalars,of R thatequals y a 1 2 6 6 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 4 0 3 7 7 7 7 5 + a 2 2 6 6 6 6 4 1 1 2 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 3 7 7 7 7 5 + a 3 2 6 6 6 6 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 7 7 5 = 2 6 6 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 7 7 7 7 5 Wewanttoknowiftherearevaluesforthescalarsthatmakethevectorequationtruesincethatisthe denitionofmembershipin h R i .ByTheoremSLSLC[93]anysuchvalueswillalsobesolutionstothe linearsystemrepresentedbytheaugmentedmatrix, 2 6 6 6 6 4 2131 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 320 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 423 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 7 7 7 7 5 Row-reducingthematrixyields, 2 6 6 6 6 6 4 1 00 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 00 1 2 0000 0000 3 7 7 7 7 7 5 FromthisweseethatthesystemofequationsisconsistentTheoremRCLS[53],andhasauniquesolution. Thissolutionwillprovidealinearcombinationofthevectorsin R thatequals y .So y 2 R C43 ContributedbyRobertBeezerStatement[123] Formalinearcombination,withunknownscalars,of R thatequals z a 1 2 6 6 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 4 0 3 7 7 7 7 5 + a 2 2 6 6 6 6 4 1 1 2 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 7 7 5 + a 3 2 6 6 6 6 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 7 7 5 = 2 6 6 6 6 4 1 1 5 3 1 3 7 7 7 7 5 Version2.02

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SubsectionSS.SOLSolutions129 Wewanttoknowiftherearevaluesforthescalarsthatmakethevectorequationtruesincethatisthe denitionofmembershipin h R i .ByTheoremSLSLC[93]anysuchvalueswillalsobesolutionstothe linearsystemrepresentedbytheaugmentedmatrix, 2 6 6 6 6 4 2131 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 3205 4233 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 3 7 7 7 7 5 Row-reducingthematrixyields, 2 6 6 6 6 6 4 1 000 0 1 00 00 1 0 000 1 0000 3 7 7 7 7 7 5 Withaleading1inthelastcolumn,thesystemisinconsistentTheoremRCLS[53],sothereareno scalars a 1 ;a 2 ;a 3 thatwillcreatealinearcombinationofthevectorsin R thatequal z .So z 62 R C44 ContributedbyRobertBeezerStatement[124] Formalinearcombination,withunknownscalars,of S thatequals y a 1 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 1 3 5 + a 2 2 4 3 1 2 3 5 + a 3 2 4 1 5 4 3 5 + a 4 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 5 1 3 5 = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 0 3 5 Wewanttoknowiftherearevaluesforthescalarsthatmakethevectorequationtruesincethatisthe denitionofmembershipin h S i .ByTheoremSLSLC[93]anysuchvalueswillalsobesolutionstothe linearsystemrepresentedbytheaugmentedmatrix, 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(131 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 21553 12410 3 5 Row-reducingthematrixyields, 2 4 1 0232 0 1 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 00000 3 5 FromthisweseethatthesystemofequationsisconsistentTheoremRCLS[53],andhasainnitelymany solutions.Anysolutionwillprovidealinearcombinationofthevectorsin R thatequals y .So y 2 S ,for example, )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 1 3 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 4 3 1 2 3 5 + 2 4 1 5 4 3 5 + 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 5 1 3 5 = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 0 3 5 C45 ContributedbyRobertBeezerStatement[124] Formalinearcombination,withunknownscalars,of S thatequals w a 1 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 1 3 5 + a 2 2 4 3 1 2 3 5 + a 3 2 4 1 5 4 3 5 + a 4 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 5 1 3 5 = 2 4 2 1 3 3 5 Version2.02

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SubsectionSS.SOLSolutions130 Wewanttoknowiftherearevaluesforthescalarsthatmakethevectorequationtruesincethatisthe denitionofmembershipin h S i .ByTheoremSLSLC[93]anysuchvalueswillalsobesolutionstothe linearsystemrepresentedbytheaugmentedmatrix, 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(131 )]TJ/F15 10.9091 Tf 8.485 0 Td [(62 21551 12413 3 5 Row-reducingthematrixyields, 2 4 1 0230 0 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 0000 1 3 5 Withaleading1inthelastcolumn,thesystemisinconsistentTheoremRCLS[53],sothereareno scalars a 1 ;a 2 ;a 3 ;a 4 thatwillcreatealinearcombinationofthevectorsin S thatequal w .So w 62h S i C50 ContributedbyRobertBeezerStatement[124] aTheoremSSNS[118]providesformulasforaset S withthisproperty,butrstwemustrow-reduce A A RREF )443()223()222()443(! 2 4 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 1 12 0000 3 5 x 3 and x 4 wouldbethefreevariablesinthehomogeneoussystem LS A; 0 andTheoremSSNS[118] providestheset S = f z 1 ; z 2 g where z 1 = 2 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 0 3 7 7 5 z 2 = 2 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 1 3 7 7 5 bSimplyemploythecomponentsofthevector z asthevariablesinthehomogeneoussystem LS A; 0 Thethreeequationsofthissystemevaluateasfollows, 2+3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5+1+4=0 1+2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5+1+3=0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1+0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5+1+1=0 Sinceeachresultiszero, z qualiesformembershipin N A cByTheoremSSNS[118]weknowthismustbepossiblethatisthemoralofthisexercise.Find scalars 1 and 2 sothat 1 z 1 + 2 z 2 = 1 2 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 0 3 7 7 5 + 2 2 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 1 3 7 7 5 = 2 6 6 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 1 2 3 7 7 5 = z TheoremSLSLC[93]allowsustoconvertthisquestionintoaquestionaboutasystemoffourequations intwovariables.Theaugmentedmatrixofthissystemrow-reducesto 2 6 6 4 1 01 0 1 2 000 000 3 7 7 5 Version2.02

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SubsectionSS.SOLSolutions131 Asolutionis 1 =1and 2 =2.Noticetoothatthissolutionisunique! C60 ContributedbyRobertBeezerStatement[124] TheoremSSNS[118]saysthatifwendthevectorformofthesolutionstothehomogeneoussystem LS A; 0 ,thenthexedvectorsoneperfreevariablewillhavethedesiredproperty.Row-reduce A viewingitastheaugmentedmatrixofahomogeneoussystemwithaninvisiblecolumnsofzerosasthelast column, 2 4 1 04 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 0 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0000 3 5 MovingtothevectorformofthesolutionsTheoremVFSLS[99],withfreevariables x 3 and x 4 ,solutions totheconsistentsystemitishomogeneous,TheoremHSC[62]canbeexpressedas 2 6 6 4 x 1 x 2 x 3 x 4 3 7 7 5 = x 3 2 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 1 0 3 7 7 5 + x 4 2 6 6 4 5 3 0 1 3 7 7 5 Thenwith S givenby S = 8 > > < > > : 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 0 3 7 7 5 ; 2 6 6 4 5 3 0 1 3 7 7 5 9 > > = > > ; TheoremSSNS[118]guaranteesthat N A = h S i = 8 > > < > > : 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 0 3 7 7 5 ; 2 6 6 4 5 3 0 1 3 7 7 5 9 > > = > > ; + M21 ContributedbyRobertBeezerStatement[124] IfthecolumnsofthecoecientmatrixfromArchetypeC[712]arenamed u 1 ; u 2 ; u 3 ; u 4 thenwecan discovertheequation )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 u 1 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 u 2 + u 3 + u 4 = 0 bybuildingahomogeneoussystemofequationsandviewingasolutiontothesystemasscalarsinalinear combinationviaTheoremSLSLC[93].Thisparticularvectorequationcanberearrangedtoread u 4 = u 1 + u 2 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 u 3 Thiscanbeinterpretedtomeanthat u 4 isunnecessaryin hf u 1 ; u 2 ; u 3 ; u 4 gi ,sothat hf u 1 ; u 2 ; u 3 ; u 4 gi = hf u 1 ; u 2 ; u 3 gi Ifwetrytorepeatthisprocessandndalinearcombinationof u 1 ; u 2 ; u 3 thatequalsthezerovector, wewillfail.TherequiredhomogeneoussystemofequationsviaTheoremSLSLC[93]hasonlyatrivial solution,whichwillnotprovidethekindofequationweneedtoremoveoneofthethreeremainingvectors. T10 ContributedbyRobertBeezerStatement[124] Thisisanequalityofsets,soDenitionSE[684]applies. Firstshowthat X = hf v 1 ; v 2 gihf v 1 ; v 2 ; 5 v 1 +3 v 2 gi = Y Choose x 2 X .Then x = a 1 v 1 + a 2 v 2 forsomescalars a 1 and a 2 .Then, x = a 1 v 1 + a 2 v 2 = a 1 v 1 + a 2 v 2 +0 v 1 +3 v 2 Version2.02

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SubsectionSS.SOLSolutions132 whichqualies x formembershipin Y ,asitisalinearcombinationof v 1 ; v 2 ; 5 v 1 +3 v 2 Nowshowtheoppositeinclusion, Y = hf v 1 ; v 2 ; 5 v 1 +3 v 2 gihf v 1 ; v 2 gi = X Choose y 2 Y .Thentherearescalars a 1 ;a 2 ;a 3 suchthat y = a 1 v 1 + a 2 v 2 + a 3 v 1 +3 v 2 Rearranging,weobtain, y = a 1 v 1 + a 2 v 2 + a 3 v 1 +3 v 2 = a 1 v 1 + a 2 v 2 +5 a 3 v 1 +3 a 3 v 2 PropertyDVAC[87] = a 1 v 1 +5 a 3 v 1 + a 2 v 2 +3 a 3 v 2 PropertyCC[86] = a 1 +5 a 3 v 1 + a 2 +3 a 3 v 2 PropertyDSAC[87] Thisisanexpressionfor y asalinearcombinationof v 1 and v 2 ,earning y membershipin X .Since X is asubsetof Y ,andviceversa,weseethat X = Y ,asdesired. T20 ContributedbyRobertBeezerStatement[125] Nomatterwhattheelementsoftheset S are,wecanchoosethescalarsinalinearcombinationtoallbe zero.Supposethat S = f v 1 ; v 2 ; v 3 ;:::; v p g .Thencompute 0 v 1 +0 v 2 +0 v 3 + +0 v p = 0 + 0 + 0 + + 0 = 0 Butwhatifwechoose S tobetheemptyset?The convention isthattheemptysuminDenitionSSCV [112]evaluatestozero,"inthiscasethisisthezerovector. Version2.02

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SectionLILinearIndependence133 SectionLI LinearIndependence SubsectionLISV LinearlyIndependentSetsofVectors TheoremSLSLC[93]tellsusthatasolutiontoahomogeneoussystemofequationsisalinearcombination ofthecolumnsofthecoecientmatrixthatequalsthezerovector.Weusedjustthissituationtoour advantagetwice!inExampleSCAD[120]wherewereducedthesetofvectorsusedinaspanconstruction fromfourdowntotwo,bydeclaringcertainvectorsassurplus.Thenexttwodenitionswillallowusto formalizethissituation. DenitionRLDCV RelationofLinearDependenceforColumnVectors Givenasetofvectors S = f u 1 ; u 2 ; u 3 ;:::; u n g ,atruestatementoftheform 1 u 1 + 2 u 2 + 3 u 3 + + n u n = 0 isa relationoflineardependence on S .Ifthisstatementisformedinatrivialfashion,i.e. i =0, 1 i n ,thenwesayitisthe trivialrelationoflineardependence on S 4 DenitionLICV LinearIndependenceofColumnVectors Thesetofvectors S = f u 1 ; u 2 ; u 3 ;:::; u n g is linearlydependent ifthereisarelationoflineardependenceon S thatisnottrivial.Inthecasewherethe only relationoflineardependenceon S isthetrivial one,then S isa linearlyindependent setofvectors. 4 Noticethatarelationoflineardependenceisan equation .Thoughmostofitisalinearcombination,it isnotalinearcombinationthatwouldbeavector.Linearindependenceisapropertyofa set ofvectors. Itiseasytotakeasetofvectors,andanequalnumberofscalars, allzero ,andformalinearcombination thatequalsthezerovector.Whentheeasywayisthe only way,thenwesaythesetislinearlyindependent. Here'sacoupleofexamples. ExampleLDS Linearlydependentsetin C 5 Considerthesetof n =4vectorsfrom C 5 S = 8 > > > > < > > > > : 2 6 6 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 1 2 3 7 7 7 7 5 ; 2 6 6 6 6 4 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 5 2 3 7 7 7 7 5 ; 2 6 6 6 6 4 2 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 6 1 3 7 7 7 7 5 ; 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 7 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 0 1 3 7 7 7 7 5 9 > > > > = > > > > ; : Todeterminelinearindependencewerstformarelationoflineardependence, 1 2 6 6 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 1 2 3 7 7 7 7 5 + 2 2 6 6 6 6 4 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 5 2 3 7 7 7 7 5 + 3 2 6 6 6 6 4 2 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 6 1 3 7 7 7 7 5 + 4 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 1 3 7 7 7 7 5 = 0 : Version2.02

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SubsectionLI.LISVLinearlyIndependentSetsofVectors134 Weknowthat 1 = 2 = 3 = 4 =0isasolutiontothisequation,butthatisofnointerestwhatsoever. Thatis always thecase,nomatterwhatfourvectorswemighthavechosen.Wearecurioustoknowifthere areother,nontrivial,solutions.TheoremSLSLC[93]tellsusthatwecanndsuchsolutionsassolutions tothehomogeneoussystem LS A; 0 wherethecoecientmatrixhasthesefourvectorsascolumns, A = 2 6 6 6 6 4 212 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1217 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1560 2211 3 7 7 7 7 5 : Row-reducingthiscoecientmatrixyields, 2 6 6 6 6 6 4 1 00 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 1 04 00 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0000 0000 3 7 7 7 7 7 5 : Wecouldsolvethishomogeneoussystemcompletely,butforthisexampleallweneedisonenontrivial solution.Settingthelonefreevariabletoanynonzerovalue,suchas x 4 =1,yieldsthenontrivialsolution x = 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 1 3 7 7 5 : completingourapplicationofTheoremSLSLC[93],wehave 2 2 6 6 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 1 2 3 7 7 7 7 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 2 6 6 6 6 4 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 5 2 3 7 7 7 7 5 +3 2 6 6 6 6 4 2 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 6 1 3 7 7 7 7 5 +1 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 1 3 7 7 7 7 5 = 0 : Thisisarelationoflineardependenceon S thatisnottrivial,soweconcludethat S islinearlydependent. ExampleLIS Linearlyindependentsetin C 5 Considerthesetof n =4vectorsfrom C 5 T = 8 > > > > < > > > > : 2 6 6 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 1 2 3 7 7 7 7 5 ; 2 6 6 6 6 4 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 5 2 3 7 7 7 7 5 ; 2 6 6 6 6 4 2 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 6 1 3 7 7 7 7 5 ; 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 7 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 1 1 3 7 7 7 7 5 9 > > > > = > > > > ; : Todeterminelinearindependencewerstformarelationoflineardependence, 1 2 6 6 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 1 2 3 7 7 7 7 5 + 2 2 6 6 6 6 4 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 5 2 3 7 7 7 7 5 + 3 2 6 6 6 6 4 2 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 6 1 3 7 7 7 7 5 + 4 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 1 3 7 7 7 7 5 = 0 : Version2.02

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SubsectionLI.LISVLinearlyIndependentSetsofVectors135 Weknowthat 1 = 2 = 3 = 4 =0isasolutiontothisequation,butthatisofnointerestwhatsoever. Thatis always thecase,nomatterwhatfourvectorswemighthavechosen.Wearecurioustoknowifthere areother,nontrivial,solutions.TheoremSLSLC[93]tellsusthatwecanndsuchsolutionsassolution tothehomogeneoussystem LS B; 0 wherethecoecientmatrixhasthesefourvectorsascolumns, B = 2 6 6 6 6 4 212 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1217 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1561 2211 3 7 7 7 7 5 : Row-reducingthiscoecientmatrixyields, 2 6 6 6 6 6 4 1 000 0 1 00 00 1 0 000 1 0000 3 7 7 7 7 7 5 : Fromtheformofthismatrix,weseethattherearenofreevariables,sothesolutionisunique,andbecause thesystemishomogeneous,thisuniquesolutionisthetrivialsolution.Sowenowknowthatthereisbut onewaytocombinethefourvectorsof T intoarelationoflineardependence,andthatonewayistheeasy andobviousway.Inthissituationwesaythattheset, T ,islinearlyindependent. ExampleLDS[132]andExampleLIS[133]reliedonsolvingahomogeneoussystemofequationsto determinelinearindependence.Wecancodifythisprocessinatime-savingtheorem. TheoremLIVHS LinearlyIndependentVectorsandHomogeneousSystems Supposethat A isan m n matrixand S = f A 1 ; A 2 ; A 3 ;:::; A n g isthesetofvectorsin C m thatare thecolumnsof A .Then S isalinearlyindependentsetifandonlyifthehomogeneoussystem LS A; 0 hasauniquesolution. Proof Supposethat LS A; 0 hasauniquesolution.Sinceitisahomogeneoussystem,thissolution mustbethetrivialsolution x = 0 .ByTheoremSLSLC[93],thismeansthattheonlyrelationoflinear dependenceon S isthetrivialone.So S islinearlyindependent. Wewillprovethecontrapositive.Supposethat LS A; 0 doesnothaveauniquesolution.Sinceit isahomogeneoussystem,itisconsistentTheoremHSC[62],andsomusthaveinnitelymanysolutions TheoremPSSLS[55].Oneoftheseinnitelymanysolutionsmustbenontrivialinfact,almostallof themare,sochooseone.ByTheoremSLSLC[93]thisnontrivialsolutionwillgiveanontrivialrelation oflineardependenceon S ,sowecanconcludethat S isalinearlydependentset. SinceTheoremLIVHS[134]isanequivalence,wecanuseittodeterminethelinearindependence ordependenceofanysetofcolumnvectors,justbycreatingacorrespondingmatrixandanalyzingthe row-reducedform.Let'sillustratethiswithtwomoreexamples. ExampleLIHS Linearlyindependent,homogeneoussystem Isthesetofvectors S = 8 > > > > < > > > > : 2 6 6 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 4 2 3 7 7 7 7 5 ; 2 6 6 6 6 4 6 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 4 3 7 7 7 7 5 ; 2 6 6 6 6 4 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 5 1 3 7 7 7 7 5 9 > > > > = > > > > ; Version2.02

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SubsectionLI.LISVLinearlyIndependentSetsofVectors136 linearlyindependentorlinearlydependent? TheoremLIVHS[134]suggestswestudythematrixwhosecolumnsarethevectorsin S A = 2 6 6 6 6 4 264 )]TJ/F15 10.9091 Tf 8.485 0 Td [(123 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 435 241 3 7 7 7 7 5 Specically,weareinterestedinthesizeofthesolutionsetforthehomogeneoussystem LS A; 0 .Rowreducing A ,weobtain 2 6 6 6 6 6 4 1 00 0 1 0 00 1 000 000 3 7 7 7 7 7 5 Now, r =3,sothereare n )]TJ/F21 10.9091 Tf 10.969 0 Td [(r =3 )]TJ/F15 10.9091 Tf 10.969 0 Td [(3=0freevariablesandweseethat LS A; 0 hasauniquesolution TheoremHSC[62],TheoremFVCS[55].ByTheoremLIVHS[134],theset S islinearlyindependent. ExampleLDHS Linearlydependent,homogeneoussystem Isthesetofvectors S = 8 > > > > < > > > > : 2 6 6 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 4 2 3 7 7 7 7 5 ; 2 6 6 6 6 4 6 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 4 3 7 7 7 7 5 ; 2 6 6 6 6 4 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 7 7 7 7 5 9 > > > > = > > > > ; linearlyindependentorlinearlydependent? TheoremLIVHS[134]suggestswestudythematrixwhosecolumnsarethevectorsin S A = 2 6 6 6 6 4 264 )]TJ/F15 10.9091 Tf 8.485 0 Td [(123 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 43 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 242 3 7 7 7 7 5 Specically,weareinterestedinthesizeofthesolutionsetforthehomogeneoussystem LS A; 0 .Rowreducing A ,weobtain 2 6 6 6 6 4 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 1 1 000 000 000 3 7 7 7 7 5 Now, r =2,sothereare n )]TJ/F21 10.9091 Tf 11.957 0 Td [(r =3 )]TJ/F15 10.9091 Tf 11.957 0 Td [(2=1freevariablesandweseethat LS A; 0 hasinnitely manysolutionsTheoremHSC[62],TheoremFVCS[55].ByTheoremLIVHS[134],theset S islinearly dependent. Asanequivalence,TheoremLIVHS[134]givesusastraightforwardwaytodetermineifasetofvectors islinearlyindependentordependent. ReviewExampleLIHS[134]andExampleLDHS[135].Theyareverysimilar,dieringonlyinthe lasttwoslotsofthethirdvector.Thisresultedinslightlydierentmatriceswhenrow-reduced,and Version2.02

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SubsectionLI.LISVLinearlyIndependentSetsofVectors137 slightlydierentvaluesof r ,thenumberofnonzerorows.Notice,too,thatwearelessinterestedinthe actualsolutionset,andmoreinterestedinitsformorsize.Theseobservationsallowustomakeaslight improvementinTheoremLIVHS[134]. TheoremLIVRN LinearlyIndependentVectors, r and n Supposethat A isan m n matrixand S = f A 1 ; A 2 ; A 3 ;:::; A n g isthesetofvectorsin C m thatare thecolumnsof A .Let B beamatrixinreducedrow-echelonformthatisrow-equivalentto A andlet r denotethenumberofnon-zerorowsin B .Then S islinearlyindependentifandonlyif n = r Proof TheoremLIVHS[134]saysthelinearindependenceof S isequivalenttothehomogeneouslinear system LS A; 0 havingauniquesolution.Since LS A; 0 isconsistentTheoremHSC[62]wecanapply TheoremCSRN[54]toseethatthesolutionisuniqueexactlywhen n = r Sonowhere'sanexampleofthemoststraightforwardwaytodetermineifasetofcolumnvectorsin linearlyindependentorlinearlydependent.Whilethismethodcanbequickandeasy,don'tforgetthe logicalprogressionfromthedenitionoflinearindependencethroughhomogeneoussystemofequations whichmakesitpossible. ExampleLDRN Linearlydependent, r > > > > > < > > > > > > : 2 6 6 6 6 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 1 0 3 3 7 7 7 7 7 7 5 ; 2 6 6 6 6 6 6 4 9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 2 1 3 7 7 7 7 7 7 5 ; 2 6 6 6 6 6 6 4 1 1 1 0 0 1 3 7 7 7 7 7 7 5 ; 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 4 2 1 2 3 7 7 7 7 7 7 5 ; 2 6 6 6 6 6 6 4 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 4 3 2 3 7 7 7 7 7 7 5 9 > > > > > > = > > > > > > ; linearlyindependentorlinearlydependent?TheoremLIVHS[134]suggestsweplacethesevectorsintoa matrixascolumnsandanalyzetherow-reducedversionofthematrix, 2 6 6 6 6 6 6 4 291 )]TJ/F15 10.9091 Tf 8.485 0 Td [(36 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(611 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2141 13024 02013 31122 3 7 7 7 7 7 7 5 RREF )443()223()222()443(! 2 6 6 6 6 6 6 6 4 1 000 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 1 001 00 1 02 000 1 1 00000 00000 3 7 7 7 7 7 7 7 5 Nowweneedonlycomputethat r =4 < 5= n torecognize,viaTheoremLIVHS[134]that S isalinearly dependentset.Boom! ExampleLLDS Largelinearlydependentsetin C 4 Considerthesetof n =9vectorsfrom C 4 R = 8 > > < > > : 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 1 2 3 7 7 5 ; 2 6 6 4 7 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 6 3 7 7 5 ; 2 6 6 4 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 5 ; 2 6 6 4 0 4 2 9 3 7 7 5 ; 2 6 6 4 5 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 4 3 3 7 7 5 ; 2 6 6 4 2 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 4 3 7 7 5 ; 2 6 6 4 3 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 3 7 7 5 ; 2 6 6 4 1 1 5 3 3 7 7 5 ; 2 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 1 1 3 7 7 5 9 > > = > > ; : ToemployTheoremLIVHS[134],weforma4 9coecientmatrix, C C = 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(17105231 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 3124 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2101 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(124 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(351 26 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2934131 3 7 7 5 : Version2.02

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SubsectionLI.LINMLinearIndependenceandNonsingularMatrices138 Todetermineifthehomogeneoussystem LS C; 0 hasauniquesolutionornot,wewouldnormallyrowreducethismatrix.Butinthisparticularexample,wecandobetter.TheoremHMVEI[64]tellsusthat sincethesystemishomogeneouswith n =9variablesin m =4equations,and n>m ,theremustbe innitelymanysolutions.Sincethereisnotauniquesolution,TheoremLIVHS[134]saysthesetislinearly dependent. ThesituationinExampleLLDS[136]isslickenoughtowarrantformulatingasatheorem. TheoremMVSLD MoreVectorsthanSizeimpliesLinearDependence Supposethat S = f u 1 ; u 2 ; u 3 ;:::; u n g isthesetofvectorsin C m ,andthat n>m .Then S isalinearly dependentset. Proof Formthe m n coecientmatrix A thathasthecolumnvectors u i ,1 i n asitscolumns. Considerthehomogeneoussystem LS A; 0 .ByTheoremHMVEI[64]thissystemhasinnitelymany solutions.Sincethesystemdoesnothaveauniquesolution,TheoremLIVHS[134]saysthecolumnsof A formalinearlydependentset,whichisthedesiredconclusion. SubsectionLINM LinearIndependenceandNonsingularMatrices Wewillnowspecializetosetsof n vectorsfrom C n .ThiswillputTheoremMVSLD[137]o-limits,while TheoremLIVHS[134]willinvolvesquarematrices.Let'sbeginbycontrastingArchetypeA[702]and ArchetypeB[707]. ExampleLDCAA LinearlydependentcolumnsinArchetypeA ArchetypeA[702]isasystemoflinearequationswithcoecientmatrix, A = 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 211 110 3 5 : Dothecolumnsofthismatrixformalinearlyindependentordependentset?ByExampleS[71]we knowthat A issingular.Accordingtothedenitionofnonsingularmatrices,DenitionNM[71],the homogeneoussystem LS A; 0 hasinnitelymanysolutions.SobyTheoremLIVHS[134],thecolumnsof A formalinearlydependentset. ExampleLICAB LinearlyindependentcolumnsinArchetypeB ArchetypeB[707]isasystemoflinearequationswithcoecientmatrix, B = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 557 104 3 5 : Dothecolumnsofthismatrixformalinearlyindependentordependentset?ByExampleNM[72]we knowthat B isnonsingular.Accordingtothedenitionofnonsingularmatrices,DenitionNM[71],the homogeneoussystem LS A; 0 hasauniquesolution.SobyTheoremLIVHS[134],thecolumnsof B form alinearlyindependentset. ThatArchetypeA[702]andArchetypeB[707]haveoppositepropertiesforthecolumnsoftheir coecientmatricesisnoaccident.Here'sthetheorem,andthenwewillupdateourequivalencesfor nonsingularmatrices,TheoremNME1[75]. Version2.02

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SubsectionLI.NSSLINullSpaces,Spans,LinearIndependence139 TheoremNMLIC NonsingularMatriceshaveLinearlyIndependentColumns Supposethat A isasquarematrix.Then A isnonsingularifandonlyifthecolumnsof A formalinearly independentset. Proof Thisisaproofwherewecanchaintogetherequivalences,ratherthanprovingthetwohalves separately. A nonsingular LS A; 0 hasauniquesolutionDenitionNM[71] columnsof A arelinearlyindependentTheoremLIVHS[134] Here'sanupdatetoTheoremNME1[75]. TheoremNME2 NonsingularMatrixEquivalences,Round2 Supposethat A isasquarematrix.Thefollowingareequivalent. 1. A isnonsingular. 2. A row-reducestotheidentitymatrix. 3.Thenullspaceof A containsonlythezerovector, N A = f 0 g 4.Thelinearsystem LS A; b hasauniquesolutionforeverypossiblechoiceof b 5.Thecolumnsof A formalinearlyindependentset. Proof TheoremNMLIC[138]isyetanotherequivalenceforanonsingularmatrix,sowecanadditto thelistinTheoremNME1[75]. SubsectionNSSLI NullSpaces,Spans,LinearIndependence InSubsectionSS.SSNS[117]weprovedTheoremSSNS[118]whichprovided n )]TJ/F21 10.9091 Tf 11.114 0 Td [(r vectorsthatcouldbe usedwiththespanconstructiontobuildtheentirenullspaceofamatrix.AswehavehintedinExample SCAD[120],andaswewillseeagaingoingforward,linearlydependentsetscarryredundantvectors withthemwhenusedinbuildingasetasaspan.Ouraimnowistoshowthatthevectorsprovidedby TheoremSSNS[118]formalinearlyindependentset,soinonesensetheyareasecientaspossiblea waytodescribethenullspace.Noticethatthevectors z j ,1 j n )]TJ/F21 10.9091 Tf 11.126 0 Td [(r rstappearinthevectorform ofsolutionstoarbitrarylinearsystemsTheoremVFSLS[99].Theexactsamevectorsappearagainin thespanconstructionintheconclusionofTheoremSSNS[118].Sincethissecondtheoremspecializes tohomogeneoussystemstheonlyrealdierenceisthatthevector c inTheoremVFSLS[99]isthezero vectorforahomogeneoussystem.Finally,TheoremBNS[139]willnowshowthatthesesamevectorsarea linearlyindependentset.We'llsetthestagefortheproofofthistheoremwithamoderatelylargeexample. Studytheexamplecarefully,asitwillmakeiteasiertounderstandtheproof. ExampleLINSB Linearindependenceofnullspacebasis Version2.02

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SubsectionLI.NSSLINullSpaces,Spans,LinearIndependence140 Supposethatweareinterestedinthenullspaceofthea3 7matrix, A ,whichrow-reducesto B = 2 4 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(24039 0 1 56071 0000 1 8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 5 Theset F = f 3 ; 4 ; 6 ; 7 g isthesetofindicesforourfourfreevariablesthatwouldbeusedinadescription ofthesolutionsetforthehomogeneoussystem LS A; 0 .ApplyingTheoremSSNS[118]wecanbeginto constructasetoffourvectorswhosespanisthenullspaceof A ,asetofvectorswewillreferenceas T N A = h T i = hf z 1 ; z 2 ; z 3 ; z 4 gi = 8 > > > > > > > > < > > > > > > > > : 2 6 6 6 6 6 6 6 6 4 1 0 0 0 3 7 7 7 7 7 7 7 7 5 ; 2 6 6 6 6 6 6 6 6 4 0 1 0 0 3 7 7 7 7 7 7 7 7 5 ; 2 6 6 6 6 6 6 6 6 4 0 0 1 0 3 7 7 7 7 7 7 7 7 5 ; 2 6 6 6 6 6 6 6 6 4 0 0 0 1 3 7 7 7 7 7 7 7 7 5 9 > > > > > > > > = > > > > > > > > ; + Sofar,wehaveconstructedasmuchoftheseindividualvectorsaswecan,basedjustontheknowledgeof thecontentsoftheset F .Thishasallowedustodeterminetheentriesinslots3,4,6and7,whilewehave leftslots1,2and5blank.Withoutdoinganymore,letsaskif T islinearlyindependent?Beginwitha relationoflineardependenceon T ,andseewhatwecanlearnaboutthescalars, 0 = 1 z 1 + 2 z 2 + 3 z 3 + 4 z 4 2 6 6 6 6 6 6 6 6 4 0 0 0 0 0 0 0 3 7 7 7 7 7 7 7 7 5 = 1 2 6 6 6 6 6 6 6 6 4 1 0 0 0 3 7 7 7 7 7 7 7 7 5 + 2 2 6 6 6 6 6 6 6 6 4 0 1 0 0 3 7 7 7 7 7 7 7 7 5 + 3 2 6 6 6 6 6 6 6 6 4 0 0 1 0 3 7 7 7 7 7 7 7 7 5 + 4 2 6 6 6 6 6 6 6 6 4 0 0 0 1 3 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 4 1 0 0 0 3 7 7 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 6 6 4 0 2 0 0 3 7 7 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 6 6 4 0 0 3 0 3 7 7 7 7 7 7 7 7 5 + 2 6 6 6 6 6 6 6 6 4 0 0 0 4 3 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 4 1 2 3 4 3 7 7 7 7 7 7 7 7 5 ApplyingDenitionCVE[84]tothetwoendsofthischainofequalities,weseethat 1 = 2 = 3 = 4 =0. Sotheonlyrelationoflineardependenceontheset T isatrivialone.ByDenitionLICV[132]theset T islinearlyindependent.Theimportantfeatureofthisexampleishowthepatternofzerosandones"in thefourvectorsledtotheconclusionoflinearindependence. TheproofofTheoremBNS[139]isreallyquitestraightforward,andreliesonthepatternofzeros andones"thatariseinthevectors z i ,1 i n )]TJ/F21 10.9091 Tf 11.153 0 Td [(r intheentriesthatcorrespondtothefreevariables. PlayalongwithExampleLINSB[138]asyoustudytheproof.Also,takealookatExampleVFSAD [95],ExampleVFSAI[102]andExampleVFSAL[103],especiallyattheconclusionofStep2temporarily ignoretheconstructionoftheconstantvector, c .Thisproofisalsoagoodrstexampleofhowtoprove aconclusionthatstatesasetislinearlyindependent. TheoremBNS BasisforNullSpaces Supposethat A isan m n matrix,and B isarow-equivalentmatrixinreducedrow-echelonformwith r Version2.02

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SubsectionLI.NSSLINullSpaces,Spans,LinearIndependence141 nonzerorows.Let D = f d 1 ;d 2 ;d 3 ;:::;d r g and F = f f 1 ;f 2 ;f 3 ;:::;f n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r g bethesetsofcolumnindices where B doesanddoesnotrespectivelyhaveleading1's.Constructthe n )]TJ/F21 10.9091 Tf 10.955 0 Td [(r vectors z j ,1 j n )]TJ/F21 10.9091 Tf 10.955 0 Td [(r ofsize n as [ z j ] i = 8 > < > : 1if i 2 F i = f j 0if i 2 F i 6 = f j )]TJ/F15 10.9091 Tf 10.303 0 Td [([ B ] k;f j if i 2 D i = d k Denetheset S = f z 1 ; z 2 ; z 3 ;:::; z n )]TJ/F22 7.9701 Tf 6.586 0 Td [(r g .Then 1. N A = h S i 2. S isalinearlyindependentset. Proof Noticerstthatthevectors z j ,1 j n )]TJ/F21 10.9091 Tf 10.52 0 Td [(r areexactlythesameasthe n )]TJ/F21 10.9091 Tf 10.52 0 Td [(r vectorsdenedin TheoremSSNS[118].Also,thehypothesesofTheoremSSNS[118]arethesameasthehypothesesofthe theoremwearecurrentlyproving.SoitisthensimplytheconclusionofTheoremSSNS[118]thattellsus that N A = h S i .Thatwastheeasyhalf,butthesecondpartisnotmuchharder.Whatisnewhereis theclaimthat S isalinearlyindependentset. Toprovethelinearindependenceofaset,weneedtostartwitharelationoflineardependenceand somehowconcludethatthescalarsinvolved mustallbezero ,i.e.thattherelationoflineardependence onlyhappensinthetrivialfashion.Sotoestablishthelinearindependenceof S ,westartwith 1 z 1 + 2 z 2 + 3 z 3 + + n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r z n )]TJ/F22 7.9701 Tf 6.586 0 Td [(r = 0 : Foreach j ,1 j n )]TJ/F21 10.9091 Tf 11.231 0 Td [(r ,considertheequalityoftheindividualentriesofthevectorsonbothsidesof thisequalityinposition f j 0=[ 0 ] f j =[ 1 z 1 + 2 z 2 + 3 z 3 + + n )]TJ/F22 7.9701 Tf 6.586 0 Td [(r z n )]TJ/F22 7.9701 Tf 6.586 0 Td [(r ] f j DenitionCVE[84] =[ 1 z 1 ] f j +[ 2 z 2 ] f j +[ 3 z 3 ] f j + +[ n )]TJ/F22 7.9701 Tf 6.586 0 Td [(r z n )]TJ/F22 7.9701 Tf 6.586 0 Td [(r ] f j DenitionCVA[84] = 1 [ z 1 ] f j + 2 [ z 2 ] f j + 3 [ z 3 ] f j + + j )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 [ z j )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 ] f j + j [ z j ] f j + j +1 [ z j +1 ] f j + + n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r [ z n )]TJ/F22 7.9701 Tf 6.586 0 Td [(r ] f j DenitionCVSM[85] = 1 + 2 + 3 + + j )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 + j + j +1 + + n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r Denitionof z j = j Soforall j ,1 j n )]TJ/F21 10.9091 Tf 11.025 0 Td [(r ,wehave j =0,whichistheconclusionthattellsusthatthe only relationof lineardependenceon S = f z 1 ; z 2 ; z 3 ;:::; z n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r g isthetrivialone.Hence,byDenitionLICV[132]the setislinearlyindependent,asdesired. ExampleNSLIL Nullspacespannedbylinearlyindependentset,ArchetypeL InExampleVFSAL[103]wepreviewedTheoremSSNS[118]byndingasetoftwovectorssuchthattheir spanwasthenullspaceforthematrixinArchetypeL[750].Writingthematrixas L ,wehave N L = 8 > > > > < > > > > : 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 1 0 3 7 7 7 7 5 ; 2 6 6 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 0 1 3 7 7 7 7 5 9 > > > > = > > > > ; + : Version2.02

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SubsectionLI.READReadingQuestions142 Solvingthehomogeneoussystem LS L; 0 resultedinrecognizing x 4 and x 5 asthefreevariables.Solook inentries4and5ofthetwovectorsaboveandnoticethepatternofzerosandonesthatprovidesthelinear independenceoftheset. SubsectionREAD ReadingQuestions 1.Let S bethesetofthreevectorsbelow. S = 8 < : 2 4 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 5 ; 2 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 2 3 5 ; 2 4 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 3 5 9 = ; Is S linearlyindependentorlinearlydependent?Explainwhy. 2.Let S bethesetofthreevectorsbelow. S = 8 < : 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 3 5 ; 2 4 3 2 2 3 5 ; 2 4 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 5 9 = ; Is S linearlyindependentorlinearlydependent?Explainwhy. 3.Basedonyouranswertothepreviousquestion,isthematrixbelowsingularornonsingular?Explain. 2 4 134 )]TJ/F15 10.9091 Tf 8.485 0 Td [(123 02 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 5 Version2.02

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SubsectionLI.EXCExercises143 SubsectionEXC Exercises DetermineifthesetsofvectorsinExercisesC20{C25arelinearlyindependentorlinearlydependent. C20 8 < : 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 3 5 ; 2 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 3 5 ; 2 4 1 5 0 3 5 9 = ; ContributedbyRobertBeezerSolution[146] C21 8 > > < > > : 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 4 2 3 7 7 5 ; 2 6 6 4 3 3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 3 3 7 7 5 ; 2 6 6 4 7 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 4 3 7 7 5 9 > > = > > ; ContributedbyRobertBeezerSolution[146] C22 8 < : 2 4 1 5 1 3 5 ; 2 4 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 5 ; 2 4 9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 8 3 5 ; 2 4 2 8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 5 ; 2 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 3 5 9 = ; ContributedbyRobertBeezerSolution[146] C23 8 > > > > < > > > > : 2 6 6 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 5 3 3 7 7 7 7 5 ; 2 6 6 6 6 4 3 3 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 7 7 7 7 5 ; 2 6 6 6 6 4 2 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 3 7 7 7 7 5 ; 2 6 6 6 6 4 1 0 1 2 2 3 7 7 7 7 5 9 > > > > = > > > > ; ContributedbyRobertBeezerSolution[146] C24 8 > > > > < > > > > : 2 6 6 6 6 4 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 1 3 7 7 7 7 5 ; 2 6 6 6 6 4 3 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 2 3 7 7 7 7 5 ; 2 6 6 6 6 4 4 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 3 3 7 7 7 7 5 ; 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 3 7 7 7 7 5 9 > > > > = > > > > ; ContributedbyRobertBeezerSolution[146] C25 8 > > > > < > > > > : 2 6 6 6 6 4 2 1 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 7 7 7 7 5 ; 2 6 6 6 6 4 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 3 2 3 7 7 7 7 5 ; 2 6 6 6 6 4 10 )]TJ/F15 10.9091 Tf 8.484 0 Td [(7 0 10 4 3 7 7 7 7 5 9 > > > > = > > > > ; ContributedbyRobertBeezerSolution[147] C30 Forthematrix B below,ndaset S thatislinearlyindependentandspansthenullspaceof B thatis, N B = h S i B = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(31 )]TJ/F15 10.9091 Tf 8.485 0 Td [(27 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1214 112 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 5 ContributedbyRobertBeezerSolution[147] C31 Forthematrix A below,ndalinearlyindependentset S sothatthenullspaceof A isspannedby Version2.02

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SubsectionLI.EXCExercises144 S ,thatis, N A = h S i A = 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2215 12115 36127 24012 3 7 7 5 ContributedbyRobertBeezerSolution[147] C32 Findasetofcolumnvectors, T ,suchthatthespanof T isthenullspaceof B h T i = N B and T isalinearlyindependentset. B = 2 4 2111 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(31 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 3 5 ContributedbyRobertBeezerSolution[148] C33 Findaset S sothat S islinearlyindependentand N A = h S i ,where N A isthenullspaceofthe matrix A below. A = 2 4 23314 11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 32 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 3 5 ContributedbyRobertBeezerSolution[148] C50 Considereacharchetypethatisasystemofequationsandconsiderthesolutionslistedforthe homogeneousversionofthearchetype.Ifonlythetrivialsolutionislisted,thenassumethisistheonly solutiontothesystem.Fromthesolutionset,determineifthecolumnsofthecoecientmatrixform alinearlyindependentorlinearlydependentset.Inthecaseofalinearlydependentset,useoneofthe samplesolutionstoprovideanontrivialrelationoflineardependenceonthesetofcolumnsofthecoecient matrixDenitionRLD[308].IndicatewhenTheoremMVSLD[137]appliesandconnectthiswiththe numberofvariablesandequationsinthesystemofequations. ArchetypeA[702] ArchetypeB[707] ArchetypeC[712] ArchetypeD[716]/ArchetypeE[720] ArchetypeF[724] ArchetypeG[729]/ArchetypeH[733] ArchetypeI[737] ArchetypeJ[741] ContributedbyRobertBeezer C51 Foreacharchetypethatisasystemofequationsconsiderthehomogeneousversion.Writeelements ofthesolutionsetinvectorformTheoremVFSLS[99]andfromthisextractthevectors z j described inTheoremBNS[139].Thesevectorsareusedinaspanconstructiontodescribethenullspaceofthe coecientmatrixforeacharchetype.Whatdoesitmeanwhenwewriteanullspaceas hfgi ? ArchetypeA[702] ArchetypeB[707] ArchetypeC[712] ArchetypeD[716]/ArchetypeE[720] ArchetypeF[724] Version2.02

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SubsectionLI.EXCExercises145 ArchetypeG[729]/ArchetypeH[733] ArchetypeI[737] ArchetypeJ[741] ContributedbyRobertBeezer C52 Foreacharchetypethatisasystemofequationsconsiderthehomogeneousversion.Samplesolutions aregivenandalinearlyindependentspanningsetisgivenforthenullspaceofthecoecientmatrix.Write eachofthesamplesolutionsindividuallyasalinearcombinationofthevectorsinthespanningsetforthe nullspaceofthecoecientmatrix. ArchetypeA[702] ArchetypeB[707] ArchetypeC[712] ArchetypeD[716]/ArchetypeE[720] ArchetypeF[724] ArchetypeG[729]/ArchetypeH[733] ArchetypeI[737] ArchetypeJ[741] ContributedbyRobertBeezer C60 Forthematrix A below,ndasetofvectors S sothat S islinearlyindependent,andthe spanof S equalsthenullspaceof A h S i = N A .SeeExerciseSS.C60[124]. A = 2 4 116 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(201 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(67 3 5 ContributedbyRobertBeezerSolution[149] M50 Considerthesetofvectorsfrom C 3 W ,givenbelow.Findaset T thatcontainsthreevectorsfrom W andsuchthat W = h T i W = hf v 1 ; v 2 ; v 3 ; v 4 ; v 5 gi = 8 < : 2 4 2 1 1 3 5 ; 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 3 5 ; 2 4 1 2 3 3 5 ; 2 4 3 1 3 3 5 ; 2 4 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 5 9 = ; + ContributedbyRobertBeezerSolution[149] T10 Provethatifasetofvectorscontainsthezerovector,thenthesetislinearlydependent.Ed.The zerovectorisdeathtolinearlyindependentsets." ContributedbyMartinJackson T12 Supposethat S isalinearlyindependentsetofvectors,and T isasubsetof S T S Denition SSET[683].Provethat T islinearlyindependent. ContributedbyRobertBeezer T13 Supposethat T isalinearlydependentsetofvectors,and T isasubsetof S T S Denition SSET[683].Provethat S islinearlydependent. ContributedbyRobertBeezer T15 Supposethat f v 1 ; v 2 ; v 3 ;:::; v n g isasetofvectors.Provethat f v 1 )]TJ/F36 10.9091 Tf 10.909 0 Td [(v 2 ; v 2 )]TJ/F36 10.9091 Tf 10.909 0 Td [(v 3 ; v 3 )]TJ/F36 10.9091 Tf 10.909 0 Td [(v 4 ;:::; v n )]TJ/F36 10.9091 Tf 10.909 0 Td [(v 1 g Version2.02

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SubsectionLI.EXCExercises146 isalinearlydependentset. ContributedbyRobertBeezerSolution[150] T20 Supposethat f v 1 ; v 2 ; v 3 ; v 4 g isalinearlyindependentsetin C 35 .Provethat f v 1 ; v 1 + v 2 ; v 1 + v 2 + v 3 ; v 1 + v 2 + v 3 + v 4 g isalinearlyindependentset. ContributedbyRobertBeezerSolution[150] T50 Supposethat A isan m n matrixwithlinearlyindependentcolumnsandthelinearsystem LS A; b isconsistent.Showthatthissystemhasauniquesolution.Noticethatwearenotrequiring A tobesquare. ContributedbyRobertBeezerSolution[151] Version2.02

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SubsectionLI.SOLSolutions147 SubsectionSOL Solutions C20 ContributedbyRobertBeezerStatement[142] Withthreevectorsfrom C 3 ,wecanformasquarematrixbymakingthesethreevectorsthecolumnsofa matrix.Wedoso,androw-reducetoobtain, 2 4 1 00 0 1 0 00 1 3 5 the3 3identitymatrix.SobyTheoremNME2[138]theoriginalmatrixisnonsingularanditscolumns arethereforealinearlyindependentset. C21 ContributedbyRobertBeezerStatement[142] TheoremLIVRN[136]sayswecananswerthisquestionbyputtingthesesvectorsintoamatrixascolumns androw-reducing.Doingthisweobtain, 2 6 6 4 1 00 0 1 0 00 1 000 3 7 7 5 With n =3vectors,3columnsand r =3leading1'swehave n = r andthetheoremsaysthe vectorsarelinearlyindependent. C22 ContributedbyRobertBeezerStatement[142] Fivevectorsfrom C 3 .TheoremMVSLD[137]saysthesetislinearlydependent.Boom. C23 ContributedbyRobertBeezerStatement[142] TheoremLIVRN[136]suggestsweanalyzeamatrixwhosecolumnsarethevectorsof S A = 2 6 6 6 6 4 1321 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2310 2121 52 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(412 3 7 7 7 7 5 Row-reducingthematrix A yields, 2 6 6 6 6 6 4 1 000 0 1 00 00 1 0 000 1 0000 3 7 7 7 7 7 5 Weseethat r =4= n ,where r isthenumberofnonzerorowsand n isthenumberofcolumns.By TheoremLIVRN[136],theset S islinearlyindependent. C24 ContributedbyRobertBeezerStatement[142] TheoremLIVRN[136]suggestsweanalyzeamatrixwhosecolumnsarethevectorsfromtheset, A = 2 6 6 6 6 4 134 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2242 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 022 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1230 3 7 7 7 7 5 Version2.02

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SubsectionLI.SOLSolutions148 Row-reducingthematrix A yields, 2 6 6 6 6 4 1 012 0 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0000 0000 0000 3 7 7 7 7 5 Weseethat r =2 6 =4= n ,where r isthenumberofnonzerorowsand n isthenumberofcolumns.By TheoremLIVRN[136],theset S islinearlydependent. C25 ContributedbyRobertBeezerStatement[142] TheoremLIVRN[136]suggestsweanalyzeamatrixwhosecolumnsarethevectorsfromtheset, A = 2 6 6 6 6 4 2410 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 310 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1310 224 3 7 7 7 7 5 Row-reducingthematrix A yields, 2 6 6 6 6 4 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 1 3 000 000 000 3 7 7 7 7 5 Weseethat r =2 6 =3= n ,where r isthenumberofnonzerorowsand n isthenumberofcolumns.By TheoremLIVRN[136],theset S islinearlydependent. C30 ContributedbyRobertBeezerStatement[142] TherequestedsetisdescribedbyTheoremBNS[139].ItiseasiesttondbyusingtheprocedureofExample VFSAL[103].Beginbyrow-reducingthematrix,viewingitasthecoecientmatrixofahomogeneous systemofequations.Weobtain, 2 4 1 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 1 11 0000 3 5 NowbuildthevectorformofthesolutionstothishomogeneoussystemTheoremVFSLS[99].Thefree variablesare x 3 and x 4 ,correspondingtothecolumnswithoutleading1's, 2 6 6 4 x 1 x 2 x 3 x 4 3 7 7 5 = x 3 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 0 3 7 7 5 + x 4 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 1 3 7 7 5 Thedesiredset S issimplytheconstantvectorsinthisexpression,andthesearethevectors z 1 and z 2 describedbyTheoremBNS[139]. S = 8 > > < > > : 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 0 3 7 7 5 ; 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 1 3 7 7 5 9 > > = > > ; C31 ContributedbyRobertBeezerStatement[142] TheoremBNS[139]providesformulasfor n )]TJ/F21 10.9091 Tf 11.118 0 Td [(r vectorsthatwillmeettherequirementsofthisquestion. Version2.02

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SubsectionLI.SOLSolutions149 ThesevectorsarethesameoneslistedinTheoremVFSLS[99]whenwesolvethehomogeneoussystem LS A; 0 ,whosesolutionsetisthenullspaceDenitionNSM[64]. ToapplyTheoremBNS[139]orTheoremVFSLS[99]werstrow-reducethematrix,resultingin B = 2 6 6 4 1 2003 00 1 06 000 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 00000 3 7 7 5 Soweseethat n )]TJ/F21 10.9091 Tf 10.909 0 Td [(r =5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3=2and F = f 2 ; 5 g ,sothevectorformofagenericsolutionvectoris 2 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 3 7 7 7 7 5 = x 2 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 0 0 0 3 7 7 7 7 5 + x 5 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 4 1 3 7 7 7 7 5 Sowehave N A = 8 > > > > < > > > > : 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 0 0 0 3 7 7 7 7 5 ; 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 4 1 3 7 7 7 7 5 9 > > > > = > > > > ; + C32 ContributedbyRobertBeezerStatement[143] TheconclusionofTheoremBNS[139]givesuseverythingthisquestionasksfor.Weneedthereduced row-echelonformofthematrixsowecandeterminethenumberofvectorsin T ,andtheirentries. 2 4 2111 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(31 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 3 5 RREF )443()223()222()443(! 2 4 1 02 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(35 0000 3 5 Wecanbuildtheset T inimmediatelyviaTheoremBNS[139],butwewillillustrateitsconstructionin twosteps.Since F = f 3 ; 4 g ,wewillhavetwovectorsandcandistributestrategicallyplacedones,and manyzeros.Thenwedistributethenegativesoftheappropriateentriesofthenon-pivotcolumnsofthe reducedrow-echelonmatrix. T = 8 > > < > > : 2 6 6 4 1 0 3 7 7 5 ; 2 6 6 4 0 1 3 7 7 5 9 > > = > > ; T = 8 > > < > > : 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 1 0 3 7 7 5 ; 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 0 1 3 7 7 5 9 > > = > > ; C33 ContributedbyRobertBeezerStatement[143] AdirectapplicationofTheoremBNS[139]willprovidethedesiredset.Werequirethereducedrow-echelon formof A 2 4 23314 11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 32 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 3 5 RREF )443()223()222()443(! 2 4 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(603 0 1 50 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 000 1 4 3 5 Thenon-pivotcolumnshaveindices F = f 3 ; 5 g .Webuildthedesiredsetintwosteps,rstplacingthe requisitezerosandonesinlocationsbasedon F ,thenplacingthenegativesoftheentriesofcolumns3and Version2.02

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SubsectionLI.SOLSolutions150 5intheproperlocations.ThisisallspeciedinTheoremBNS[139]. S = 8 > > > > < > > > > : 2 6 6 6 6 4 1 0 3 7 7 7 7 5 ; 2 6 6 6 6 4 0 1 3 7 7 7 7 5 9 > > > > = > > > > ; = 8 > > > > < > > > > : 2 6 6 6 6 4 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 1 0 0 3 7 7 7 7 5 ; 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 2 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 1 3 7 7 7 7 5 9 > > > > = > > > > ; C60 ContributedbyRobertBeezerStatement[144] TheoremBNS[139]saysthatifwendthevectorformofthesolutionstothehomogeneoussystem LS A; 0 ,thenthexedvectorsoneperfreevariablewillhavethedesiredproperties.Row-reduce A viewingitastheaugmentedmatrixofahomogeneoussystemwithaninvisiblecolumnsofzerosasthelast column, 2 4 1 04 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 0 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0000 3 5 MovingtothevectorformofthesolutionsTheoremVFSLS[99],withfreevariables x 3 and x 4 ,solutions totheconsistentsystemitishomogeneous,TheoremHSC[62]canbeexpressedas 2 6 6 4 x 1 x 2 x 3 x 4 3 7 7 5 = x 3 2 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 1 0 3 7 7 5 + x 4 2 6 6 4 5 3 0 1 3 7 7 5 Thenwith S givenby S = 8 > > < > > : 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 0 3 7 7 5 ; 2 6 6 4 5 3 0 1 3 7 7 5 9 > > = > > ; TheoremBNS[139]guaranteesthesethasthedesiredproperties. M50 ContributedbyRobertBeezerStatement[144] Wewanttorstndsomerelationsoflineardependenceon f v 1 ; v 2 ; v 3 ; v 4 ; v 5 g thatwillallowusto kickout"somevectors,inthespiritofExampleSCAD[120].Tondrelationsoflineardependence,we formulateamatrix A whosecolumnsare v 1 ; v 2 ; v 3 ; v 4 ; v 5 .Thenweconsiderthehomogeneoussystemof equations LS A; 0 byrow-reducingitscoecientmatrixrememberthatifweformulatedtheaugmented matrixwewouldjustaddacolumnofzeros.Afterrow-reducing,weobtain 2 4 1 002 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 0 1 01 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 00 1 00 3 5 Fromthiswethatsolutionscanbeobtainedemployingthefreevariables x 4 and x 5 .Withappropriate choiceswewillbeabletoconcludethatvectors v 4 and v 5 areunnecessaryforcreating W viaaspan.By TheoremSLSLC[93]thechoiceoffreevariablesbelowleadtosolutionsandlinearcombinations,which arethenrearranged. x 4 =1 ;x 5 =0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 v 1 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 v 2 + v 3 + v 4 + v 5 = 0 v 4 =2 v 1 + v 2 x 4 =0 ;x 5 =1 v 1 + v 2 + v 3 + v 4 + v 5 = 0 v 5 = )]TJ/F36 10.9091 Tf 8.485 0 Td [(v 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 v 2 Version2.02

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SubsectionLI.SOLSolutions151 Since v 4 and v 5 canbeexpressedaslinearcombinationsof v 1 and v 2 wecansaythat v 4 and v 5 arenot neededforthelinearcombinationsusedtobuild W aclaimthatwecouldestablishcarefullywithapair ofsetequalityarguments.Thus W = hf v 1 ; v 2 ; v 3 gi = 8 < : 2 4 2 1 1 3 5 ; 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 3 5 ; 2 4 1 2 3 3 5 9 = ; + Thatthe f v 1 ; v 2 ; v 3 g islinearlyindependentsetcanbeestablishedquicklywithTheoremLIVRN[136]. Thereareotheranswerstothisquestion,butnoticethatanynontriviallinearcombinationof v 1 ; v 2 ; v 3 ; v 4 ; v 5 willhaveazerocoecienton v 3 ,sothisvectorcanneverbeeliminatedfromthesetusedtobuildthe span. T15 ContributedbyRobertBeezerStatement[144] Considerthefollowinglinearcombination 1 v 1 )]TJ/F36 10.9091 Tf 10.909 0 Td [(v 2 +1 v 2 )]TJ/F36 10.9091 Tf 10.909 0 Td [(v 3 +1 v 3 )]TJ/F36 10.9091 Tf 10.909 0 Td [(v 4 + +1 v n )]TJ/F36 10.9091 Tf 10.909 0 Td [(v 1 = v 1 )]TJ/F36 10.9091 Tf 10.909 0 Td [(v 2 + v 2 )]TJ/F36 10.9091 Tf 10.909 0 Td [(v 3 + v 3 )]TJ/F36 10.9091 Tf 10.909 0 Td [(v 4 + + v n )]TJ/F36 10.9091 Tf 10.909 0 Td [(v 1 = v 1 + 0 + 0 + + 0 )]TJ/F36 10.9091 Tf 10.909 0 Td [(v 1 = 0 ThisisanontrivialrelationoflineardependenceDenitionRLDCV[132],sobyDenitionLICV[132] thesetislinearlydependent. T20 ContributedbyRobertBeezerStatement[145] Ourhypothesisandourconclusionusethetermlinearindependence,soitwillgetaworkout.Toestablish linearindependence,webeginwiththedenitionDenitionLICV[132]andwritearelationoflinear dependenceDenitionRLDCV[132], 1 v 1 + 2 v 1 + v 2 + 3 v 1 + v 2 + v 3 + 4 v 1 + v 2 + v 3 + v 4 = 0 UsingthedistributiveandcommutativepropertiesofvectoradditionandscalarmultiplicationTheorem VSPCV[86]thisequationcanberearrangedas 1 + 2 + 3 + 4 v 1 + 2 + 3 + 4 v 2 + 3 + 4 v 3 + 4 v 4 = 0 However,thisisarelationoflineardependenceDenitionRLDCV[132]onalinearlyindependentset, f v 1 ; v 2 ; v 3 ; v 4 g thiswasourlonehypothesis.BythedenitionoflinearindependenceDenitionLICV [132]thescalarsmustallbezero.Thisisthehomogeneoussystemofequations, 1 + 2 + 3 + 4 =0 2 + 3 + 4 =0 3 + 4 =0 4 =0 Row-reducingthecoecientmatrixofthissystemorbacksolvinggivestheconclusion 1 =0 2 =0 3 =0 4 =0 Thismeans,byDenitionLICV[132],thattheoriginalset f v 1 ; v 1 + v 2 ; v 1 + v 2 + v 3 ; v 1 + v 2 + v 3 + v 4 g islinearlyindependent. Version2.02

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SubsectionLI.SOLSolutions152 T50 ContributedbyRobertBeezerStatement[145] Let A =[ A 1 j A 2 j A 3 j ::: j A n ]. LS A; b isconsistent,soweknowthesystemhasatleastonesolution DenitionCS[50].Wewouldliketoshowthattherearenomorethanonesolutiontothesystem. EmployingTechniqueU[693],supposethat x and y aretwosolutionvectorsfor LS A; b .ByTheorem SLSLC[93]weknowwecanwrite, b =[ x ] 1 A 1 +[ x ] 2 A 2 +[ x ] 3 A 3 + +[ x ] n A n b =[ y ] 1 A 1 +[ y ] 2 A 2 +[ y ] 3 A 3 + +[ y ] n A n Then 0 = b )]TJ/F36 10.9091 Tf 10.909 0 Td [(b =[ x ] 1 A 1 +[ x ] 2 A 2 + +[ x ] n A n )]TJ/F15 10.9091 Tf 10.909 0 Td [([ y ] 1 A 1 +[ y ] 2 A 2 + +[ y ] n A n =[ x ] 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [([ y ] 1 A 1 +[ x ] 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [([ y ] 2 A 2 + +[ x ] n )]TJ/F15 10.9091 Tf 10.91 0 Td [([ y ] n A n ThisisarelationoflineardependenceDenitionRLDCV[132]onalinearlyindependentsetthecolumns of A .Sothescalars must allbezero, [ x ] 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [([ y ] 1 =0[ x ] 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [([ y ] 2 =0 ::: [ x ] n )]TJ/F15 10.9091 Tf 10.909 0 Td [([ y ] n =0 Rearrangingtheseequationsyieldsthestatementthat[ x ] i =[ y ] i ,for1 i n .However,thisisexactly howwedenevectorequalityDenitionCVE[84],so x = y andthesystemhasonlyonesolution. Version2.02

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SectionLDSLinearDependenceandSpans153 SectionLDS LinearDependenceandSpans Inanylinearlydependentsetthereisalwaysonevectorthatcanbewrittenasalinearcombinationof theothers.ThisisthesubstanceoftheupcomingTheoremDLDS[152].Perhapsthiswillexplaintheuse oftheworddependent."Inalinearlydependentset,atleastonevectordepends"ontheothersviaa linearcombination. Indeed,becauseTheoremDLDS[152]isanequivalenceTechniqueE[690]someauthorsusethis conditionasadenitionTechniqueD[687]oflineardependence.Thenlinearindependenceisdenedas thelogicaloppositeoflineardependence.Ofcourse,wehave chosen totakeDenitionLICV[132]asour denition,andthenfollowwithTheoremDLDS[152]asatheorem. SubsectionLDSS LinearlyDependentSetsandSpans Ifweusealinearlydependentsettoconstructaspan,thenwecan always createthesameinnitesetwith astartingsetthatisonevectorsmallerinsize.WewillillustratethisbehaviorinExampleRSC5[153]. However,thiswillnotbepossibleifwebuildaspanfromalinearlyindependentset.Soinacertainsense, usingalinearlyindependentsettoformulateaspanisthebestpossibleway|therearen'tanyextra vectorsbeingusedtobuildupallthenecessarylinearcombinations.OK,here'sthetheorem,andthen theexample. TheoremDLDS DependencyinLinearlyDependentSets Supposethat S = f u 1 ; u 2 ; u 3 ;:::; u n g isasetofvectors.Then S isalinearlydependentsetifandonlyif thereisanindex t ,1 t n suchthat u t isalinearcombinationofthevectors u 1 ; u 2 ; u 3 ;:::; u t )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 ; u t +1 ;:::; u n Proof Supposethat S islinearlydependent,sothereexistsanontrivialrelationoflineardependence byDenitionLICV[132].Thatis,therearescalars, i ,1 i n ,whicharenotallzero,suchthat 1 u 1 + 2 u 2 + 3 u 3 + + n u n = 0 : Sincethe i cannotallbezero,chooseone,say t ,thatisnonzero.Then, u t = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 t )]TJ/F21 10.9091 Tf 8.485 0 Td [( t u t PropertyMICN[681] = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 t 1 u 1 + + t )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 u t )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 + t +1 u t +1 + + n u n TheoremVSPCV[86] = )]TJ/F21 10.9091 Tf 8.485 0 Td [( 1 t u 1 + + )]TJ/F21 10.9091 Tf 8.485 0 Td [( t )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 t u t )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 + )]TJ/F21 10.9091 Tf 8.485 0 Td [( t +1 t u t +1 + + )]TJ/F21 10.9091 Tf 8.485 0 Td [( n t u n TheoremVSPCV[86] Sincethevaluesof i t areagainscalars,wehaveexpressed u t asalinearcombinationoftheotherelements of S Assumethatthevector u t isalinearcombinationoftheothervectorsin S .Writethislinear combination,denotingtherelevantscalarsas 1 2 ,..., t )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 t +1 ,... n ,as u t = 1 u 1 + 2 u 2 + + t )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 u t )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 + t +1 u t +1 + + n u n Thenwehave 1 u 1 + + t )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 u t )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 + )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 u t + t +1 u t +1 + + n u n Version2.02

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SubsectionLDS.LDSSLinearlyDependentSetsandSpans154 = u t + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 u t TheoremVSPCV[86] =+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 u t PropertyDSAC[87] =0 u t PropertyAICN[681] = 0 DenitionCVSM[85] Sothescalars 1 ; 2 ; 3 ;:::; t )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 ; t = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; t +1 ;:::; n providea nontrivial linearcombinationofthe vectorsin S ,thusestablishingthat S isalinearlydependentsetDenitionLICV[132]. Thistheoremcanbeused,sometimesrepeatedly,towhittledownthesizeofasetofvectorsusedina spanconstruction.WehaveseensomeofthisalreadyinExampleSCAD[120],butinthenextexample wewilldetailsomeofthesubtleties. ExampleRSC5 Reducingaspanin C 5 Considerthesetof n =4vectorsfrom C 5 R = f v 1 ; v 2 ; v 3 ; v 4 g = 8 > > > > < > > > > : 2 6 6 6 6 4 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 2 3 7 7 7 7 5 ; 2 6 6 6 6 4 2 1 3 1 2 3 7 7 7 7 5 ; 2 6 6 6 6 4 0 )]TJ/F15 10.9091 Tf 8.484 0 Td [(7 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 3 7 7 7 7 5 ; 2 6 6 6 6 4 4 1 2 1 6 3 7 7 7 7 5 9 > > > > = > > > > ; anddene V = h R i ToemployTheoremLIVHS[134],weforma5 4coecientmatrix, D D = 2 6 6 6 6 4 1204 21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(71 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1362 31 )]TJ/F15 10.9091 Tf 8.485 0 Td [(111 22 )]TJ/F15 10.9091 Tf 8.485 0 Td [(26 3 7 7 7 7 5 androw-reducetounderstandsolutionstothehomogeneoussystem LS D; 0 2 6 6 6 6 6 4 1 004 0 1 00 00 1 1 0000 0000 3 7 7 7 7 7 5 : Wecanndinnitelymanysolutionstothissystem,mostofthemnontrivial,andwechooseanyonewe liketobuildarelationoflineardependenceon R .Let'sbeginwith x 4 =1,tondthesolution 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 3 7 7 5 : Sowecanwritetherelationoflineardependence, )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 v 1 +0 v 2 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 v 3 +1 v 4 = 0 : TheoremDLDS[152]guaranteesthatwecansolvethisrelationoflineardependencefor some vectorin R ,butthechoiceofwhichoneisuptous.Noticehoweverthat v 2 hasazerocoecient.Inthiscase,we cannotchoosetosolvefor v 2 .Maybesomeotherrelationoflineardependencewouldproduceanonzero Version2.02

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SubsectionLDS.COVCastingOutVectors155 coecientfor v 2 ifwejusthadtosolveforthisvector.Unfortunately,thisexamplehasbeenengineered to always produceazerocoecienthere,asyoucanseefromsolvingthehomogeneoussystem.Every solutionhas x 2 =0! OK,ifweareconvincedthatwecannotsolvefor v 2 ,let'sinsteadsolvefor v 3 v 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 v 1 +0 v 2 +1 v 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 v 1 +1 v 4 : Wenowclaimthatthisparticularequationwillallowustowrite V = h R i = hf v 1 ; v 2 ; v 3 ; v 4 gi = hf v 1 ; v 2 ; v 4 gi inessencedeclaring v 3 assurplusforthetaskofbuilding V asaspan.Thisclaimisanequalityoftwo sets,sowewilluseDenitionSE[684]toestablishitcarefully.Let R 0 = f v 1 ; v 2 ; v 4 g and V 0 = h R 0 i .We wanttoshowthat V = V 0 Firstshowthat V 0 V .Sinceeveryvectorof R 0 isin R ,anyvectorwecanconstructin V 0 asalinear combinationofvectorsfrom R 0 canalsobeconstructedasavectorin V bythesamelinearcombination ofthesamevectorsin R .Thatwaseasy,nowturnitaround. Nextshowthat V V 0 .Chooseany v from V .Thentherearescalars 1 ; 2 ; 3 ; 4 sothat v = 1 v 1 + 2 v 2 + 3 v 3 + 4 v 4 = 1 v 1 + 2 v 2 + 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 v 1 +1 v 4 + 4 v 4 = 1 v 1 + 2 v 2 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 v 1 + 3 v 4 + 4 v 4 = 1 )]TJ/F15 10.9091 Tf 10.91 0 Td [(4 3 v 1 + 2 v 2 + 3 + 4 v 4 : Thisequationsaysthat v canthenbewrittenasalinearcombinationofthevectorsin R 0 andhence qualiesformembershipin V 0 .So V V 0 andwehaveestablishedthat V = V 0 If R 0 wasalsolinearlydependentitisnot,wecouldreducethesetevenfurther.Noticethatwecould havechosentoeliminateanyoneof v 1 v 3 or v 4 ,butsomehow v 2 isessentialtothecreationof V sinceit cannotbereplacedbyanylinearcombinationof v 1 v 3 or v 4 SubsectionCOV CastingOutVectors InExampleRSC5[153]weusedfourvectorstocreateaspan.Witharelationoflineardependencein hand,wewereabletotoss-out"oneofthesefourvectorsandcreatethesamespanfromasubsetof justthreevectorsfromtheoriginalsetoffour.Wedidhavetotakesomecareastojustwhichvector wetossed-out.Inthenextexample,wewillbemoremethodicalaboutjusthowwechoosetoeliminate vectorsfromalinearlydependentsetwhilepreservingaspan. ExampleCOV Castingoutvectors Webeginwithaset S containingsevenvectorsfrom C 4 S = 8 > > < > > : 2 6 6 4 1 2 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 5 ; 2 6 6 4 4 8 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 7 7 5 ; 2 6 6 4 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 2 3 7 7 5 ; 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 4 3 7 7 5 ; 2 6 6 4 0 9 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 8 3 7 7 5 ; 2 6 6 4 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(31 3 7 7 5 ; 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 37 3 7 7 5 9 > > = > > ; anddene W = h S i .Theset S isobviouslylinearlydependentbyTheoremMVSLD[137],sincewehave n =7vectorsfrom C 4 .Sowecanslimdown S some,andstillcreate W asthespanofasmallersetof Version2.02

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SubsectionLDS.COVCastingOutVectors156 vectors.Asadeviceforidentifyingrelationsoflineardependenceamongthevectorsof S ,weplacethe sevencolumnvectorsof S intoamatrixascolumns, A =[ A 1 j A 2 j A 3 j ::: j A 7 ]= 2 6 6 4 140 )]TJ/F15 10.9091 Tf 8.485 0 Td [(107 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 28 )]TJ/F15 10.9091 Tf 8.485 0 Td [(139 )]TJ/F15 10.9091 Tf 8.485 0 Td [(137 002 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(412 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4248 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3137 3 7 7 5 ByTheoremSLSLC[93]anontrivialsolutionto LS A; 0 willgiveusanontrivialrelationoflinear dependenceDenitionRLDCV[132]onthecolumnsof A whicharetheelementsoftheset S .The row-reducedformfor A isthematrix B = 2 6 6 4 1 40021 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 00 1 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(35 000 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(66 0000000 3 7 7 5 sowecaneasilycreatesolutionstothehomogeneoussystem LS A; 0 usingthefreevariables x 2 ;x 5 ;x 6 ;x 7 Anysuchsolutionwillcorrespondtoarelationoflineardependenceonthecolumnsof B .Thesesolutions willallowustosolveforonecolumnvectorasalinearcombinationofsomeothers,inthespiritofTheorem DLDS[152],andremovethatvectorfromtheset.We'llsetaboutformingtheselinearcombinations methodically.Setthefreevariable x 2 toone,andsettheotherfreevariablestozero.Thenasolutionto LS A; 0 is x = 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 1 0 0 0 0 0 3 7 7 7 7 7 7 7 7 5 whichcanbeusedtocreatethelinearcombination )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 A 1 +1 A 2 +0 A 3 +0 A 4 +0 A 5 +0 A 6 +0 A 7 = 0 Thiscanthenbearrangedandsolvedfor A 2 ,resultingin A 2 expressedasalinearcombinationof f A 1 ; A 3 ; A 4 g A 2 =4 A 1 +0 A 3 +0 A 4 Thismeansthat A 2 issurplus,andwecancreate W justaswellwithasmallersetwiththisvector removed, W = hf A 1 ; A 3 ; A 4 ; A 5 ; A 6 ; A 7 gi Technically,thissetequalityfor W requiresaproof,inthespiritofExampleRSC5[153],butwewill bypassthisrequirementhere,andinthenextfewparagraphs. Now,setthefreevariable x 5 toone,andsettheotherfreevariablestozero.Thenasolutionto LS B; 0 is x = 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 0 0 3 7 7 7 7 7 7 7 7 5 Version2.02

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SubsectionLDS.COVCastingOutVectors157 whichcanbeusedtocreatethelinearcombination )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 A 1 +0 A 2 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 A 3 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 A 4 +1 A 5 +0 A 6 +0 A 7 = 0 Thiscanthenbearrangedandsolvedfor A 5 ,resultingin A 5 expressedasalinearcombinationof f A 1 ; A 3 ; A 4 g A 5 =2 A 1 +1 A 3 +2 A 4 Thismeansthat A 5 issurplus,andwecancreate W justaswellwithasmallersetwiththisvector removed, W = hf A 1 ; A 3 ; A 4 ; A 6 ; A 7 gi Doitagain,setthefreevariable x 6 toone,andsettheotherfreevariablestozero.Thenasolutionto LS B; 0 is x = 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 3 6 0 1 0 3 7 7 7 7 7 7 7 7 5 whichcanbeusedtocreatethelinearcombination )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 A 1 +0 A 2 +3 A 3 +6 A 4 +0 A 5 +1 A 6 +0 A 7 = 0 Thiscanthenbearrangedandsolvedfor A 6 ,resultingin A 6 expressedasalinearcombinationof f A 1 ; A 3 ; A 4 g A 6 =1 A 1 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 A 3 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 A 4 Thismeansthat A 6 issurplus,andwecancreate W justaswellwithasmallersetwiththisvector removed, W = hf A 1 ; A 3 ; A 4 ; A 7 gi Setthefreevariable x 7 toone,andsettheotherfreevariablestozero.Thenasolutionto LS B; 0 is x = 2 6 6 6 6 6 6 6 6 4 3 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 0 0 1 3 7 7 7 7 7 7 7 7 5 whichcanbeusedtocreatethelinearcombination 3 A 1 +0 A 2 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 A 3 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 A 4 +0 A 5 +0 A 6 +1 A 7 = 0 Thiscanthenbearrangedandsolvedfor A 7 ,resultingin A 7 expressedasalinearcombinationof f A 1 ; A 3 ; A 4 g A 7 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 A 1 +5 A 3 +6 A 4 Thismeansthat A 7 issurplus,andwecancreate W justaswellwithasmallersetwiththisvector removed, W = hf A 1 ; A 3 ; A 4 gi Version2.02

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SubsectionLDS.COVCastingOutVectors158 Youmightthinkwecouldkeepthisup,butwehaverunoutoffreevariables.Andnotcoincidentally, theset f A 1 ; A 3 ; A 4 g islinearlyindependentcheckthis!.Itshouldbeclearhoweachfreevariablewas usedtoeliminatethecorrespondingcolumnfromthesetusedtospanthecolumnspace,asthiswillbethe essenceoftheproofofthenexttheorem.Thecolumnvectorsin S werenotchosenentirelyatrandom,they arethecolumnsofArchetypeI[737].SeeifyoucanmimicthisexampleusingthecolumnsofArchetype J[741].Goahead,we'llgograbacupofcoeeandbebackbeforeyounishup. Forextracredit,noticethatthevector b = 2 6 6 4 3 9 1 4 3 7 7 5 isthevectorofconstantsinthedenitionofArchetypeI[737].Sincethesystem LS A; b isconsistent, weknowbyTheoremSLSLC[93]that b isalinearcombinationofthecolumnsof A ,orstatedequivalently, b 2 W .Thismeansthat b mustalsobealinearcombinationofjustthethreecolumns A 1 ; A 3 ; A 4 .Can youndsuchalinearcombination?Didyounoticethatthereisjustasingleuniqueanswer?Hmmmm. ExampleCOV[154]deservesyourcarefulattention,sincethisimportantexamplemotivatesthefollowingveryfundamentaltheorem. TheoremBS BasisofaSpan Supposethat S = f v 1 ; v 2 ; v 3 ;:::; v n g isasetofcolumnvectors.Dene W = h S i andlet A bethe matrixwhosecolumnsarethevectorsfrom S .Let B bethereducedrow-echelonformof A ,with D = f d 1 ;d 2 ;d 3 ;:::;d r g thesetofcolumnindicescorrespondingtothepivotcolumnsof B .Then 1. T = f v d 1 ; v d 2 ; v d 3 ;::: v d r g isalinearlyindependentset. 2. W = h T i Proof Toprovethat T islinearlyindependent,beginwitharelationoflineardependenceon T 0 = 1 v d 1 + 2 v d 2 + 3 v d 3 + ::: + r v d r andwewilltrytoconcludethattheonlypossibilityforthescalars i isthattheyareallzero.Denotethe non-pivotcolumnsof B by F = f f 1 ;f 2 ;f 3 ;:::;f n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r g .Thenwecanpreservetheequalitybyaddingabig fatzerotothelinearcombination, 0 = 1 v d 1 + 2 v d 2 + 3 v d 3 + ::: + r v d r +0 v f 1 +0 v f 2 +0 v f 3 + ::: +0 v f n )]TJ/F23 5.9776 Tf 5.756 0 Td [(r ByTheoremSLSLC[93],thescalarsinthislinearcombinationsuitablyreorderedareasolutiontothe homogeneoussystem LS A; 0 .Butnoticethatthisisthesolutionobtainedbysettingeachfreevariable tozero.IfweconsiderthedescriptionofasolutionvectorintheconclusionofTheoremVFSLS[99],in thecaseofahomogeneoussystem,thenweseethatifallthefreevariablesaresettozerotheresulting solutionvectoristrivialallzeros.Soitmustbethat i =0,1 i r .ThisimpliesbyDenitionLICV [132]that T isalinearlyindependentset. ThesecondconclusionofthistheoremisanequalityofsetsDenitionSE[684].Since T isasubsetof S ,anylinearcombinationofelementsoftheset T canalsobeviewedasalinearcombinationofelements oftheset S .So h T ih S i = W .Itremainstoprovethat W = h S ih T i Foreach k ,1 k n )]TJ/F21 10.9091 Tf 10.909 0 Td [(r ,formasolution x to LS A; 0 bysettingthefreevariablesasfollows: x f 1 =0 x f 2 =0 x f 3 =0 :::x f k =1 :::x f n )]TJ/F23 5.9776 Tf 5.756 0 Td [(r =0 Version2.02

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SubsectionLDS.COVCastingOutVectors159 ByTheoremVFSLS[99],theremainderofthissolutionvectorisgivenby, x d 1 = )]TJ/F15 10.9091 Tf 10.303 0 Td [([ B ] 1 ;f k x d 2 = )]TJ/F15 10.9091 Tf 10.303 0 Td [([ B ] 2 ;f k x d 3 = )]TJ/F15 10.9091 Tf 10.303 0 Td [([ B ] 3 ;f k :::x d r = )]TJ/F15 10.9091 Tf 10.303 0 Td [([ B ] r;f k Fromthissolution,weobtainarelationoflineardependenceonthecolumnsof A )]TJ/F15 10.9091 Tf 10.303 0 Td [([ B ] 1 ;f k v d 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [([ B ] 2 ;f k v d 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [([ B ] 3 ;f k v d 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(::: )]TJ/F15 10.9091 Tf 10.909 0 Td [([ B ] r;f k v d r +1 v f k = 0 whichcanbearrangedastheequality v f k =[ B ] 1 ;f k v d 1 +[ B ] 2 ;f k v d 2 +[ B ] 3 ;f k v d 3 + ::: +[ B ] r;f k v d r Now,supposewetakeanarbitraryelement, w ,of W = h S i andwriteitasalinearcombinationofthe elementsof S ,butwiththetermsorganizedaccordingtotheindicesin D and F w = 1 v d 1 + 2 v d 2 + 3 v d 3 + ::: + r v d r + 1 v f 1 + 2 v f 2 + 3 v f 3 + ::: + n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r v f n )]TJ/F23 5.9776 Tf 5.757 0 Td [(r Fromtheabove,wecanreplaceeach v f j byalinearcombinationofthe v d i w = 1 v d 1 + 2 v d 2 + 3 v d 3 + ::: + r v d r + 1 [ B ] 1 ;f 1 v d 1 +[ B ] 2 ;f 1 v d 2 +[ B ] 3 ;f 1 v d 3 + ::: +[ B ] r;f 1 v d r + 2 [ B ] 1 ;f 2 v d 1 +[ B ] 2 ;f 2 v d 2 +[ B ] 3 ;f 2 v d 3 + ::: +[ B ] r;f 2 v d r + 3 [ B ] 1 ;f 3 v d 1 +[ B ] 2 ;f 3 v d 2 +[ B ] 3 ;f 3 v d 3 + ::: +[ B ] r;f 3 v d r + n )]TJ/F22 7.9701 Tf 6.586 0 Td [(r [ B ] 1 ;f n )]TJ/F23 5.9776 Tf 5.756 0 Td [(r v d 1 +[ B ] 2 ;f n )]TJ/F23 5.9776 Tf 5.757 0 Td [(r v d 2 +[ B ] 3 ;f n )]TJ/F23 5.9776 Tf 5.756 0 Td [(r v d 3 + ::: +[ B ] r;f n )]TJ/F23 5.9776 Tf 5.757 0 Td [(r v d r WithrepeatedapplicationsofseveralofthepropertiesofTheoremVSPCV[86]wecanrearrangethis expressionas, = 1 + 1 [ B ] 1 ;f 1 + 2 [ B ] 1 ;f 2 + 3 [ B ] 1 ;f 3 + ::: + n )]TJ/F22 7.9701 Tf 6.586 0 Td [(r [ B ] 1 ;f n )]TJ/F23 5.9776 Tf 5.756 0 Td [(r v d 1 + 2 + 1 [ B ] 2 ;f 1 + 2 [ B ] 2 ;f 2 + 3 [ B ] 2 ;f 3 + ::: + n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r [ B ] 2 ;f n )]TJ/F23 5.9776 Tf 5.756 0 Td [(r v d 2 + 3 + 1 [ B ] 3 ;f 1 + 2 [ B ] 3 ;f 2 + 3 [ B ] 3 ;f 3 + ::: + n )]TJ/F22 7.9701 Tf 6.587 0 Td [(r [ B ] 3 ;f n )]TJ/F23 5.9776 Tf 5.756 0 Td [(r v d 3 + r + 1 [ B ] r;f 1 + 2 [ B ] r;f 2 + 3 [ B ] r;f 3 + ::: + n )]TJ/F22 7.9701 Tf 6.586 0 Td [(r [ B ] r;f n )]TJ/F23 5.9776 Tf 5.756 0 Td [(r v d r Thismessexpressesthevector w asalinearcombinationofthevectorsin T = f v d 1 ; v d 2 ; v d 3 ;::: v d r g thussayingthat w 2h T i .Therefore, W = h S ih T i InExampleCOV[154],wetossed-outvectorsoneatatime.Butineachinstance,werewrotethe oendingvectorasalinearcombinationofthosevectorsthatcorrespondedtothepivotcolumnsofthe reducedrow-echelonformofthematrixofcolumns.IntheproofofTheoremBS[157],weaccomplishthis reductioninonebigstep.InExampleCOV[154]wearrivedatalinearlyindependentsetatexactlythe samemomentthatweranoutoffreevariablestoexploit.Thiswasnotacoincidence,itisthesubstance ofourconclusionoflinearindependenceinTheoremBS[157]. Version2.02

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SubsectionLDS.COVCastingOutVectors160 Here'sastraightforwardapplicationofTheoremBS[157]. ExampleRSC4 Reducingaspanin C 4 Beginwithasetofvevectorsfrom C 4 S = 8 > > < > > : 2 6 6 4 1 1 2 1 3 7 7 5 ; 2 6 6 4 2 2 4 2 3 7 7 5 ; 2 6 6 4 2 0 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 1 3 7 7 5 ; 2 6 6 4 7 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4 3 7 7 5 ; 2 6 6 4 0 2 5 1 3 7 7 5 9 > > = > > ; andlet W = h S i .Toarriveatasmallerlinearlyindependentset,followtheproceduredescribedin TheoremBS[157].Placethevectorsfrom S intoamatrixascolumns,androw-reduce, 2 6 6 4 12270 12012 24 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 12141 3 7 7 5 RREF )443()223()222()443(! 2 6 6 4 1 2012 00 1 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 00000 00000 3 7 7 5 Columns1and3arethepivotcolumns D = f 1 ; 3 g sotheset T = 8 > > < > > : 2 6 6 4 1 1 2 1 3 7 7 5 ; 2 6 6 4 2 0 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 1 3 7 7 5 9 > > = > > ; islinearlyindependentand h T i = h S i = W .Boom! Sincethereducedrow-echelonformofamatrixisuniqueTheoremRREFU[32],theprocedureof TheoremBS[157]leadsustoauniqueset T .However,thereisawidevarietyofpossibilitiesforsets T thatarelinearlyindependentandwhichcanbeemployedinaspantocreate W .Withoutproof,welist twootherpossibilities: T 0 = 8 > > < > > : 2 6 6 4 2 2 4 2 3 7 7 5 ; 2 6 6 4 2 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 3 7 7 5 9 > > = > > ; T = 8 > > < > > : 2 6 6 4 3 1 1 2 3 7 7 5 ; 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 3 0 3 7 7 5 9 > > = > > ; Canyouprovethat T 0 and T arelinearlyindependentsetsand W = h S i = h T 0 i = h T i ? ExampleRES Reworkingelementsofaspan Beginwithasetofvevectorsfrom C 4 R = 8 > > < > > : 2 6 6 4 2 1 3 2 3 7 7 5 ; 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 0 1 3 7 7 5 ; 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 7 7 5 ; 2 6 6 4 3 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 5 ; 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4 3 7 7 5 9 > > = > > ; Version2.02

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SubsectionLDS.COVCastingOutVectors161 Itiseasytocreateelementsof X = h R i |wewillcreateoneatrandom, y =6 2 6 6 4 2 1 3 2 3 7 7 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 0 1 3 7 7 5 +1 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 7 7 5 +6 2 6 6 4 3 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 5 +2 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4 3 7 7 5 = 2 6 6 4 9 2 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 7 7 5 Weknowwecanreplace R byasmallersetsinceitisobviouslylinearlydependentbyTheoremMVSLD [137]thatwillcreatethesamespan.Heregoes, 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(83 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 30 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(24 3 7 7 5 RREF )443()223()222()443(! 2 6 6 4 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(30 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 1 202 000 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 00000 3 7 7 5 So,ifwecollecttherst,secondandfourthvectorsfrom R P = 8 > > < > > : 2 6 6 4 2 1 3 2 3 7 7 5 ; 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 0 1 3 7 7 5 ; 2 6 6 4 3 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 5 9 > > = > > ; then P islinearlyindependentand h P i = h R i = X byTheoremBS[157].Sincewebuilt y asanelement of h R i itmustalsobeanelementof h P i .Canwewrite y asalinearcombinationofjustthethreevectors in P ?Theansweris,ofcourse,yes.Butlet'scomputeanexplicitlinearcombinationjustforfun.By TheoremSLSLC[93]wecangetsuchalinearcombinationbysolvingasystemofequationswiththe columnvectorsof R asthecolumnsofacoecientmatrix,and y asthevectorofconstants.Employing anaugmentedmatrixtosolvethissystem, 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(139 1112 30 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 7 7 5 RREF )443()223()222()443(! 2 6 6 4 1 001 0 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 00 1 2 0000 3 7 7 5 Sowesee,asexpected,that 1 2 6 6 4 2 1 3 2 3 7 7 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 1 0 1 3 7 7 5 +2 2 6 6 4 3 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 3 7 7 5 = 2 6 6 4 9 2 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 7 7 5 = y Akeyfeatureofthisexampleisthatthelinearcombinationthatexpresses y asalinearcombinationofthe vectorsin P isunique.Thisisaconsequenceofthelinearindependenceof P .Thelinearlyindependent set P issmallerthan R ,butstilljustbarelybigenoughtocreateelementsoftheset X = h R i .There aremany,manywaystowrite y asalinearcombinationofthevevectorsin R theappropriatesystem ofequationstoverifythisclaimhastwofreevariablesinthedescriptionofthesolutionset,yetthereis preciselyonewaytowrite y asalinearcombinationofthethreevectorsin P Version2.02

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SubsectionLDS.READReadingQuestions162 SubsectionREAD ReadingQuestions 1.Let S bethelinearlydependentsetofthreevectorsbelow. S = 8 > > < > > : 2 6 6 4 1 10 100 1000 3 7 7 5 ; 2 6 6 4 1 1 1 1 3 7 7 5 ; 2 6 6 4 5 23 203 2003 3 7 7 5 9 > > = > > ; Writeonevectorfrom S asalinearcombinationoftheothertwoyoushouldbeabletodothison sight,ratherthandoingsomecomputations.Convertthisexpressionintoanontrivialrelationof lineardependenceon S 2.Explainwhytheworddependent"isusedinthedenitionoflineardependence. 3.Supposethat Y = h P i = h Q i ,where P isalinearlydependentsetand Q islinearlyindependent. Wouldyouratheruse P or Q todescribe Y ?Why? Version2.02

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SubsectionLDS.EXCExercises163 SubsectionEXC Exercises C20 Let T bethesetofcolumnsofthematrix B below.Dene W = h T i .Findaset R sothat R has3vectors, R isasubsetof T ,and W = h R i B = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(31 )]TJ/F15 10.9091 Tf 8.484 0 Td [(27 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1214 112 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 3 5 ContributedbyRobertBeezerSolution[164] C40 Verifythattheset R 0 = f v 1 ; v 2 ; v 4 g attheendofExampleRSC5[153]islinearlyindependent. ContributedbyRobertBeezer C50 Considerthesetofvectorsfrom C 3 W ,givenbelow.Findalinearlyindependentset T thatcontains threevectorsfrom W andsuchthat h W i = h T i W = f v 1 ; v 2 ; v 3 ; v 4 ; v 5 g = 8 < : 2 4 2 1 1 3 5 ; 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 3 5 ; 2 4 1 2 3 3 5 ; 2 4 3 1 3 3 5 ; 2 4 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 5 9 = ; ContributedbyRobertBeezerSolution[164] C51 Giventheset S below,ndalinearlyindependentset T sothat h T i = h S i S = 8 < : 2 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 5 ; 2 4 3 0 1 3 5 ; 2 4 1 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 3 5 ; 2 4 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 3 5 9 = ; ContributedbyRobertBeezerSolution[164] C52 Let W bethespanofthesetofvectors S below, W = h S i .Findaset T sothat1thespanof T is W h T i = W T isalinearlyindependentset,and T isasubsetof S .points S = 8 < : 2 4 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 5 ; 2 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 3 5 ; 2 4 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 5 ; 2 4 3 1 1 3 5 ; 2 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 3 5 9 = ; ContributedbyRobertBeezerSolution[164] C55 Let T bethesetofvectors T = 8 < : 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 5 ; 2 4 3 0 1 3 5 ; 2 4 4 2 3 3 5 ; 2 4 3 0 6 3 5 9 = ; .Findtwodierentsubsetsof T ,named R and S ,sothat R and S eachcontainthreevectors,andsothat h R i = h T i and h S i = h T i .Provethat both R and S arelinearlyindependent. ContributedbyRobertBeezerSolution[165] C70 RepriseExampleRES[159]bycreatinganewversionofthevector y .Inotherwords,formanew, dierentlinearcombinationofthevectorsin R tocreateanewvector y butdonotsimplifytheproblem toomuchbychoosinganyofthevenewscalarstobezero.Thenexpressthisnew y asacombination ofthevectorsin P ContributedbyRobertBeezer Version2.02

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SubsectionLDS.EXCExercises164 M10 AttheconclusionofExampleRSC4[159]twoalternativesolutions,sets T 0 and T ,areproposed. Verifytheseclaimsbyprovingthat h T i = h T 0 i and h T i = h T i ContributedbyRobertBeezer T40 Supposethat v 1 and v 2 areanytwovectorsfrom C m .Provethefollowingsetequality. hf v 1 ; v 2 gi = hf v 1 + v 2 ; v 1 )]TJ/F36 10.9091 Tf 10.909 0 Td [(v 2 gi ContributedbyRobertBeezerSolution[166] Version2.02

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SubsectionLDS.SOLSolutions165 SubsectionSOL Solutions C20 ContributedbyRobertBeezerStatement[162] Let T = f w 1 ; w 2 ; w 3 ; w 4 g .Thevector 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 1 3 7 7 5 isasolutiontothehomogeneoussystemwiththematrix B as thecoecientmatrixcheckthis!.ByTheoremSLSLC[93]itprovidesthescalarsforalinearcombination ofthecolumnsof B thevectorsin T thatequalsthezerovector,arelationoflineardependenceon T 2 w 1 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 w 2 + w 4 = 0 Wecanrearrangethisequationbysolvingfor w 4 w 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 w 1 + w 2 Thisequationtellsusthatthevector w 4 issuperuousinthespanconstructionthatcreates W .So W = hf w 1 ; w 2 ; w 3 gi .Therequestedsetis R = f w 1 ; w 2 ; w 3 g C50 ContributedbyRobertBeezerStatement[162] ToapplyTheoremBS[157],weformulateamatrix A whosecolumnsare v 1 ; v 2 ; v 3 ; v 4 ; v 5 .Thenwe row-reduce A .Afterrow-reducing,weobtain 2 4 1 002 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 0 1 01 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 00 1 00 3 5 Fromthiswethatthepivotcolumnsare D = f 1 ; 2 ; 3 g .Thus T = f v 1 ; v 2 ; v 3 g = 8 < : 2 4 2 1 1 3 5 ; 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 3 5 ; 2 4 1 2 3 3 5 9 = ; isalinearlyindependentsetand h T i = W .ComparethisproblemwithExerciseLI.M50[144]. C51 ContributedbyRobertBeezerStatement[162] TheoremBS[157]sayswecanmakeamatrixwiththesefourvectorsascolumns,row-reduce,andjust keepthecolumnswithindicesintheset D .Herewego,formingtherelevantmatrixandrow-reducing, 2 4 2315 )]TJ/F15 10.9091 Tf 8.485 0 Td [(101 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 3 5 RREF )443()223()222()443(! 2 4 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 0 1 11 0000 3 5 Analyzingtherow-reducedversionofthismatrix,weseethatthersttwocolumnsarepivotcolumns,so D = f 1 ; 2 g .TheoremBS[157]saysweneedonlykeep"thersttwocolumnstocreateasetwiththe requisiteproperties, T = 8 < : 2 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 5 ; 2 4 3 0 1 3 5 9 = ; C52 ContributedbyRobertBeezerStatement[162] Version2.02

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SubsectionLDS.SOLSolutions166 ThisisastraightsetupfortheconclusionofTheoremBS[157].Thehypothesesofthistheoremtellus topackthevectorsof W intothecolumnsofamatrixandrow-reduce, 2 4 12433 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(311 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(110 3 5 RREF )443()223()222()443(! 2 4 1 0201 0 1 101 000 1 0 3 5 Pivotcolumnshaveindices D = f 1 ; 2 ; 4 g .TheoremBS[157]tellsustoform T withcolumns1 ; 2and4 of S S = 8 < : 2 4 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 5 ; 2 4 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 1 3 5 ; 2 4 3 1 1 3 5 9 = ; C55 ContributedbyRobertBeezerStatement[162] Let A bethematrixwhosecolumnsarethevectorsin T .Thenrow-reduce A A RREF )443()223()222()443(! B = 2 4 1 002 0 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 00 1 1 3 5 FromTheoremBS[157]wecanform R bychoosingthecolumnsof A thatcorrespondtothepivotcolumns of B .TheoremBS[157]alsoguaranteesthat R willbelinearlyindependent. R = 8 < : 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 5 ; 2 4 3 0 1 3 5 ; 2 4 4 2 3 3 5 9 = ; Thatwaseasy.Tond S willrequireabitmorework.From B wecanobtainasolutionto LS A; 0 whichbyTheoremSLSLC[93]willprovideanontrivialrelationoflineardependenceonthecolumnsof A whicharethevectorsin T .Towit,choosethefreevariable x 4 tobe1,then x 1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2, x 2 =1, x 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1, andso )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 5 + 2 4 3 0 1 3 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 4 4 2 3 3 5 + 2 4 3 0 6 3 5 = 2 4 0 0 0 3 5 thisequationcanberewrittenwiththesecondvectorstayingput,andtheotherthreemovingtotheother sideoftheequality, 2 4 3 0 1 3 5 = 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 5 + 2 4 4 2 3 3 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 4 3 0 6 3 5 Wecouldhavechosenothervectorstostayput,butmayhavethenneededtodividebyanonzeroscalar. Thisequationisenoughtoconcludethatthesecondvectorin T issurplus"andcanbereplacedseethe carefulargumentinExampleRSC5[153].Soset S = 8 < : 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 5 ; 2 4 4 2 3 3 5 ; 2 4 3 0 6 3 5 9 = ; andthen h S i = h T i T isalsoalinearlyindependentset,whichwecanshowdirectly.Makeamatrix C whosecolumnsarethevectorsin S .Row-reduce B andyouwillobtaintheidentitymatrix I 3 .By TheoremLIVRN[136],theset S islinearlyindependent. Version2.02

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SubsectionLDS.SOLSolutions167 T40 ContributedbyRobertBeezerStatement[163] Thisisanequalityofsets,soDenitionSE[684]applies. Theeasy"halfrst.Showthat X = hf v 1 + v 2 ; v 1 )]TJ/F36 10.9091 Tf 10.909 0 Td [(v 2 gihf v 1 ; v 2 gi = Y Choose x 2 X .Then x = a 1 v 1 + v 2 + a 2 v 1 )]TJ/F36 10.9091 Tf 10.909 0 Td [(v 2 forsomescalars a 1 and a 2 .Then, x = a 1 v 1 + v 2 + a 2 v 1 )]TJ/F36 10.9091 Tf 10.909 0 Td [(v 2 = a 1 v 1 + a 1 v 2 + a 2 v 1 + )]TJ/F21 10.9091 Tf 8.485 0 Td [(a 2 v 2 = a 1 + a 2 v 1 + a 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(a 2 v 2 whichqualies x formembershipin Y ,asitisalinearcombinationof v 1 ; v 2 Nowshowtheoppositeinclusion, Y = hf v 1 ; v 2 gihf v 1 + v 2 ; v 1 )]TJ/F36 10.9091 Tf 10.909 0 Td [(v 2 gi = X Choose y 2 Y .Thentherearescalars b 1 ;b 2 suchthat y = b 1 v 1 + b 2 v 2 .Rearranging,weobtain, y = b 1 v 1 + b 2 v 2 = b 1 2 [ v 1 + v 2 + v 1 )]TJ/F36 10.9091 Tf 10.91 0 Td [(v 2 ]+ b 2 2 [ v 1 + v 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [( v 1 )]TJ/F36 10.9091 Tf 10.909 0 Td [(v 2 ] = b 1 + b 2 2 v 1 + v 2 + b 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 2 2 v 1 )]TJ/F36 10.9091 Tf 10.909 0 Td [(v 2 Thisisanexpressionfor y asalinearcombinationof v 1 + v 2 and v 1 )]TJ/F36 10.9091 Tf 11.104 0 Td [(v 2 ,earning y membershipin X Since X isasubsetof Y ,andviceversa,weseethat X = Y ,asdesired. Version2.02

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SectionOOrthogonality168 SectionO Orthogonality Inthissectionwedeneacouplemoreoperationswithvectors,andproveafewtheorems.Atrstblush thesedenitionsandresultswillnotappearcentraltowhatfollows,butwewillmakeuseofthematkey pointsintheremainderofthecoursesuchasSectionMINM[226],SectionOD[601].Becausewehave chosentouse C asoursetofscalars,thissubsectionisabitmore,uh,...complexthanitwouldbeforthe realnumbers.We'llexplainaswegoalonghowthingsgeteasierfortherealnumbers R .Ifyouhaven't already,nowwouldbeagoodtimetoreviewsomeofthebasicpropertiesofarithmeticwithcomplex numbersdescribedinSectionCNO[679].Withthatdone,wecanextendthebasicsofcomplexnumber arithmetictoourstudyofvectorsin C m SubsectionCAV ComplexArithmeticandVectors Weknowhowtheadditionandmultiplicationofcomplexnumbersisemployedindeningtheoperations forvectorsin C m DenitionCVA[84]andDenitionCVSM[85].Wecanalsoextendtheideaofthe conjugatetovectors. DenitionCCCV ComplexConjugateofaColumnVector Supposethat u isavectorfrom C m .Thentheconjugateofthevector, u ,isdenedby [ u ] i = [ u ] i 1 i m ThisdenitioncontainsNotationCCCV. 4 Withthisdenitionwecanshowthattheconjugateofacolumnvectorbehavesaswewouldexpect withregardtovectoradditionandscalarmultiplication. TheoremCRVA ConjugationRespectsVectorAddition Suppose x and y aretwovectorsfrom C m .Then x + y = x + y Proof Foreach1 i m [ x + y ] i = [ x + y ] i DenitionCCCV[167] = [ x ] i +[ y ] i DenitionCVA[84] = [ x ] i + [ y ] i TheoremCCRA[681] =[ x ] i +[ y ] i DenitionCCCV[167] =[ x + y ] i DenitionCVA[84] ThenbyDenitionCVE[84]wehave x + y = x + y TheoremCRSM ConjugationRespectsVectorScalarMultiplication Version2.02

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SubsectionO.IPInnerproducts169 Suppose x isavectorfrom C m ,and 2 C isascalar.Then x = x Proof For1 i m [ x ] i = [ x ] i DenitionCCCV[167] = [ x ] i DenitionCVSM[85] = [ x ] i TheoremCCRM[682] = [ x ] i DenitionCCCV[167] =[ x ] i DenitionCVSM[85] ThenbyDenitionCVE[84]wehave x = x Thesetwotheoremstogethertellushowwecanpush"complexconjugationthroughlinearcombinations. SubsectionIP Innerproducts DenitionIP InnerProduct Giventhevectors u ; v 2 C m the innerproduct of u and v isthescalarquantityin C h u ; v i =[ u ] 1 [ v ] 1 +[ u ] 2 [ v ] 2 +[ u ] 3 [ v ] 3 + +[ u ] m [ v ] m = m X i =1 [ u ] i [ v ] i ThisdenitioncontainsNotationIP. 4 Thisoperationisabitdierentinthatwebeginwithtwovectorsbutproduceascalar.Computing oneisstraightforward. ExampleCSIP Computingsomeinnerproducts Thescalarproductof u = 2 4 2+3 i 5+2 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(3+ i 3 5 and v = 2 4 1+2 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(4+5 i 0+5 i 3 5 is h u ; v i =+3 i 1+2 i ++2 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(4+5 i + )]TJ/F15 10.9091 Tf 8.484 0 Td [(3+ i 0+5 i =+3 i )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i ++2 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 i + )]TJ/F15 10.9091 Tf 8.485 0 Td [(3+ i )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 i = )]TJ/F21 10.9091 Tf 10.91 0 Td [(i + )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(33 i ++15 i =3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(19 i Version2.02

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SubsectionO.IPInnerproducts170 Thescalarproductof w = 2 6 6 6 6 4 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 2 8 3 7 7 7 7 5 and x = 2 6 6 6 6 4 3 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 7 7 5 is h w ; x i =2 3+4 1+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0+2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1+8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2=2+4+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3+2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1+8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2= )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 : InthecasewheretheentriesofourvectorsareallrealnumbersasinthesecondpartofExampleCSIP [168],thecomputationoftheinnerproductmaylookfamiliarandbeknowntoyouasa dotproduct or scalarproduct .Soyoucanviewtheinnerproductasageneralizationofthescalarproducttovectors from C m ratherthan R m Also,notethatwehavechosentoconjugatetheentriesofthe second vectorlistedintheinnerproduct, whilemanyauthorschoosetoconjugateentriesfromthe rst component.Itreallymakesnodierence whichchoiceismade,itjustrequiresthatsubsequentdenitionsandtheoremsareconsistentwiththe choice.YoucanstudytheconclusionofTheoremIPAC[170]asanexplanationofthemagnitudeofthe dierencethatresultsfromthischoice.Butbecarefulasyoureadothertreatmentsoftheinnerproduct oritsuseinapplications,andbesureyouknowaheadoftime which choicehasbeenmade. Thereareseveralquicktheoremswecannowprove,andtheywilleachbeusefullater. TheoremIPVA InnerProductandVectorAddition Suppose u ; v ; w 2 C m .Then 1. h u + v ; w i = h u ; w i + h v ; w i 2. h u ; v + w i = h u ; v i + h u ; w i Proof Theproofsofthetwopartsareverysimilar,withthesecondonerequiringjustabitmoreeort duetotheconjugationthatoccurs.Wewillprovepart2andyoucanprovepart1ExerciseO.T10[179]. h u ; v + w i = m X i =1 [ u ] i [ v + w ] i DenitionIP[168] = m X i =1 [ u ] i [ v ] i +[ w ] i DenitionCVA[84] = m X i =1 [ u ] i [ v ] i + [ w ] i TheoremCCRA[681] = m X i =1 [ u ] i [ v ] i +[ u ] i [ w ] i PropertyDCN[681] = m X i =1 [ u ] i [ v ] i + m X i =1 [ u ] i [ w ] i PropertyCACN[680] = h u ; v i + h u ; w i DenitionIP[168] Version2.02

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SubsectionO.IPInnerproducts171 TheoremIPSM InnerProductandScalarMultiplication Suppose u ; v 2 C m and 2 C .Then 1. h u ; v i = h u ; v i 2. h u ; v i = h u ; v i Proof Theproofsofthetwopartsareverysimilar,withthesecondonerequiringjustabitmoreeort duetotheconjugationthatoccurs.Wewillprovepart2andyoucanprovepart1ExerciseO.T11[179]. h u ; v i = m X i =1 [ u ] i [ v ] i DenitionIP[168] = m X i =1 [ u ] i [ v ] i DenitionCVSM[85] = m X i =1 [ u ] i [ v ] i TheoremCCRM[682] = m X i =1 [ u ] i [ v ] i PropertyCMCN[680] = m X i =1 [ u ] i [ v ] i PropertyDCN[681] = h u ; v i DenitionIP[168] TheoremIPAC InnerProductisAnti-Commutative Supposethat u and v arevectorsin C m .Then h u ; v i = h v ; u i Proof h u ; v i = m X i =1 [ u ] i [ v ] i DenitionIP[168] = m X i =1 [ u ] i [ v ] i TheoremCCT[682] = m X i =1 [ u ] i [ v ] i TheoremCCRM[682] = m X i =1 [ u ] i [ v ] i TheoremCCRA[681] = m X i =1 [ v ] i [ u ] i PropertyCMCN[680] = h v ; u i DenitionIP[168] Version2.02

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SubsectionO.NNorm172 SubsectionN Norm Iftreatinglinearalgebrainamoregeometricfashion,thelengthofavectoroccursnaturally,andiswhat youwouldexpectfromitsname.Withcomplexnumbers,wewilldeneasimilarfunction.Recallthatif c isacomplexnumber,then j c j denotesitsmodulusDenitionMCN[682]. DenitionNV NormofaVector The norm ofthevector u isthescalarquantityin C k u k = q j [ u ] 1 j 2 + j [ u ] 2 j 2 + j [ u ] 3 j 2 + + j [ u ] m j 2 = v u u t m X i =1 j [ u ] i j 2 ThisdenitioncontainsNotationNV. 4 Computinganormisalsoeasytodo. ExampleCNSV Computingthenormofsomevectors Thenormof u = 2 6 6 4 3+2 i 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 i 2+4 i 2+ i 3 7 7 5 is k u k = q j 3+2 i j 2 + j 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 i j 2 + j 2+4 i j 2 + j 2+ i j 2 = p 13+37+20+5= p 75=5 p 3 : Thenormof v = 2 6 6 6 6 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 7 7 7 7 5 is k v k = q j 3 j 2 + j)]TJ/F15 10.9091 Tf 11.516 0 Td [(1 j 2 + j 2 j 2 + j 4 j 2 + j)]TJ/F15 10.9091 Tf 11.516 0 Td [(3 j 2 = p 3 2 +1 2 +2 2 +4 2 +3 2 = p 39 : Noticehowthenormofavectorwithrealnumberentriesisjustthelengthofthevector.Innerproducts andnormsarerelatedbythefollowingtheorem. TheoremIPN InnerProductsandNorms Supposethat u isavectorin C m .Then k u k 2 = h u ; u i Proof k u k 2 = 0 @ v u u t m X i =1 j [ u ] i j 2 1 A 2 DenitionNV[171] = m X i =1 j [ u ] i j 2 Version2.02

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SubsectionO.OVOrthogonalVectors173 = m X i =1 [ u ] i [ u ] i DenitionMCN[682] = h u ; u i DenitionIP[168] WhenourvectorshaveentriesonlyfromtherealnumbersTheoremIPN[171]saysthatthedotproduct ofavectorwithitselfisequaltothelengthofthevectorsquared. TheoremPIP PositiveInnerProducts Supposethat u isavectorin C m .Then h u ; u i 0withequalityifandonlyif u = 0 Proof FromtheproofofTheoremIPN[171]weseethat h u ; u i = j [ u ] 1 j 2 + j [ u ] 2 j 2 + j [ u ] 3 j 2 + + j [ u ] m j 2 Sinceeachmodulusissquared,everytermispositive,andthesummustalsobepositive.Noticethatin generaltheinnerproductisacomplexnumberandcannotbecomparedwithzero,butinthespecialcase of h u ; u i theresultisarealnumber.Thephrase,withequalityifandonlyif"meansthatwewantto showthatthestatement h u ; u i =0i.e.withequalityisequivalentifandonlyif"tothestatement u = 0 If u = 0 ,thenitisastraightforwardcomputationtoseethat h u ; u i =0.Intheotherdirection,assume that h u ; u i =0.Asbefore, h u ; u i isasumofmoduli.Sowehave 0= h u ; u i = j [ u ] 1 j 2 + j [ u ] 2 j 2 + j [ u ] 3 j 2 + + j [ u ] m j 2 Nowwehaveasumofsquaresequalingzero,soeachtermmustbezero.Thenbysimilarlogic, j [ u ] i j =0 willimplythat[ u ] i =0,since0+0 i istheonlycomplexnumberwithzeromodulus.Thuseveryentryof u iszeroandso u = 0 ,asdesired. NoticethatTheoremPIP[172]contains three implications: u 2 C m h u ; u i 0 u = 0 h u ; u i =0 h u ; u i =0 u = 0 TheresultscontainedinTheoremPIP[172]aresummarizedbysayingtheinnerproductis positive denite ." SubsectionOV OrthogonalVectors Orthogonal"isageneralizationofperpendicular."Youmayhaveusedmutuallyperpendicularvectorsin aphysicsclass,oryoumayrecallfromacalculusclassthatperpendicularvectorshaveazerodotproduct. Wewillnowextendtheseideasintotherealmofhigherdimensionsandcomplexscalars. DenitionOV OrthogonalVectors Apairofvectors, u and v ,from C m are orthogonal iftheirinnerproductiszero,thatis, h u ; v i =0. 4 ExampleTOV Twoorthogonalvectors Version2.02

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SubsectionO.OVOrthogonalVectors174 Thevectors u = 2 6 6 4 2+3 i 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i 1+ i 1+ i 3 7 7 5 v = 2 6 6 4 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 2+3 i 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 i 1 3 7 7 5 areorthogonalsince h u ; v i =+3 i + i + )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 i ++ i +6 i ++ i = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1+5 i + )]TJ/F15 10.9091 Tf 10.909 0 Td [(16 i + )]TJ/F15 10.9091 Tf 8.485 0 Td [(2+10 i ++ i =0+0 i: Weextendthisdenitiontowholesetsbyrequiringvectorstobepairwiseorthogonal.Despiteusing thesameword,carefulthoughtaboutwhatobjectsyouareusingwilleliminateanysourceofconfusion. DenitionOSV OrthogonalSetofVectors Supposethat S = f u 1 ; u 2 ; u 3 ;:::; u n g isasetofvectorsfrom C m .Then S isan orthogonalset ifevery pairofdierentvectorsfrom S isorthogonal,thatis h u i ; u j i =0whenever i 6 = j 4 Wenowdenetheprototypicalorthogonalset,whichwewillreferencerepeatedly. DenitionSUV StandardUnitVectors Let e j 2 C m ,1 j m denotethecolumnvectorsdenedby [ e j ] i = 0if i 6 = j 1if i = j Thentheset f e 1 ; e 2 ; e 3 ;:::; e m g = f e j j 1 j m g isthesetof standardunitvectors in C m ThisdenitioncontainsNotationSUV. 4 Noticethat e j isidenticaltocolumn j ofthe m m identitymatrix I m DenitionIM[72].This observationwilloftenbeuseful.Itisnothardtoseethatthesetofstandardunitvectorsisanorthogonal set.Wewillreservethenotation e i forthesevectors. ExampleSUVOS StandardUnitVectorsareanOrthogonalSet ComputetheinnerproductoftwodistinctvectorsfromthesetofstandardunitvectorsDenitionSUV [173],say e i e j ,where i 6 = j h e i ; e j i =0 0+0 0+ +1 0+ +0 0+ +0 1+ +0 0+0 0 =0+0+ +1+ +0+ +0+0 =0 Sotheset f e 1 ; e 2 ; e 3 ;:::; e m g isanorthogonalset. ExampleAOS Anorthogonalset Version2.02

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SubsectionO.OVOrthogonalVectors175 Theset f x 1 ; x 2 ; x 3 ; x 4 g = 8 > > < > > : 2 6 6 4 1+ i 1 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i i 3 7 7 5 ; 2 6 6 4 1+5 i 6+5 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 i 3 7 7 5 ; 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7+34 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 10.909 0 Td [(23 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(10+22 i 30+13 i 3 7 7 5 ; 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 i 6+ i 4+3 i 6 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 3 7 7 5 9 > > = > > ; isanorthogonalset.Sincetheinnerproductisanti-commutativeTheoremIPAC[170]wecantestpairs ofdierentvectorsinanyorder.Iftheresultiszero,thenitwillalsobezeroiftheinnerproductis computedintheoppositeorder.Thismeanstherearesixpairsofdierentvectorstouseinaninner productcomputation.We'lldotwoandyoucanpracticeyourinnerproductsontheotherfour. h x 1 ; x 3 i =+ i )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F15 10.9091 Tf 10.909 0 Td [(34 i + )]TJ/F15 10.9091 Tf 8.485 0 Td [(8+23 i + )]TJ/F21 10.9091 Tf 10.909 0 Td [(i )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 10.909 0 Td [(22 i + i )]TJ/F15 10.9091 Tf 10.909 0 Td [(13 i = )]TJ/F15 10.9091 Tf 10.909 0 Td [(41 i + )]TJ/F15 10.9091 Tf 8.485 0 Td [(8+23 i + )]TJ/F15 10.9091 Tf 8.485 0 Td [(32 )]TJ/F15 10.9091 Tf 10.91 0 Td [(12 i ++30 i =0+0 i and h x 2 ; x 4 i =+5 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(2+4 i ++5 i )]TJ/F21 10.9091 Tf 10.909 0 Td [(i + )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 i + )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 i + i = )]TJ/F15 10.9091 Tf 8.485 0 Td [(22 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 i ++24 i + )]TJ/F15 10.9091 Tf 8.485 0 Td [(31+17 i + )]TJ/F15 10.9091 Tf 10.91 0 Td [(35 i =0+0 i Sofar,thissectionhasseenlotsofdenitions,andlotsoftheoremsestablishingun-surprisingconsequencesofthosedenitions.Buthereisourrsttheoremthatsuggeststhatinnerproductsandorthogonal vectorshavesomeutility.Itisalsooneofourrstillustrationsofhowtoarriveatlinearindependenceas theconclusionofatheorem. TheoremOSLI OrthogonalSetsareLinearlyIndependent Supposethat S isanorthogonalsetofnonzerovectors.Then S islinearlyindependent. Proof Let S = f u 1 ; u 2 ; u 3 ;:::; u n g beanorthogonalsetofnonzerovectors.Toprovethelinear independenceof S ,wecanappealtothedenitionDenitionLICV[132]andbeginwithanarbitrary relationoflineardependenceDenitionRLDCV[132], 1 u 1 + 2 u 2 + 3 u 3 + + n u n = 0 : Then,forevery1 i n ,wehave i = 1 h u i ; u i i i h u i ; u i i TheoremPIP[172] = 1 h u i ; u i i 1 + 2 + + i h u i ; u i i + + n PropertyZCN[681] = 1 h u i ; u i i 1 h u 1 ; u i i + + i h u i ; u i i + + n h u n ; u i i DenitionOSV[173] = 1 h u i ; u i i h 1 u 1 ; u i i + h 2 u 2 ; u i i + + h n u n ; u i i TheoremIPSM[170] = 1 h u i ; u i i h 1 u 1 + 2 u 2 + 3 u 3 + + n u n ; u i i TheoremIPVA[169] = 1 h u i ; u i i h 0 ; u i i DenitionRLDCV[132] = 1 h u i ; u i i 0DenitionIP[168] Version2.02

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SubsectionO.GSPGram-SchmidtProcedure176 =0PropertyZCN[681] Soweconcludethat i =0forall1 i n inanyrelationoflineardependenceon S .Butthissaysthat S isalinearlyindependentsetsincetheonlywaytoformarelationoflineardependenceisthetrivialway DenitionLICV[132].Boom! SubsectionGSP Gram-SchmidtProcedure TheGram-SchmidtProcedureisreallyatheorem.Itsaysthatifwebeginwithalinearlyindependentset of p vectors, S ,thenwecandoanumberofcalculationswiththesevectorsandproduceanorthogonal setof p vectors, T ,sothat h S i = h T i .Giventhelargenumberofcomputationsinvolved,itisindeeda proceduretodoallthenecessarycomputations,anditisbestemployedonacomputer.However,italso hasvalueinproofswherewemayonoccasionwishtoreplacealinearlyindependentsetbyanorthogonal set. Thisisourrstoccasiontousethetechniqueofmathematicalinduction"foraproof,atechnique wewillseeagainseveraltimes,especiallyinChapterD[370].Sostudythesimpleexampledescribedin TechniqueI[694]rst. TheoremGSP Gram-SchmidtProcedure Supposethat S = f v 1 ; v 2 ; v 3 ;:::; v p g isalinearlyindependentsetofvectorsin C m .Denethevectors u i ,1 i p by u i = v i )]TJ 12.955 7.38 Td [(h v i ; u 1 i h u 1 ; u 1 i u 1 )]TJ 12.954 7.38 Td [(h v i ; u 2 i h u 2 ; u 2 i u 2 )]TJ 12.954 7.38 Td [(h v i ; u 3 i h u 3 ; u 3 i u 3 )-222()]TJ 43.749 7.38 Td [(h v i ; u i )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 i h u i )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 ; u i )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 i u i )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 Thenif T = f u 1 ; u 2 ; u 3 ;:::; u p g ,then T isanorthogonalsetofnon-zerovectors,and h T i = h S i Proof Wewillprovetheresultbyusinginductionon p TechniqueI[694].Tobegin,weprovethat T hasthedesiredpropertieswhen p =1.Inthiscase u 1 = v 1 and T = f u 1 g = f v 1 g = S .Because S and T areequal, h S i = h T i .Equallytrivial, T isanorthogonalset.If u 1 = 0 ,then S wouldbealinearly dependentset,acontradiction. Supposethatthetheoremistrueforanysetof p )]TJ/F15 10.9091 Tf 8.485 0 Td [(1linearlyindependentvectors.Let S = f v 1 ; v 2 ; v 3 ;:::; v p g bealinearlyindependentsetof p vectors.Then S 0 = f v 1 ; v 2 ; v 3 ;:::; v p )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 g isalsolinearlyindependent. Sowecanapplythetheoremto S 0 andconstructthevectors T 0 = f u 1 ; u 2 ; u 3 ;:::; u p )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 g T 0 istherefore anorthogonalsetofnonzerovectorsand h S 0 i = h T 0 i .Dene u p = v p )]TJ 12.264 7.441 Td [(h v p ; u 1 i h u 1 ; u 1 i u 1 )]TJ 12.265 7.441 Td [(h v p ; u 2 i h u 2 ; u 2 i u 2 )]TJ 12.265 7.441 Td [(h v p ; u 3 i h u 3 ; u 3 i u 3 )-222()]TJ 43.749 7.441 Td [(h v p ; u p )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 i h u p )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 ; u p )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 i u p )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 andlet T = T 0 [f u p g .Weneedtonowshowthat T hasseveralpropertiesbybuildingonwhatweknow about T 0 .Butrstnoticethattheaboveequationhasnoproblemswiththedenominators h u i ; u i i being zero,sincethe u i arefrom T 0 ,whichiscomposedofnonzerovectors. Weshowthat h T i = h S i ,byrstestablishingthat h T ih S i .Suppose x 2h T i ,so x = a 1 u 1 + a 2 u 2 + a 3 u 3 + + a p u p Theterm a p u p isalinearcombinationofvectorsfrom T 0 andthevector v p ,whiletheremainingtermsare alinearcombinationofvectorsfrom T 0 .Since h T 0 i = h S 0 i ,anytermthatisamultipleofavectorfrom T 0 canberewrittenasalinearcombinationofvectorsfrom S 0 .Theremainingterm a p v p isamultipleofa vectorin S .Soweseethat x canberewrittenasalinearcombinationofvectorsfrom S ,i.e. x 2h S i Version2.02

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SubsectionO.GSPGram-SchmidtProcedure177 Toshowthat h S ih T i ,beginwith y 2h S i ,so y = a 1 v 1 + a 2 v 2 + a 3 v 3 + + a p v p Rearrangeourdeningequationfor u p bysolvingfor v p .Thentheterm a p v p isamultipleofalinear combinationofelementsof T .Theremainingtermsarealinearcombinationof v 1 ; v 2 ; v 3 ;:::; v p )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 henceanelementof h S 0 i = h T 0 i .Thustheseremainingtermscanbewrittenasalinearcombinationof thevectorsin T 0 .So y isalinearcombinationofvectorsfrom T ,i.e. y 2h T i Theelementsof T 0 arenonzero,butwhatabout u p ?Supposetothecontrarythat u p = 0 0 = u p = v p )]TJ 12.264 7.441 Td [(h v p ; u 1 i h u 1 ; u 1 i u 1 )]TJ 12.265 7.441 Td [(h v p ; u 2 i h u 2 ; u 2 i u 2 )]TJ 12.265 7.441 Td [(h v p ; u 3 i h u 3 ; u 3 i u 3 )-222()]TJ 43.749 7.441 Td [(h v p ; u p )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 i h u p )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 ; u p )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 i u p )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 v p = h v p ; u 1 i h u 1 ; u 1 i u 1 + h v p ; u 2 i h u 2 ; u 2 i u 2 + h v p ; u 3 i h u 3 ; u 3 i u 3 + + h v p ; u p )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 i h u p )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 ; u p )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 i u p )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 Since h S 0 i = h T 0 i wecanwritethevectors u 1 ; u 2 ; u 3 ;:::; u p )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 ontherightsideofthisequationinterms ofthevectors v 1 ; v 2 ; v 3 ;:::; v p )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 andwethenhavethevector v p expressedasalinearcombinationof theother p )]TJ/F15 10.9091 Tf 10.853 0 Td [(1vectorsin S ,implyingthat S isalinearlydependentsetTheoremDLDS[152],contrary toourlonehypothesisabout S Finally,itisasimplemattertoestablishthat T isanorthogonalset,thoughitwillnotappearso simplelooking.Thinkaboutyourobjectsasyouworkthroughthefollowing|whatisavectorandwhat isascalar.Since T 0 isanorthogonalsetbyinduction,mostpairsofelementsin T arealreadyknownto beorthogonal.Wejustneedtotestnew"innerproducts,between u p and u i ,for1 i p )]TJ/F15 10.9091 Tf 10.837 0 Td [(1.Herewe go,usingsummationnotation, h u p ; u i i = v p )]TJ/F22 7.9701 Tf 11.247 14.216 Td [(p )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 X k =1 h v p ; u k i h u k ; u k i u k ; u i + = h v p ; u i i)]TJ/F27 10.9091 Tf 17.576 18.655 Td [(* p )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 X k =1 h v p ; u k i h u k ; u k i u k ; u i + TheoremIPVA[169] = h v p ; u i i)]TJ/F22 7.9701 Tf 17.913 14.216 Td [(p )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 X k =1 h v p ; u k i h u k ; u k i u k ; u i TheoremIPVA[169] = h v p ; u i i)]TJ/F22 7.9701 Tf 17.913 14.216 Td [(p )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 X k =1 h v p ; u k i h u k ; u k i h u k ; u i i TheoremIPSM[170] = h v p ; u i i)]TJ 18.771 7.44 Td [(h v p ; u i i h u i ; u i i h u i ; u i i)]TJ/F27 10.9091 Tf 17.575 10.364 Td [(X k 6 = i h v p ; u k i h u k ; u k i InductionHypothesis = h v p ; u i i)-222(h v p ; u i i)]TJ/F27 10.9091 Tf 17.576 10.364 Td [(X k 6 = i 0 =0 ExampleGSTV Gram-Schmidtofthreevectors WewillillustratetheGram-Schmidtprocesswiththreevectors.Beginwiththelinearlyindependentcheck this!set S = f v 1 ; v 2 ; v 3 g = 8 < : 2 4 1 1+ i 1 3 5 ; 2 4 )]TJ/F21 10.9091 Tf 8.485 0 Td [(i 1 1+ i 3 5 ; 2 4 0 i i 3 5 9 = ; Version2.02

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SubsectionO.GSPGram-SchmidtProcedure178 Then u 1 = v 1 = 2 4 1 1+ i 1 3 5 u 2 = v 2 )]TJ 12.279 7.38 Td [(h v 2 ; u 1 i h u 1 ; u 1 i u 1 = 1 4 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 i 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 2+5 i 3 5 u 3 = v 3 )]TJ 12.279 7.38 Td [(h v 3 ; u 1 i h u 1 ; u 1 i u 1 )]TJ 12.278 7.38 Td [(h v 3 ; u 2 i h u 2 ; u 2 i u 2 = 1 11 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 1+3 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 3 5 and T = f u 1 ; u 2 ; u 3 g = 8 < : 2 4 1 1+ i 1 3 5 ; 1 4 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 i 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 2+5 i 3 5 ; 1 11 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 1+3 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 3 5 9 = ; isanorthogonalsetwhichyoucancheckofnonzerovectorsand h T i = h S i allbyTheoremGSP[175]. Ofcourse,asaby-productoforthogonality,theset T isalsolinearlyindependentTheoremOSLI[174]. Onenaldenitionrelatedtoorthogonalvectors. DenitionONS OrthoNormalSet Suppose S = f u 1 ; u 2 ; u 3 ;:::; u n g isanorthogonalsetofvectorssuchthat k u i k =1forall1 i n Then S isan orthonormal setofvectors. 4 Onceyouhaveanorthogonalset,itiseasytoconvertittoanorthonormalset|multiplyeachvector bythereciprocalofitsnorm,andtheresultingvectorwillhavenorm1.Thisscalingofeachvectorwill notaecttheorthogonalitypropertiesapplyTheoremIPSM[170]. ExampleONTV Orthonormalset,threevectors Theset T = f u 1 ; u 2 ; u 3 g = 8 < : 2 4 1 1+ i 1 3 5 ; 1 4 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 i 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 2+5 i 3 5 ; 1 11 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 1+3 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 3 5 9 = ; fromExampleGSTV[176]isanorthogonalset.Wecomputethenormofeachvector, k u 1 k =2 k u 2 k = 1 2 p 11 k u 3 k = p 2 p 11 Convertingeachvectortoanormof1,yieldsanorthonormalset, w 1 = 1 2 2 4 1 1+ i 1 3 5 w 2 = 1 1 2 p 11 1 4 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 i 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 2+5 i 3 5 = 1 2 p 11 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 i 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 2+5 i 3 5 w 3 = 1 p 2 p 11 1 11 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 1+3 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 3 5 = 1 p 22 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 1+3 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 3 5 Version2.02

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SubsectionO.READReadingQuestions179 ExampleONFV Orthonormalset,fourvectors Asanexerciseconvertthelinearlyindependentset S = 8 > > < > > : 2 6 6 4 1+ i 1 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i i 3 7 7 5 ; 2 6 6 4 i 1+ i )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F21 10.9091 Tf 8.485 0 Td [(i 3 7 7 5 ; 2 6 6 4 i )]TJ/F21 10.9091 Tf 8.485 0 Td [(i )]TJ/F15 10.9091 Tf 8.485 0 Td [(1+ i 1 3 7 7 5 ; 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i i 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 5 9 > > = > > ; toanorthogonalsetviatheGram-SchmidtProcessTheoremGSP[175]andthenscalethevectorsto norm1tocreateanorthonormalset.Youshouldgetthesamesetyouwouldifyouscaledtheorthogonal setofExampleAOS[173]tobecomeanorthonormalset. ItiscrazytodoallbutthesimplestandsmallestinstancesoftheGram-Schmidtprocedurebyhand. Well,OK,maybejustonceortwicetogetagoodunderstandingofTheoremGSP[175].Afterthat,leta machinedotheworkforyou.That'swhattheyarefor.See:ComputationGSP.MMA[670]. WewillseeorthonormalsetsagaininSubsectionMINM.UM[229].Theyareintimatelyrelatedto unitarymatricesDenitionUM[229]throughTheoremCUMOS[230].Someoftheutilityoforthonormal setsiscapturedbyTheoremCOB[332]inSubsectionB.OBC[331].Orthonormalsetsappearonceagain inSectionOD[601]wheretheyarekeyinorthonormaldiagonalization. SubsectionREAD ReadingQuestions 1.Istheset 8 < : 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 5 ; 2 4 5 3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 3 5 ; 2 4 8 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 5 9 = ; anorthogonalset?Why? 2.Whatisthedistinctionbetweenanorthogonalsetandanorthonormalset? 3.WhatisniceabouttheoutputoftheGram-Schmidtprocess? Version2.02

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SubsectionO.EXCExercises180 SubsectionEXC Exercises C20 CompleteExampleAOS[173]byverifyingthatthefourremaininginnerproductsarezero. ContributedbyRobertBeezer C21 Verifythattheset T createdinExampleGSTV[176]bytheGram-SchmidtProcedureisanorthogonalset. ContributedbyRobertBeezer T10 Provepart1oftheconclusionofTheoremIPVA[169]. ContributedbyRobertBeezer T11 Provepart1oftheconclusionofTheoremIPSM[170]. ContributedbyRobertBeezer T20 Supposethat u ; v ; w 2 C n ; 2 C and u isorthogonaltoboth v and w .Provethat u is orthogonalto v + w ContributedbyRobertBeezerSolution[180] T30 Supposethattheset S inthehypothesisofTheoremGSP[175]isnotjustlinearlyindependent, butisalsoorthogonal.Provethattheset T createdbytheGram-Schmidtprocedureisequalto S .Note thatwearegettingastrongerconclusionthan h T i = h S i |theconclusionisthat T = S .Inotherwords, itispointlesstoapplytheGram-Schmidtproceduretoasetthatisalreadyorthogonal. ContributedbySteveCaneld Version2.02

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SubsectionO.SOLSolutions181 SubsectionSOL Solutions T20 ContributedbyRobertBeezerStatement[179] VectorsareorthogonaliftheirinnerproductiszeroDenitionOV[172],sowecompute, h v + w ; u i = h v ; u i + h w ; u i TheoremIPVA[169] = h v ; u i + h w ; u i TheoremIPSM[170] = + DenitionOV[172] =0 SobyDenitionOV[172], u and v + w areanorthogonalpairofvectors. Version2.02

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AnnotatedAcronymsO.VVectors182 AnnotatedAcronymsV Vectors TheoremVSPCV[86] Thesearethefundamentalrulesforworkingwiththeaddition,andscalarmultiplication,ofcolumnvectors. WewillseesomethingverysimilarinthenextchapterTheoremVSPM[184]andthenthiswillbe generalizedintowhatisarguablyourmostimportantdenition,DenitionVS[279]. TheoremSLSLC[93] Vectoradditionandscalarmultiplicationarethetwofundamentaloperationsonvectors,andlinearcombinationsrollthembothintoone.TheoremSLSLC[93]connectslinearcombinationswithsystemsof equations.Thisonewewillseeoftenenoughthatitisworthmemorizing. TheoremPSPHS[105] Thistheoremisinterestinginitsownright,andsometimesthevaugenesssurroundingthechoiceof z can seemmysterious.ButwelistitherebecausewewillseeanimportanttheoreminSectionILT[477]which willgeneralizethisresultTheoremKPI[483]. TheoremLIVRN[136] Ifyouhaveasetofcolumnvectors,thisisthefastestcomputationalapproachtodetermineifthesetis linearlyindependent.Makethevectorsthecolumnsofamatrix,row-reduce,compare r and n .That'sit |andyoualwaysgetananswer.Putthisoneinyourtoolkit. TheoremBNS[139] WewillhaveseveraltheoremsalllistedintheseAnnotatedAcronyms"sectionswhoseconclusionswill providealinearlyindependentsetofvectorswhosespanequalssomesetofinterestthenullspacehere. Whilethenotationinthistheoremmightappearagruesome,inpracticeitcanbecomeveryroutineto apply.Sopracticethisone|we'llbeusingitallthroughthebook. TheoremBS[157] Aspromised,anothertheoremthatprovidesalinearlyindependentsetofvectorswhosespanequalssome setofinterestaspannow.Youcanusethisonetocleanup any span. Version2.02

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ChapterM Matrices Wehavemadefrequentuseofmatricesforsolvingsystemsofequations,andwehavebeguntoinvestigate afewoftheirproperties,suchasthenullspaceandnonsingularity.Inthischapter,wewilltakeamore systematicapproachtothestudyofmatrices. SectionMO MatrixOperations Inthissectionwewillbackupandstartsimple.Firstadenitionofatotallygeneralsetofmatrices. DenitionVSM VectorSpaceof m n Matrices Thevectorspace M mn isthesetofall m n matriceswithentriesfromthesetofcomplexnumbers. ThisdenitioncontainsNotationVSM. 4 SubsectionMEASM MatrixEquality,Addition,ScalarMultiplication Justaswemade,andused,acarefuldenitionofequalityforcolumnvectors,sotoo,wehaveprecise denitionsformatrices. DenitionME MatrixEquality The m n matrices A and B are equal ,written A = B provided[ A ] ij =[ B ] ij forall1 i m ,1 j n ThisdenitioncontainsNotationME. 4 Soequalityofmatricestranslatestotheequalityofcomplexnumbers,onanentry-by-entrybasis.Notice thatwenowhaveyetanotherdenitionthatusesthesymbol="forshorthand.Wheneveratheorem hasaconclusionsayingtwomatricesareequalthinkaboutyourobjects,wewillconsiderappealing tothisdenitionasawayofformulatingthetop-levelstructureoftheproof.Wewillnowdenetwo operationsontheset M mn .Again,wewilloverloadasymbol`+'andaconventionjuxtapositionfor scalarmultiplication. DenitionMA MatrixAddition Giventhe m n matrices A and B ,denethe sum of A and B asan m n matrix,written A + B 183

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SubsectionMO.MEASMMatrixEquality,Addition,ScalarMultiplication184 accordingto [ A + B ] ij =[ A ] ij +[ B ] ij 1 i m; 1 j n ThisdenitioncontainsNotationMA. 4 Somatrixadditiontakestwomatricesofthesamesizeandcombinestheminanaturalway!tocreate anewmatrixofthesamesize.Perhapsthisistheobvious"thingtodo,butitdoesn'trelieveusfrom theobligationtostateitcarefully. ExampleMA Additionoftwomatricesin M 23 If A = 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(34 10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 B = 62 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 352 then A + B = 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(34 10 )]TJ/F15 10.9091 Tf 8.484 0 Td [(7 + 62 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 352 = 2+6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3+24+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 1+30+5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7+2 = 8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 45 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 Oursecondoperationtakestwoobjectsofdierenttypes,specicallyanumberandamatrix,and combinesthemtocreateanothermatrix.Aswithvectors,inthiscontextwecallanumbera scalar in ordertoemphasizethatitisnotamatrix. DenitionMSM MatrixScalarMultiplication Giventhe m n matrix A andthescalar 2 C ,the scalarmultiple of A isan m n matrix,written A anddenedaccordingto [ A ] ij = [ A ] ij 1 i m; 1 j n ThisdenitioncontainsNotationMSM. 4 Noticeagainthatwehaveyetanotherkindofmultiplication,anditisagainwrittenputtingtwo symbolsside-by-side.Computationally,scalarmatrixmultiplicationisveryeasy. ExampleMSM Scalarmultiplicationin M 32 If A = 2 4 28 )]TJ/F15 10.9091 Tf 8.485 0 Td [(35 01 3 5 and =7,then A =7 2 4 28 )]TJ/F15 10.9091 Tf 8.485 0 Td [(35 01 3 5 = 2 4 77 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(37 77 3 5 = 2 4 1456 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2135 07 3 5 Version2.02

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SubsectionMO.VSPVectorSpaceProperties185 SubsectionVSP VectorSpaceProperties Withdenitionsofmatrixadditionandscalarmultiplicationwecannowstate,andprove,severalproperties ofeachoperation,andsomepropertiesthatinvolvetheirinterplay.Wenowcollecttenofthemherefor laterreference. TheoremVSPM VectorSpacePropertiesofMatrices Supposethat M mn isthesetofall m n matricesDenitionVSM[182]withadditionandscalar multiplicationasdenedinDenitionMA[182]andDenitionMSM[183].Then ACMAdditiveClosure,Matrices If A;B 2 M mn ,then A + B 2 M mn SCMScalarClosure,Matrices If 2 C and A 2 M mn ,then A 2 M mn CMCommutativity,Matrices If A;B 2 M mn ,then A + B = B + A AAMAdditiveAssociativity,Matrices If A;B;C 2 M mn ,then A + B + C = A + B + C ZMZeroVector,Matrices Thereisamatrix, O ,calledthe zeromatrix ,suchthat A + O = A forall A 2 M mn AIMAdditiveInverses,Matrices If A 2 M mn ,thenthereexistsamatrix )]TJ/F21 10.9091 Tf 8.485 0 Td [(A 2 M mn sothat A + )]TJ/F21 10.9091 Tf 8.485 0 Td [(A = O SMAMScalarMultiplicationAssociativity,Matrices If ; 2 C and A 2 M mn ,then A = A DMAMDistributivityacrossMatrixAddition,Matrices If 2 C and A;B 2 M mn ,then A + B = A + B DSAMDistributivityacrossScalarAddition,Matrices If ; 2 C and A 2 M mn ,then + A = A + A OMOne,Matrices If A 2 M mn ,then1 A = A Proof Whilesomeofthesepropertiesseemveryobvious,theyallrequireproof.However,theproofsare notveryinteresting,andborderontedious.We'llproveoneversionofdistributivityverycarefully,and youcantestyourproof-buildingskillsonsomeoftheothers.We'llgiveournewnotationformatrixentries aworkouthere.ComparethestyleoftheproofsherewiththosegivenforvectorsinTheoremVSPCV[86] |whiletheobjectsherearemorecomplicated,ournotationmakestheproofscleaner. ToprovePropertyDSAM[184], + A = A + A ,weneedtoestablishtheequalityoftwomatrices seeTechniqueGS[689].DenitionME[182]saysweneedtoestablishtheequalityoftheirentries, one-by-one.Howdowedothis,whenwedonotevenknowhowmanyentriesthetwomatricesmight have?ThisiswhereNotationME[182]comesintoplay.Ready?Herewego. Version2.02

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SubsectionMO.TSMTransposesandSymmetricMatrices186 For any i and j ,1 i m ,1 j n [ + A ] ij = + [ A ] ij DenitionMSM[183] = [ A ] ij + [ A ] ij Distributivityin C =[ A ] ij +[ A ] ij DenitionMSM[183] =[ A + A ] ij DenitionMA[182] Thereareseveralthingstonoticehere.Eachequalssignisanequalityofnumbers.Thetwoends oftheequation,beingtrueforany i and j ,allowustoconcludetheequalityofthematricesbyDenition ME[182].Thereareseveralplussigns,andseveralinstancesofjuxtaposition.Identifyeachone,and stateexactlywhatoperationisbeingrepresentedbyeach. Fornow,notethesimilaritiesbetweenTheoremVSPM[184]aboutmatricesandTheoremVSPCV[86] aboutvectors. Thezeromatrixdescribedinthistheorem, O ,iswhatyouwouldexpect|amatrixfullofzeros. DenitionZM ZeroMatrix The m n zeromatrix iswrittenas O = O m n anddenedby[ O ] ij =0,forall1 i m ,1 j n ThisdenitioncontainsNotationZM. 4 SubsectionTSM TransposesandSymmetricMatrices Wedescribeonemorecommonoperationwecanperformonmatrices.Informally,totransposeamatrix istobuildanewmatrixbyswappingitsrowsandcolumns. DenitionTM TransposeofaMatrix Givenan m n matrix A ,its transpose isthe n m matrix A t givenby A t ij =[ A ] ji ; 1 i n; 1 j m: ThisdenitioncontainsNotationTM. 4 ExampleTM Transposeofa 3 4 matrix Suppose D = 2 4 372 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1428 03 )]TJ/F15 10.9091 Tf 8.485 0 Td [(25 3 5 : Wecouldformulatethetranspose,entry-by-entry,usingthedenition.Butitiseasiertojustsystematically rewriterowsascolumnsorvice-versa.Theformofthedenitiongivenwillbemoreusefulinproofs.So wehave D t = 2 6 6 4 3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(10 743 22 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(385 3 7 7 5 Version2.02

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SubsectionMO.TSMTransposesandSymmetricMatrices187 Itwillsometimeshappenthatamatrixisequaltoitstranspose.Inthiscase,wewillcallamatrix symmetric .Thesematricesoccurnaturallyincertainsituations,andalsohavesomeniceproperties,so itisworthstatingthedenitioncarefully.Informallyamatrixissymmetricifwecanip"itaboutthe maindiagonalupper-leftcorner,runningdowntothelower-rightcornerandhaveitlookunchanged. DenitionSYM SymmetricMatrix Thematrix A is symmetric if A = A t 4 ExampleSYM Asymmetric 5 5 matrix Thematrix E = 2 6 6 6 6 4 23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(957 316 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(960 )]TJ/F15 10.9091 Tf 8.485 0 Td [(19 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(39 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 7 7 7 7 5 issymmetric. YoumighthavenoticedthatDenitionSYM[186]didnotspecifythesizeofthematrix A ,ashasbeen ourcustom.That'sbecauseitwasn'tnecessary.Analternativewouldhavebeentostatethedenition justforsquarematrices,butthisisthesubstanceofthenextproof.Beforereadingthenextproof,we wanttooeryousomeadviceabouthowtobecomemoreprocientatconstructingproofs.Perhapsyou canapplythisadvicetothenexttheorem.HaveapeekatTechniqueP[695]now. TheoremSMS SymmetricMatricesareSquare Supposethat A isasymmetricmatrix.Then A issquare. Proof Westartbyspecifying A 'ssize,withoutassumingitissquare,sincewearetryingto prove that, sowecan'talsoassumeit.Suppose A isan m n matrix.Because A issymmetric,weknowbyDenition SM[375]that A = A t .So,inparticular,DenitionME[182]requiresthat A and A t musthavethesame size.Thesizeof A t is n m .Because A has m rowsand A t has n rows,weconcludethat m = n ,and hence A mustbesquarebyDenitionSQM[71]. Wenishthissectionwiththreeeasytheorems,buttheyillustratetheinterplayofourthreenew operations,ournewnotation,andthetechniquesusedtoprovematrixequalities. TheoremTMA TransposeandMatrixAddition Supposethat A and B are m n matrices.Then A + B t = A t + B t Proof Thestatementtobeprovedisanequalityofmatrices,soweworkentry-by-entryanduseDenition ME[182].Thinkcarefullyabouttheobjectsinvolvedhere,andthemanyusesoftheplussign.For 1 i m ,1 j n A + B t ij =[ A + B ] ji DenitionTM[185] =[ A ] ji +[ B ] ji DenitionMA[182] = A t ij + B t ij DenitionTM[185] = A t + B t ij DenitionMA[182] Version2.02

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SubsectionMO.MCCMatricesandComplexConjugation188 Sincethematrices A + B t and A t + B t agreeateachentry,DenitionME[182]tellsusthetwomatrices areequal. TheoremTMSM TransposeandMatrixScalarMultiplication Supposethat 2 C and A isan m n matrix.Then A t = A t Proof Thestatementtobeprovedisanequalityofmatrices,soweworkentry-by-entryanduseDenition ME[182].Noticethatthedesiredequalityisof n m matrices,andthinkcarefullyabouttheobjects involvedhere,plusthemanyusesofjuxtaposition.For1 i m ,1 j n A t ji =[ A ] ij DenitionTM[185] = [ A ] ij DenitionMSM[183] = A t ji DenitionTM[185] = A t ji DenitionMSM[183] Sincethematrices A t and A t agreeateachentry,DenitionME[182]tellsusthetwomatricesare equal. TheoremTT TransposeofaTranspose Supposethat A isan m n matrix.Then )]TJ/F21 10.9091 Tf 5 -8.837 Td [(A t t = A Proof Weagainwanttoproveanequalityofmatrices,soweworkentry-by-entryanduseDenitionME [182].For1 i m ,1 j n h )]TJ/F21 10.9091 Tf 5 -8.836 Td [(A t t i ij = A t ji DenitionTM[185] =[ A ] ij DenitionTM[185] Itsusuallystraightforwardtocoaxthetransposeofamatrixoutofacomputationaldevice.See:ComputationTM.MMA[671]ComputationTM.TI86[673]ComputationTM.SAGE[677]. SubsectionMCC MatricesandComplexConjugation AswedidwithvectorsDenitionCCCV[167],wecandenewhatitmeanstotaketheconjugateofa matrix. DenitionCCM ComplexConjugateofaMatrix Suppose A isan m n matrix.Thenthe conjugate of A ,written A isan m n matrixdenedby A ij = [ A ] ij ThisdenitioncontainsNotationCCM. 4 ExampleCCM Complexconjugateofamatrix If A = 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 35+4 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(3+6 i 2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(3 i 0 Version2.02

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SubsectionMO.MCCMatricesandComplexConjugation189 then A = 2+ i 35 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 i )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 i 2+3 i 0 Theinterplaybetweentheconjugateofamatrixandthetwooperationsonmatricesiswhatyoumight expect. TheoremCRMA ConjugationRespectsMatrixAddition Supposethat A and B are m n matrices.Then A + B = A + B Proof For1 i m ,1 j n A + B ij = [ A + B ] ij DenitionCCM[187] = [ A ] ij +[ B ] ij DenitionMA[182] = [ A ] ij + [ B ] ij TheoremCCRA[681] = A ij + B ij DenitionCCM[187] = A + B ij DenitionMA[182] Sincethematrices A + B and A + B areequalineachentry,DenitionME[182]saysthat A + B = A + B TheoremCRMSM ConjugationRespectsMatrixScalarMultiplication Supposethat 2 C and A isan m n matrix.Then A = A Proof For1 i m ,1 j n A ij = [ A ] ij DenitionCCM[187] = [ A ] ij DenitionMSM[183] = [ A ] ij TheoremCCRM[682] = A ij DenitionCCM[187] = A ij DenitionMSM[183] Sincethematrices A and A areequalineachentry,DenitionME[182]saysthat A = A TheoremCCM ConjugateoftheConjugateofaMatrix Supposethat A isan m n matrix.Then )]TJETq1 0 0 1 264.672 199.682 cm[]0 d 0 J 0.436 w 0 0 m 8.182 0 l SQBT/F21 10.9091 Tf 264.672 190.701 Td [(A = A Proof For1 i m ,1 j n h )]TJETq1 0 0 1 173.015 154.252 cm[]0 d 0 J 0.436 w 0 0 m 8.182 0 l SQBT/F21 10.9091 Tf 173.015 145.271 Td [(A i ij = A ij DenitionCCM[187] = [ A ] ij DenitionCCM[187] =[ A ] ij TheoremCCT[682] Sincethematrices )]TJETq1 0 0 1 150 82.329 cm[]0 d 0 J 0.436 w 0 0 m 8.182 0 l SQBT/F21 10.9091 Tf 150 73.347 Td [(A and A areequalineachentry,DenitionME[182]saysthat )]TJETq1 0 0 1 464.425 82.329 cm[]0 d 0 J 0.436 w 0 0 m 8.182 0 l SQBT/F21 10.9091 Tf 464.425 73.347 Td [(A = A Finally,wewillneedthefollowingresultaboutmatrixconjugationandtransposeslater. Version2.02

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SubsectionMO.AMAdjointofaMatrix190 TheoremMCT MatrixConjugationandTransposes Supposethat A isan m n matrix.Then A t = )]TJETq1 0 0 1 299.44 708.924 cm[]0 d 0 J 0.436 w 0 0 m 8.182 0 l SQBT/F21 10.9091 Tf 299.44 699.943 Td [(A t Proof For1 i m ,1 j n h A t i ji = [ A t ] ji DenitionCCM[187] = [ A ] ij DenitionTM[185] = A ij DenitionCCM[187] = h )]TJETq1 0 0 1 219.749 597.596 cm[]0 d 0 J 0.436 w 0 0 m 8.182 0 l SQBT/F21 10.9091 Tf 219.749 588.615 Td [(A t i ji DenitionTM[185] Sincethematrices A t and )]TJETq1 0 0 1 195.071 547.135 cm[]0 d 0 J 0.436 w 0 0 m 8.182 0 l SQBT/F21 10.9091 Tf 195.071 538.153 Td [(A t areequalineachentry,DenitionME[182]saysthat A t = )]TJETq1 0 0 1 514.79 547.135 cm[]0 d 0 J 0.436 w 0 0 m 8.182 0 l SQBT/F21 10.9091 Tf 514.79 538.153 Td [(A t SubsectionAM AdjointofaMatrix Thecombinationoftransposingandconjugatingamatrixwillbeimportantinsubsequentsections,such asSubsectionMINM.UM[229]andSectionOD[601].Wemakeakeydenitionhereandprovesomebasic resultsinthesamespiritasthoseabove. DenitionA Adjoint If A isasquarematrix,thenits adjoint is A = )]TJETq1 0 0 1 294.922 380.862 cm[]0 d 0 J 0.436 w 0 0 m 8.182 0 l SQBT/F21 10.9091 Tf 294.922 371.88 Td [(A t ThisdenitioncontainsNotationA. 4 Youwillseetheadjointwrittenelsewherevariouslyas A H A or A y .NoticethatTheoremMCT[189] saysitdoesnotreallymatterifweconjugateandthentranspose,ortransposeandthenconjugate. TheoremAMA AdjointandMatrixAddition Suppose A and B arematricesofthesamesize.Then A + B = A + B Proof A + B = )]TJETq1 0 0 1 211.169 241.629 cm[]0 d 0 J 0.436 w 0 0 m 30.337 0 l SQBT/F21 10.9091 Tf 211.169 232.648 Td [(A + B t DenitionA[189] = )]TJETq1 0 0 1 211.169 222.196 cm[]0 d 0 J 0.436 w 0 0 m 8.182 0 l SQBT/F21 10.9091 Tf 211.169 213.214 Td [(A + B t TheoremCRMA[188] = )]TJETq1 0 0 1 211.169 202.762 cm[]0 d 0 J 0.436 w 0 0 m 8.182 0 l SQBT/F21 10.9091 Tf 211.169 193.78 Td [(A t + )]TJETq1 0 0 1 246.24 202.762 cm[]0 d 0 J 0.436 w 0 0 m 8.822 0 l SQBT/F21 10.9091 Tf 246.24 193.78 Td [(B t TheoremTMA[186] = A + B DenitionA[189] TheoremAMSM AdjointandMatrixScalarMultiplication Suppose 2 C isascalarand A isamatrix.Then A = A Proof A = )]TJETq1 0 0 1 207.41 62.982 cm[]0 d 0 J 0.436 w 0 0 m 15.201 0 l SQBT/F21 10.9091 Tf 207.41 54 Td [(A t DenitionA[189] Version2.02

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SubsectionMO.READReadingQuestions191 = )]TJETq1 0 0 1 207.41 732.84 cm[]0 d 0 J 0.436 w 0 0 m 7.019 0 l SQBT/F21 10.9091 Tf 207.41 726.616 Td [( A t TheoremCRMSM[188] = )]TJETq1 0 0 1 216.247 716.165 cm[]0 d 0 J 0.436 w 0 0 m 8.182 0 l SQBT/F21 10.9091 Tf 216.247 707.183 Td [(A t TheoremTMSM[187] = A DenitionA[189] TheoremAA AdjointofanAdjoint Supposethat A isamatrix.Then A = A Proof A = A t DenitionA[189] = )]TJ/F15 10.9091 Tf 5 -8.836 Td [( A t TheoremMCT[189] = )]TJETq1 0 0 1 217.899 527.504 cm[]0 d 0 J 0.436 w 0 0 m 8.182 0 l SQBT/F21 10.9091 Tf 217.899 518.522 Td [(A t t DenitionA[189] = )]TJETq1 0 0 1 203.353 501.337 cm[]0 d 0 J 0.436 w 0 0 m 8.182 0 l SQBT/F21 10.9091 Tf 203.353 492.355 Td [(A TheoremTT[187] = A TheoremCCM[188] Takenoteofhowthetheoremsinthissection,whilesimple,buildonearliertheoremsanddenitions andneverdescendtothelevelofentry-by-entryproofsbasedonDenitionME[182].Inotherwords,the equalsignsthatappearinthepreviousproofsareequalitiesofmatrices,notscalarswhichistheopposite ofaprooflikethatofTheoremTMA[186]. SubsectionREAD ReadingQuestions 1.Performthefollowingmatrixcomputation. 2 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(281 45 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(302 3 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 4 2712 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(105 1733 3 5 2.TheoremVSPM[184]remindsyouofwhatprevioustheorem?Howstrongisthesimilarity? 3.Computethetransposeofthematrixbelow. 2 4 684 )]TJ/F15 10.9091 Tf 8.485 0 Td [(210 9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(56 3 5 Version2.02

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SubsectionMO.EXCExercises192 SubsectionEXC Exercises InChapterV[83]wedenedtheoperationsofvectoradditionandvectorscalarmultiplicationinDenition CVA[84]andDenitionCVSM[85].Thesetwooperationsformedtheunderpinningsoftheremainderof thechapter.WehavenowdenedsimilaroperationsformatricesinDenitionMA[182]andDenition MSM[183].YouwillhavenoticedtheresultingsimilaritiesbetweenTheoremVSPCV[86]andTheorem VSPM[184]. InExercisesM20{M25,youwillbeaskedtoextendthesesimilaritiestootherfundamentaldenitions andconceptswerstsawinChapterV[83].ThissequenceofproblemswassuggestedbyMartinJackson. M20 Suppose S = f B 1 ;B 2 ;B 3 ;:::;B p g isasetofmatricesfrom M mn .Formulateappropriatedefinitionsforthefollowingtermsandgiveanexampleoftheuseofeach. 1.Alinearcombinationofelementsof S 2.Arelationoflineardependenceon S ,bothtrivialandnon-trivial. 3. S isalinearlyindependentset. 4. h S i ContributedbyRobertBeezer M21 Showthattheset S islinearlyindependentin M 2 ; 2 S = 10 00 ; 01 00 ; 00 10 ; 00 01 ContributedbyRobertBeezer M22 Determineiftheset S = )]TJ/F15 10.9091 Tf 8.485 0 Td [(234 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(22 0 )]TJ/F15 10.9091 Tf 8.484 0 Td [(11 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 222 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(110 )]TJ/F15 10.9091 Tf 8.484 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 islinearlyindependentin M 2 ; 3 ContributedbyRobertBeezer M23 Determineifthematrix A isinthespanof S .Inotherwords,is A 2h S i ?Ifsowrite A asalinear combinationoftheelementsof S A = )]TJ/F15 10.9091 Tf 8.485 0 Td [(13242 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 S = )]TJ/F15 10.9091 Tf 8.484 0 Td [(234 )]TJ/F15 10.9091 Tf 8.484 0 Td [(13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(22 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 222 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(110 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 ContributedbyRobertBeezer M24 Suppose Y isthesetofall3 3symmetricmatricesDenitionSYM[186].Findaset T sothat T islinearlyindependentand h T i = Y ContributedbyRobertBeezer Version2.02

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SubsectionMO.EXCExercises193 M25 Deneasubsetof M 3 ; 3 by U 33 = n A 2 M 3 ; 3 j [ A ] ij =0whenever i>j o Findaset R sothat R islinearlyindependentand h R i = U 33 ContributedbyRobertBeezer T13 ProvePropertyCM[184]ofTheoremVSPM[184].Writeyourproofinthestyleoftheproofof PropertyDSAM[184]giveninthissection. ContributedbyRobertBeezerSolution[193] T14 ProvePropertyAAM[184]ofTheoremVSPM[184].Writeyourproofinthestyleoftheproofof PropertyDSAM[184]giveninthissection. ContributedbyRobertBeezer T17 ProvePropertySMAM[184]ofTheoremVSPM[184].Writeyourproofinthestyleoftheproofof PropertyDSAM[184]giveninthissection. ContributedbyRobertBeezer T18 ProvePropertyDMAM[184]ofTheoremVSPM[184].Writeyourproofinthestyleoftheproofof PropertyDSAM[184]giveninthissection. ContributedbyRobertBeezer Version2.02

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SubsectionMO.SOLSolutions194 SubsectionSOL Solutions T13 ContributedbyRobertBeezerStatement[192] Forall A;B 2 M mn andforall1 i m ,1 i n [ A + B ] ij =[ A ] ij +[ B ] ij DenitionMA[182] =[ B ] ij +[ A ] ij Commutativityin C =[ B + A ] ij DenitionMA[182] Withequalityofeachentryofthematrices A + B and B + A beingequalDenitionME[182]tellsusthe twomatricesareequal. Version2.02

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SectionMMMatrixMultiplication195 SectionMM MatrixMultiplication Weknowhowtoaddvectorsandhowtomultiplythembyscalars.Together,theseoperationsgiveusthe possibilityofmakinglinearcombinations.Similarly,weknowhowtoaddmatricesandhowtomultiply matricesbyscalars.Inthissectionwemixalltheseideastogetherandproduceanoperationknownas matrixmultiplication."Thiswillleadtosomeresultsthatarebothsurprisingandcentral.Webegin withadenitionofhowtomultiplyavectorbyamatrix. SubsectionMVP Matrix-VectorProduct Wehaverepeatedlyseentheimportanceofforminglinearcombinationsofthecolumnsofamatrix.Asone exampleofthis,theoft-usedTheoremSLSLC[93],saidthateverysolutiontoasystemoflinearequations givesrisetoalinearcombinationofthecolumnvectorsofthecoecientmatrixthatequalsthevectorof constants.Thistheorem,andothers,motivatethefollowingcentraldenition. DenitionMVP Matrix-VectorProduct Suppose A isan m n matrixwithcolumns A 1 ; A 2 ; A 3 ;:::; A n and u isavectorofsize n .Thenthe matrix-vectorproduct of A with u isthelinearcombination A u =[ u ] 1 A 1 +[ u ] 2 A 2 +[ u ] 3 A 3 + +[ u ] n A n ThisdenitioncontainsNotationMVP. 4 So,thematrix-vectorproductisyetanotherversionofmultiplication,"atleastinthesensethatwe haveyetagainoverloadedjuxtapositionoftwosymbolsasournotation.Rememberyourobjects,an m n matrixtimesavectorofsize n willcreateavectorofsize m .Soif A isrectangular,thenthesizeofthe vectorchanges.Withallthelinearcombinationswehaveperformedsofar,thiscomputationshouldnow seemsecondnature. ExampleMTV Amatrixtimesavector Consider A = 2 4 14234 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3201 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 16 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 3 5 u = 2 6 6 6 6 4 2 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 7 7 5 Then A u =2 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 3 5 +1 2 4 4 2 6 3 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 4 2 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 5 +3 2 4 3 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 4 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 5 3 5 = 2 4 7 1 6 3 5 : Wecannowrepresentsystemsoflinearequationscompactlywithamatrix-vectorproductDenition MVP[194]andcolumnvectorequalityDenitionCVE[84].Thisnallyyieldsaverypopularalternative toourunconventional LS A; b notation. Version2.02

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SubsectionMM.MVPMatrix-VectorProduct196 TheoremSLEMM SystemsofLinearEquationsasMatrixMultiplication Thesetofsolutionstothelinearsystem LS A; b equalsthesetofsolutionsfor x inthevectorequation A x = b Proof Thistheoremsaysthattwosetsofsolutionsareequal.Soweneedtoshowthatonesetof solutionsisasubsetoftheother,andviceversaDenitionSE[684].Let A 1 ; A 2 ; A 3 ;:::; A n bethe columnsof A .BothofthesesetinclusionsthenfollowfromthefollowingchainofequivalencesTechnique E[690], x isasolutionto LS A; b [ x ] 1 A 1 +[ x ] 2 A 2 +[ x ] 3 A 3 + +[ x ] n A n = b TheoremSLSLC[93] x isasolutionto A x = b DenitionMVP[194] ExampleMNSLE Matrixnotationforsystemsoflinearequations ConsiderthesystemoflinearequationsfromExampleNSLE[26]. 2 x 1 +4 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 +5 x 4 + x 5 =9 3 x 1 + x 2 + x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 5 =0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 1 +7 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 3 +2 x 4 +2 x 5 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 hascoecientmatrix A = 2 4 24 )]TJ/F15 10.9091 Tf 8.485 0 Td [(351 3101 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(27 )]TJ/F15 10.9091 Tf 8.485 0 Td [(522 3 5 andvectorofconstants b = 2 4 9 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 5 andsowillbedescribedcompactlybythevectorequation A x = b Thematrix-vectorproductisaverynaturalcomputation.Wehavemotivateditbyitsconnections withsystemsofequations,buthereisaanotherexample. ExampleMBC Money'sbestcities Everyyear Money magazineselectsseveralcitiesintheUnitedStatesasthebest"citiestolivein,based onawidearrayofstatisticsabouteachcity.Thisisanexampleofhowtheeditorsof Money mightarrive atasinglenumberthatconsolidatesthestatisticsaboutacity.WewillanalyzeLosAngeles,Chicagoand NewYorkCity,basedonfourcriteria:averagehightemperatureinJulyFarenheit,numberofcolleges anduniversitiesina30-mileradius,numberoftoxicwastesitesintheSuperfundenvironmentalclean-up programandapersonalcrimeindexbasedonFBIstatisticsaverage=100,smallerissafer.Itshouldbe apparenthowtogeneralizetheexampletoagreaternumberofcitiesandagreaternumberofstatistics. Webeginbybuildingatableofstatistics.Therowswillbelabeledwiththecities,andthecolumns withstatisticalcategories.Thesevaluesarefrom Money 'swebsiteinearly2005. City Temp Colleges Superfund Crime LosAngeles 77 28 93 254 Chicago 84 38 85 363 NewYork 84 99 1 193 Version2.02

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SubsectionMM.MVPMatrix-VectorProduct197 Conceivablythesedatamightresideinaspreadsheet.Nowwemustcombinethestatisticsforeachcity. Wecouldaccomplishthisbyweightingeachcategory,scalingthevaluesandsummingthem.Thesizesof theweightswoulddependuponthenumericalsizeofeachstatisticgenerally,butmoreimportantly,they wouldreecttheeditorsopinionsorbeliefsaboutwhichstatisticsweremostimportanttotheirreaders.Is thecrimeindexmoreimportantthanthenumberofcollegesanduniversities?Ofcourse,thereisnoright answertothisquestion. Supposetheeditorsnallydecideonthefollowingweightstoemploy:temperature,0 : 23;colleges,0 : 46; Superfund, )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 05;crime, )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 20.Noticehownegativeweightsareusedforundesirablestatistics.Then, forexample,theeditorswouldcomputeforLosAngeles, : 23+ : 46+ )]TJ/F15 10.9091 Tf 8.484 0 Td [(0 : 05+ )]TJ/F15 10.9091 Tf 8.484 0 Td [(0 : 20= )]TJ/F15 10.9091 Tf 8.485 0 Td [(24 : 86 Thiscomputationmightremindyouofaninnerproduct,butwewillproducethecomputationsforallof thecitiesasamatrix-vectorproduct.Writethetableofrawstatisticsasamatrix T = 2 4 772893254 843885363 84991193 3 5 andtheweightsasavector w = 2 6 6 4 0 : 23 0 : 46 )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 05 )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 20 3 7 7 5 thenthematrix-vectorproductDenitionMVP[194]yields T w = : 23 2 4 77 84 84 3 5 + : 46 2 4 28 38 99 3 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 05 2 4 93 85 1 3 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 : 20 2 4 254 363 193 3 5 = 2 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(24 : 86 )]TJ/F15 10.9091 Tf 8.484 0 Td [(40 : 05 26 : 21 3 5 Thisvectorcontainsasinglenumberforeachofthecitiesbeingstudied,sotheeditorswouldrankNew Yorkbest : 21,LosAngelesnext )]TJ/F15 10.9091 Tf 8.485 0 Td [(24 : 86,andChicagothird )]TJ/F15 10.9091 Tf 8.484 0 Td [(40 : 05.Ofcourse,themayor'soces inChicagoandLosAngelesarefreetocounterwithadierentsetofweightsthatcausetheircitytobe rankedbest.Thesealternativeweightswouldbechosentoplaytoeachcities'strengths,andminimize theirproblemareas. Ifaspeadsheetwereusedtomakethesecomputations,arowofweightswouldbeenteredsomewhere nearthetableofdataandtheformulasinthespreadsheetwouldeectamatrix-vectorproduct.This exampleismeanttoillustratehowlinear"computationsaddition,multiplicationcanbeorganizedasa matrix-vectorproduct. Anotherexamplewouldbethematrixofnumericalscoresonexaminationsandexercisesforstudentsin aclass.Therowswouldcorrespondtostudentsandthecolumnstoexamsandassignments.Theinstructor couldthenassignweightstothedierentexamsandassignments,andviaamatrix-vectorproduct,compute asinglescoreforeachstudent. Latermuchlaterwewillneedthefollowingtheorem,whichisreallyatechnicallemmaseeTechnique LC[696].Sinceweareinapositiontoproveitnow,wewill.Butyoucansafelyskipitforthemoment, ifyoupromisetocomebacklatertostudytheproofwhenthetheoremisemployed.Atthatpointyou willalsobeabletounderstandthecommentsintheparagraphfollowingtheproof. TheoremEMMVP EqualMatricesandMatrix-VectorProducts Supposethat A and B are m n matricessuchthat A x = B x forevery x 2 C n .Then A = B Proof Weareassuming A x = B x forall x 2 C n ,sowecanemploythisequalityfor any choiceof thevector x .However,we'lllimitouruseofthisequalitytothestandardunitvectors, e j ,1 j n Version2.02

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SubsectionMM.MMMatrixMultiplication198 DenitionSUV[173].Forall1 j n ,1 i m [ A ] ij =0[ A ] i 1 + +0[ A ] i;j )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 +1[ A ] ij +0[ A ] i;j +1 + +0[ A ] in =[ A ] i 1 [ e j ] 1 +[ A ] i 2 [ e j ] 2 +[ A ] i 3 [ e j ] 3 + +[ A ] in [ e j ] n DenitionSUV[173] =[ A e j ] i DenitionMVP[194] =[ B e j ] i DenitionCVE[84] =[ B ] i 1 [ e j ] 1 +[ B ] i 2 [ e j ] 2 +[ B ] i 3 [ e j ] 3 + +[ B ] in [ e j ] n DenitionMVP[194] =0[ B ] i 1 + +0[ B ] i;j )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 +1[ B ] ij +0[ B ] i;j +1 + +0[ B ] in DenitionSUV[173] =[ B ] ij SobyDenitionME[182]thematrices A and B areequal,asdesired. Youmightnoticethatthehypothesesofthistheoremcouldbeweakened"i.e.madelessrestrictive. Wecouldsupposetheequalityofthematrix-vectorproductsforjustthestandardunitvectorsDenition SUV[173]oranyotherspanningsetDenitionTSVS[313]of C n ExerciseLISS.T40[320].However, inpractice,whenweapplythistheoremwewillonlyneedthisweakerform.Ifwemadethehypothesis lessrestrictive,wewouldcallthetheoremstronger." SubsectionMM MatrixMultiplication Wenowdenehowtomultiplytwomatricestogether.Stopforaminuteandthinkabouthowyoumight denethisnewoperation. Manybookswouldpresentthisdenitionmuchearlierinthecourse.However,wehavetakengreat caretodelayitaslongaspossibleandtopresentasmanyideasaspracticalbasedmostlyonthenotion oflinearcombinations.Towardstheconclusionofthecourse,orwhenyouperhapstakeasecondcourse inlinearalgebra,youmaybeinapositiontoappreciatethereasonsforthis.Fornow,understandthat matrixmultiplicationisacentraldenitionandperhapsyouwillappreciateitsimportancemorebyhaving saveditforlater. DenitionMM MatrixMultiplication Suppose A isan m n matrixand B isan n p matrixwithcolumns B 1 ; B 2 ; B 3 ;:::; B p .Thenthe matrixproduct of A with B isthe m p matrixwherecolumn i isthematrix-vectorproduct A B i Symbolically, AB = A [ B 1 j B 2 j B 3 j ::: j B p ]=[ A B 1 j A B 2 j A B 3 j ::: j A B p ] : 4 ExamplePTM Productoftwomatrices Set A = 2 4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(146 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4123 )]TJ/F15 10.9091 Tf 8.485 0 Td [(512 )]TJ/F15 10.9091 Tf 8.485 0 Td [(34 3 5 B = 2 6 6 6 6 4 1621 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1432 1123 64 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(230 3 7 7 7 7 5 Version2.02

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SubsectionMM.MMEEMatrixMultiplication,Entry-by-Entry199 Then AB = 2 6 6 6 6 4 A 2 6 6 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 6 1 3 7 7 7 7 5 A 2 6 6 6 6 4 6 4 1 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 7 7 5 A 2 6 6 6 6 4 2 3 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 3 7 7 7 7 5 A 2 6 6 6 6 4 1 2 3 2 0 3 7 7 7 7 5 3 7 7 7 7 5 = 2 4 28172010 20 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(18 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4412 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 5 : Isthisthedenitionofmatrixmultiplicationyouexpected?Perhapsourpreviousoperationsfor matricescausedyoutothinkthatwemightmultiplytwomatricesofthe same size, entry-by-entry ?Notice thatourcurrentdenitionusesmatricesofdierentsizesthoughthenumberofcolumnsintherstmust equalthenumberofrowsinthesecond,andtheresultisofathirdsize.Noticetoointheprevious examplethatwecannotevenconsidertheproduct BA ,sincethesizesofthetwomatricesinthisorder aren'tright. Butitgetsweirderthanthat.Manyofyouroldideasaboutmultiplication"won'tapplytomatrix multiplication,butsomestillwill.Somakenoassumptions,anddon'tdoanythinguntilyouhavea theoremthatsaysyoucan.Evenifthesizesareright,matrixmultiplicationisnotcommutative|order matters. ExampleMMNC Matrixmultiplicationisnotcommutative Set A = 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 B = 40 51 : Thenwehavetwosquare,2 2matrices,soDenitionMM[197]allowsustomultiplythemineither order.Wend AB = 193 62 BA = 412 417 and AB 6 = BA .Notevenclose.Itshouldnotbehardforyoutoconstructotherpairsofmatricesthatdo notcommutetryacoupleof3 3's.Canyoundapairofnon-identicalmatricesthat do commute? Matrixmultiplicationisfundamental,soitisanaturalprocedureforanycomputationaldevice.See: ComputationMM.MMA[671]. SubsectionMMEE MatrixMultiplication,Entry-by-Entry Whilecertainnatural"propertiesofmultiplicationdon'thold,manymoredo.Inthenextsubsection, we'llstateandprovetherelevanttheorems.Butrst,weneedatheoremthatprovidesanalternatemeans ofmultiplyingtwomatrices.Inmanytexts,thiswouldbegivenasthe denition ofmatrixmultiplication. Weprefertoturnitaroundandhavethefollowingformulaasaconsequenceofourdenition.Itwillprove usefulforproofsofmatrixequality,whereweneedtoexamineproductsofmatrices,entry-by-entry. TheoremEMP EntriesofMatrixProducts Suppose A isan m n matrixand B isan n p matrix.Thenfor1 i m ,1 j p ,theindividual entriesof AB aregivenby [ AB ] ij =[ A ] i 1 [ B ] 1 j +[ A ] i 2 [ B ] 2 j +[ A ] i 3 [ B ] 3 j + +[ A ] in [ B ] nj Version2.02

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SubsectionMM.MMEEMatrixMultiplication,Entry-by-Entry200 = n X k =1 [ A ] ik [ B ] kj Proof Denotethecolumnsof A asthevectors A 1 ; A 2 ; A 3 ;:::; A n andthecolumnsof B asthevectors B 1 ; B 2 ; B 3 ;:::; B p .Thenfor1 i m ,1 j p [ AB ] ij =[ A B j ] i DenitionMM[197] = [ B j ] 1 A 1 +[ B j ] 2 A 2 +[ B j ] 3 A 3 + +[ B j ] n A n i DenitionMVP[194] = [ B j ] 1 A 1 i + [ B j ] 2 A 2 i + [ B j ] 3 A 3 i + + [ B j ] n A n i DenitionCVA[84] =[ B j ] 1 [ A 1 ] i +[ B j ] 2 [ A 2 ] i +[ B j ] 3 [ A 3 ] i + +[ B j ] n [ A n ] i DenitionCVSM[85] =[ B ] 1 j [ A ] i 1 +[ B ] 2 j [ A ] i 2 +[ B ] 3 j [ A ] i 3 + +[ B ] nj [ A ] in NotationME[182] =[ A ] i 1 [ B ] 1 j +[ A ] i 2 [ B ] 2 j +[ A ] i 3 [ B ] 3 j + +[ A ] in [ B ] nj PropertyCMCN[680] = n X k =1 [ A ] ik [ B ] kj ExamplePTMEE Productoftwomatrices,entry-by-entry ConsideragainthetwomatricesfromExamplePTM[197] A = 2 4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(146 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4123 )]TJ/F15 10.9091 Tf 8.485 0 Td [(512 )]TJ/F15 10.9091 Tf 8.485 0 Td [(34 3 5 B = 2 6 6 6 6 4 1621 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1432 1123 64 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(230 3 7 7 7 7 5 Thensupposewejustwantedtheentryof AB inthesecondrow,thirdcolumn: [ AB ] 23 =[ A ] 21 [ B ] 13 +[ A ] 22 [ B ] 23 +[ A ] 23 [ B ] 33 +[ A ] 24 [ B ] 43 +[ A ] 25 [ B ] 53 =+ )]TJ/F15 10.9091 Tf 8.484 0 Td [(4++ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1+= )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 Noticehowthereare5termsinthesum,since5isthecommondimensionofthetwomatricescolumn countfor A ,rowcountfor B .IntheconclusionofTheoremEMP[198],itwouldbetheindex k that wouldrunfrom1to5inthiscomputation.Here'sabitmorepractice. Theentryofthirdrow,rstcolumn: [ AB ] 31 =[ A ] 31 [ B ] 11 +[ A ] 32 [ B ] 21 +[ A ] 33 [ B ] 31 +[ A ] 34 [ B ] 41 +[ A ] 35 [ B ] 51 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1++ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3+= )]TJ/F15 10.9091 Tf 8.485 0 Td [(18 Togetsomemorepracticeonyourown,completethecomputationoftheother10entriesofthisproduct. Constructsomeotherpairsofmatricesofcompatiblesizesandcomputetheirproducttwoways.First useDenitionMM[197].Sincelinearcombinationsarestraightforwardforyounow,thisshouldbeeasy todoandtodocorrectly.Thendoitagain,usingTheoremEMP[198].Sincethisprocessmaytakesome practice,useyourrstcomputationtocheckyourwork. TheoremEMP[198]isthewaymanypeoplecomputematrixproductsbyhand.Itwillalsobevery usefulforthetheoremswearegoingtoproveshortly.However,thedenitionDenitionMM[197]is frequentlythemostusefulforitsconnectionswithdeeperideaslikethenullspaceandtheupcoming columnspace. Version2.02

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SubsectionMM.PMMPropertiesofMatrixMultiplication201 SubsectionPMM PropertiesofMatrixMultiplication Inthissubsection,wecollectpropertiesofmatrixmultiplicationanditsinteractionwiththezeromatrix DenitionZM[185],theidentitymatrixDenitionIM[72],matrixadditionDenitionMA[182], scalarmatrixmultiplicationDenitionMSM[183],theinnerproductDenitionIP[168],conjugation TheoremMMCC[203],andthetransposeDenitionTM[185].Whew!Herewego.Thesearegreat proofstopracticewith,sotrytoconcocttheproofsbeforereadingthem,they'llgetprogressivelymore complicatedaswego. TheoremMMZM MatrixMultiplicationandtheZeroMatrix Suppose A isan m n matrix.Then 1. A O n p = O m p 2. O p m A = O p n Proof We'llproveandleavetoyou.Entry-by-entry,for1 i m ,1 j p [ A O n p ] ij = n X k =1 [ A ] ik [ O n p ] kj TheoremEMP[198] = n X k =1 [ A ] ik 0DenitionZM[185] = n X k =1 0 =0PropertyZCN[681] =[ O m p ] ij DenitionZM[185] SobythedenitionofmatrixequalityDenitionME[182],thematrices A O n p and O m p areequal. TheoremMMIM MatrixMultiplicationandIdentityMatrix Suppose A isan m n matrix.Then 1. AI n = A 2. I m A = A Proof Again,we'llproveandleavetoyou.Entry-by-entry,For1 i m ,1 j n [ AI n ] ij = n X k =1 [ A ] ik [ I n ] kj TheoremEMP[198] =[ A ] ij [ I n ] jj + n X k =1 k 6 = j [ A ] ik [ I n ] kj PropertyCACN[680] =[ A ] ij + n X k =1 ;k 6 = j [ A ] ik DenitionIM[72] =[ A ] ij + n X k =1 ;k 6 = j 0 =[ A ] ij Version2.02

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SubsectionMM.PMMPropertiesofMatrixMultiplication202 Sothematrices A and AI n areequal,entry-by-entry,andbythedenitionofmatrixequalityDenition ME[182]wecansaytheyareequalmatrices. Itisthistheoremthatgivestheidentitymatrixitsname.Itisamatrixthatbehaveswithmatrix multiplicationlikethescalar1doeswithscalarmultiplication.Tomultiplybytheidentitymatrixisto havenoeectontheothermatrix. TheoremMMDAA MatrixMultiplicationDistributesAcrossAddition Suppose A isan m n matrixand B and C are n p matricesand D isa p s matrix.Then 1. A B + C = AB + AC 2. B + C D = BD + CD Proof We'lldo,youdo.Entry-by-entry,for1 i m ,1 j p [ A B + C ] ij = n X k =1 [ A ] ik [ B + C ] kj TheoremEMP[198] = n X k =1 [ A ] ik [ B ] kj +[ C ] kj DenitionMA[182] = n X k =1 [ A ] ik [ B ] kj +[ A ] ik [ C ] kj PropertyDCN[681] = n X k =1 [ A ] ik [ B ] kj + n X k =1 [ A ] ik [ C ] kj PropertyCACN[680] =[ AB ] ij +[ AC ] ij TheoremEMP[198] =[ AB + AC ] ij DenitionMA[182] Sothematrices A B + C and AB + AC areequal,entry-by-entry,andbythedenitionofmatrixequality DenitionME[182]wecansaytheyareequalmatrices. TheoremMMSMM MatrixMultiplicationandScalarMatrixMultiplication Suppose A isan m n matrixand B isan n p matrix.Let beascalar.Then AB = A B = A B Proof Theseareequalitiesofmatrices.We'lldotherstone,thesecondissimilarandwillbegood practiceforyou.For1 i m ,1 j p [ AB ] ij = [ AB ] ij DenitionMSM[183] = n X k =1 [ A ] ik [ B ] kj TheoremEMP[198] = n X k =1 [ A ] ik [ B ] kj PropertyDCN[681] = n X k =1 [ A ] ik [ B ] kj DenitionMSM[183] =[ A B ] ij TheoremEMP[198] Sothematrices AB and A B areequal,entry-by-entry,andbythedenitionofmatrixequality DenitionME[182]wecansaytheyareequalmatrices. TheoremMMA Version2.02

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SubsectionMM.PMMPropertiesofMatrixMultiplication203 MatrixMultiplicationisAssociative Suppose A isan m n matrix, B isan n p matrixand D isa p s matrix.Then A BD = AB D Proof Amatrixequality,sowe'llgoentry-by-entry,nosurprisethere.For1 i m ,1 j s [ A BD ] ij = n X k =1 [ A ] ik [ BD ] kj TheoremEMP[198] = n X k =1 [ A ] ik p X ` =1 [ B ] k` [ D ] `j TheoremEMP[198] = n X k =1 p X ` =1 [ A ] ik [ B ] k` [ D ] `j PropertyDCN[681] Wecanswitchtheorderofthesummationsincethesearenitesums, = p X ` =1 n X k =1 [ A ] ik [ B ] k` [ D ] `j PropertyCACN[680] As[ D ] `j doesnotdependontheindex k ,wecanusedistributivitytomoveitoutsideoftheinnersum, = p X ` =1 [ D ] `j n X k =1 [ A ] ik [ B ] k` PropertyDCN[681] = p X ` =1 [ D ] `j [ AB ] i` TheoremEMP[198] = p X ` =1 [ AB ] i` [ D ] `j PropertyCMCN[680] =[ AB D ] ij TheoremEMP[198] Sothematrices AB D and A BD areequal,entry-by-entry,andbythedenitionofmatrixequality DenitionME[182]wecansaytheyareequalmatrices. Thestatementofournexttheoremistechnicallyinaccurate.Ifweupgradethevectors u ; v tomatrices withasinglecolumn,thentheexpression u t v isa1 1matrix,thoughwewilltreatthissmallmatrixas ifitwassimplythescalarquantityinitsloneentry.WhenweapplyTheoremMMIP[202]thereshould notbeanyconfusion. TheoremMMIP MatrixMultiplicationandInnerProducts Ifweconsiderthevectors u ; v 2 C m as m 1matricesthen h u ; v i = u t v Proof h u ; v i = m X k =1 [ u ] k [ v ] k DenitionIP[168] = m X k =1 [ u ] k 1 [ v ] k 1 Columnvectorsasmatrices Version2.02

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SubsectionMM.PMMPropertiesofMatrixMultiplication204 = m X k =1 u t 1 k [ v ] k 1 DenitionTM[185] = m X k =1 u t 1 k [ v ] k 1 DenitionCCCV[167] = u t v 11 TheoremEMP[198] Tonishwejustblurthedistinctionbetweena1 1matrix u t v anditsloneentry. TheoremMMCC MatrixMultiplicationandComplexConjugation Suppose A isan m n matrixand B isan n p matrix.Then AB = A B Proof Toobtainthismatrixequality,wewillworkentry-by-entry.For1 i m ,1 j p AB ij = [ AB ] ij DenitionCCM[187] = n X k =1 [ A ] ik [ B ] kj TheoremEMP[198] = n X k =1 [ A ] ik [ B ] kj TheoremCCRA[681] = n X k =1 [ A ] ik [ B ] kj TheoremCCRM[682] = n X k =1 A ik B kj DenitionCCM[187] = A B ij TheoremEMP[198] Sothematrices AB and A B areequal,entry-by-entry,andbythedenitionofmatrixequalityDenition ME[182]wecansaytheyareequalmatrices. Anothertheoreminthisstyle,anditsagoodone.Ifyou'vebeenpracticingwiththepreviousproofs youshouldbeabletodothisoneyourself. TheoremMMT MatrixMultiplicationandTransposes Suppose A isan m n matrixand B isan n p matrix.Then AB t = B t A t Proof Thistheoremmaybesurprisingbutifwecheckthesizesofthematricesinvolved,thenmaybe itwillnotseemsofar-fetched.First, AB hassize m p ,soitstransposehassize p m .Theproduct of B t with A t isa p n matrixtimesan n m matrix,alsoresultingina p m matrix.Soatleast ourobjectsarecompatibleforequalityandwouldnotbe,ingeneral,ifwedidn'treversetheorderofthe matrixmultiplication. Herewegoagain,entry-by-entry.For1 i m ,1 j p AB t ji =[ AB ] ij DenitionTM[185] = n X k =1 [ A ] ik [ B ] kj TheoremEMP[198] = n X k =1 [ B ] kj [ A ] ik PropertyCMCN[680] Version2.02

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SubsectionMM.HMHermitianMatrices205 = n X k =1 B t jk A t ki DenitionTM[185] = B t A t ji TheoremEMP[198] Sothematrices AB t and B t A t areequal,entry-by-entry,andbythedenitionofmatrixequalityDenitionME[182]wecansaytheyareequalmatrices. Thistheoremseemsoddatrstglance,sincewehavetoswitchtheorderof A and B .Butifwesimply considerthesizesofthematricesinvolved,wecanseethattheswitchisnecessaryforthisreasonalone. Thattheindividualentriesoftheproductsthencomealongtobeequalisabonus. Astheadjointofamatrixisacompositionofaconjugateandatranspose,itsinteractionwithmatrix multiplicationissimilartothatofatranspose.Here'sthelastofourlonglistofbasicpropertiesofmatrix multiplication. TheoremMMAD MatrixMultiplicationandAdjoints Suppose A isan m n matrixand B isan n p matrix.Then AB = B A Proof AB = )]TJETq1 0 0 1 205.563 476.621 cm[]0 d 0 J 0.436 w 0 0 m 17.004 0 l SQBT/F21 10.9091 Tf 205.563 467.64 Td [(AB t DenitionA[189] = )]TJETq1 0 0 1 205.563 457.188 cm[]0 d 0 J 0.436 w 0 0 m 8.182 0 l SQBT/F21 10.9091 Tf 205.563 448.206 Td [(A B t TheoremMMCC[203] = )]TJETq1 0 0 1 205.563 437.754 cm[]0 d 0 J 0.436 w 0 0 m 8.822 0 l SQBT/F21 10.9091 Tf 205.563 428.772 Td [(B t )]TJETq1 0 0 1 229.759 437.754 cm[]0 d 0 J 0.436 w 0 0 m 8.182 0 l SQBT/F21 10.9091 Tf 229.759 428.772 Td [(A t TheoremMMT[203] = B A DenitionA[189] Noticehownoneoftheseproofsabovereliedonwritingouthugegeneralmatriceswithlotsofellipses ..."andtryingtoformulatetheequalitiesawholematrixatatime.Thismessybusinessisaproof technique"tobeavoidedatallcosts.NoticetoohowtheproofofTheoremMMAD[204]doesnotusean entry-by-entryapproach,butsimplybuildsonpreviousresultsaboutmatrixmultiplication'sinteraction withconjugationandtransposes. Thesetheorems,alongwithTheoremVSPM[184]andtheotherresultsinSectionMO[182],giveyou therules"forhowmatricesinteractwiththevariousoperationswehavedenedonmatricesaddition, scalarmultiplication,matrixmultiplication,conjugation,transposesandadjoints.Usethemandusethem often.Butdon'ttrytodoanythingwithamatrixthatyoudon'thavearulefor.Together,wewould informallycallalltheseoperations,andtheattendanttheorems,thealgebraofmatrices."Notice,too, thateverycolumnvectorisjusta n 1matrix,sothesetheoremsapplytocolumnvectorsalso.Finally, theseresults,takenasawhole,maymakeusfeelthatthedenitionofmatrixmultiplicationisnotso unnatural. SubsectionHM HermitianMatrices Theadjointofamatrixhasabasicpropertywhenemployedinamatrix-vectorproductaspartofaninner product.Atthispoint,youcouldevenusethefollowingresultasamotivationforthedenitionofan adjoint. TheoremAIP AdjointandInnerProduct Version2.02

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SubsectionMM.HMHermitianMatrices206 Supposethat A isan m n matrixand x 2 C n y 2 C m .Then h A x ; y i = h x ;A y i Proof h A x ; y i = A x t y TheoremMMIP[202] = x t A t y TheoremMMT[203] = x t A t y TheoremCCM[188] = x t )]TJETq1 0 0 1 224.673 625.718 cm[]0 d 0 J 0.436 w 0 0 m 8.182 0 l SQBT/F21 10.9091 Tf 224.673 616.736 Td [(A t y TheoremMCT[189] = x t A y DenitionA[189] = x t A y TheoremMMCC[203] = h x ;A y i TheoremMMIP[202] SometimesamatrixisequaltoitsadjointDenitionA[189],andthesematriceshaveinteresting properties.Oneofthemostcommonsituationswherethisoccursiswhenamatrixhasonlyrealnumber entries.ThenwearesimplytalkingaboutsymmetricmatricesDenitionSYM[186],soyoucanview thisasageneralizationofasymmetricmatrix. DenitionHM HermitianMatrix Thesquarematrix A is Hermitian or self-adjoint if A = A 4 Again,thesetofrealmatricesthatareHermitianisexactlythesetofsymmetricmatrices.InSection PEE[419]wewilluncoversomeamazingpropertiesofHermitianmatrices,sowhenyougetthere,run backheretoremindyourselfofthisdenition.Furtherpropertieswillalsoappearinvarioussectionsofthe TopicsPartT[793].RightnowweproveafundamentalresultaboutHermitianmatrices,matrixvector productsandinnerproducts.Asacharacterization,thiscouldbeemployedasadenitionofaHermitian matrixandsomeauthorstakethisapproach. TheoremHMIP HermitianMatricesandInnerProducts Supposethat A isasquarematrixofsize n .Then A isHermitianifandonlyif h A x ; y i = h x ;A y i forall x ; y 2 C n Proof Thisistheeasyhalf"oftheproof,andmakestherationaleforadenitionofHermitian matricesmostobvious.Assume A isHermitian, h A x ; y i = h x ;A y i TheoremAIP[204] = h x ;A y i DenitionHM[205] Thishalf"willtakeabitmorework.Assumethat h A x ; y i = h x ;A y i forall x ; y 2 C n .Chooseany x 2 C n .Wewanttoshowthat A = A byestablishingthat A x = A x .Withonlythismuchmotivation, considertheinnerproduct, h A x )]TJ/F21 10.9091 Tf 10.909 0 Td [(A x ;A x )]TJ/F21 10.9091 Tf 10.909 0 Td [(A x i = h A x )]TJ/F21 10.9091 Tf 10.909 0 Td [(A x ;A x i)-222(h A x )]TJ/F21 10.9091 Tf 10.909 0 Td [(A x ;A x i TheoremIPVA[169] = h A x )]TJ/F21 10.9091 Tf 10.909 0 Td [(A x ;A x i)-222(h A A x )]TJ/F21 10.9091 Tf 10.909 0 Td [(A x ; x i TheoremAIP[204] = h A A x )]TJ/F21 10.9091 Tf 10.909 0 Td [(A x ; x i)-222(h A A x )]TJ/F21 10.9091 Tf 10.91 0 Td [(A x ; x i Hypothesis =0PropertyAICN[681] Version2.02

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SubsectionMM.READReadingQuestions207 Becausethisinnerproductequalszero,andhasthesamevectorineachargument A x )]TJ/F21 10.9091 Tf 10.99 0 Td [(A x ,Theorem PIP[172]givestheconclusionthat A x )]TJ/F21 10.9091 Tf 11.295 0 Td [(A x = 0 .With A x = A x forall x 2 C n ,TheoremEMMVP [196]says A = A ,whichisthedeningpropertyofaHermitianmatrixDenitionHM[205]. So,informally,Hermitianmatricesarethosethatcanbetossedaroundfromonesideofaninner producttotheotherwithrecklessabandon.We'llseelaterwhatthisbuysus. SubsectionREAD ReadingQuestions 1.Formthematrixvectorproductof 2 4 23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(273 1532 3 5 with 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 5 3 7 7 5 2.Multiplytogetherthetwomatricesbelowintheordergiven. 2 4 23 )]TJ/F15 10.9091 Tf 8.484 0 Td [(10 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(273 1532 3 5 2 6 6 4 26 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 02 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 5 3.Rewritethesystemoflinearequationsbelowasavectorequalityandusingamatrix-vectorproduct. Thisquestiondoesnotaskforasolutiontothesystem.Butitdoesaskyoutoexpressthesystem ofequationsinanewformusingtoolsfromthissection. 2 x 1 +3 x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 3 =0 x 1 +2 x 2 + x 3 =3 x 1 +3 x 2 +3 x 3 =7 Version2.02

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SubsectionMM.EXCExercises208 SubsectionEXC Exercises C20 Computetheproductofthetwomatricesbelow, AB .DothisusingthedenitionsofthematrixvectorproductDenitionMVP[194]andthedenitionofmatrixmultiplicationDenitionMM[197]. A = 2 4 25 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 5 B = 15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(34 202 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ContributedbyRobertBeezerSolution[209] T10 Supposethat A isasquarematrixandthereisavector, b ,suchthat LS A; b hasauniquesolution. Provethat A isnonsingular.GiveadirectproofperhapsappealingtoTheoremPSPHS[105]ratherthan justnegatingasentencefromthetextdiscussingasimilarsituation. ContributedbyRobertBeezerSolution[209] T20 ProvethesecondpartofTheoremMMZM[200]. ContributedbyRobertBeezer T21 ProvethesecondpartofTheoremMMIM[200]. ContributedbyRobertBeezer T22 ProvethesecondpartofTheoremMMDAA[201]. ContributedbyRobertBeezer T23 ProvethesecondpartofTheoremMMSMM[201]. ContributedbyRobertBeezerSolution[209] T31 Supposethat A isan m n matrixand x ; y 2N A .Provethat x + y 2N A ContributedbyRobertBeezer T32 Supposethat A isan m n matrix, 2 C ,and x 2N A .Provethat x 2N A ContributedbyRobertBeezer T40 Supposethat A isan m n matrixand B isan n p matrix.Provethatthenullspaceof B isa subsetofthenullspaceof AB ,thatis N B N AB .Provideanexamplewheretheoppositeisfalse, inotherwordsgiveanexamplewhere N AB 6N B ContributedbyRobertBeezerSolution[209] T41 Supposethat A isan n n nonsingularmatrixand B isan n p matrix.Provethatthenullspace of B isequaltothenullspaceof AB ,thatis N B = N AB .ComparewithExerciseMM.T40[207]. ContributedbyRobertBeezerSolution[210] T50 Suppose u and v areanytwosolutionsofthelinearsystem LS A; b .Provethat u )]TJ/F36 10.9091 Tf 11.673 0 Td [(v isan elementofthenullspaceof A ,thatis, u )]TJ/F36 10.9091 Tf 10.909 0 Td [(v 2N A ContributedbyRobertBeezer T51 GiveanewproofofTheoremPSPHS[105]replacingapplicationsofTheoremSLSLC[93]with matrix-vectorproductsTheoremSLEMM[195]. ContributedbyRobertBeezerSolution[210] T52 Supposethat x ; y 2 C n b 2 C m and A isan m n matrix.If x y and x + y areeachasolutionto thelinearsystem LS A; b ,whatinterestingcanyousayabout b ?Formanimplicationwiththeexistence Version2.02

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SubsectionMM.EXCExercises209 ofthethreesolutionsasthehypothesisandaninterestingstatementabout LS A; b astheconclusion, andthengiveaproof. ContributedbyRobertBeezerSolution[210] Version2.02

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SubsectionMM.SOLSolutions210 SubsectionSOL Solutions C20 ContributedbyRobertBeezerStatement[207] ByDenitionMM[197], AB = 2 4 2 4 25 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 3 5 1 2 2 4 25 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 5 5 0 2 4 25 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 2 2 4 25 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 5 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 5 RepeatedapplicationsofDenitionMVP[194]give = 2 4 1 2 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 5 +2 2 4 5 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 5 5 2 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 5 +0 2 4 5 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 5 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 2 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 5 +2 2 4 5 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 5 4 2 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 2 4 5 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 5 3 5 = 2 4 12104 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(59 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(210 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1014 3 5 T10 ContributedbyRobertBeezerStatement[207] Since LS A;b hasatleastonesolution,wecanapplyTheoremPSPHS[105].Becausethesolutionis assumedtobeunique,thenullspaceof A mustbetrivial.ThenTheoremNMTNS[74]impliesthat A is nonsingular. TheconverseofthisstatementisatrivialapplicationofTheoremNMUS[74].Thatsaid,wecould extendourNSMxxseriesoftheoremswithanaddedequivalencefornonsingularity,Givenasinglevector ofconstants, b ,thesystem LS A; b hasauniquesolution." T23 ContributedbyRobertBeezerStatement[207] We'llruntheproofentry-by-entry. [ AB ] ij = [ AB ] ij DenitionMSM[183] = n X k =1 [ A ] ik [ B ] kj TheoremEMP[198] = n X k =1 [ A ] ik [ B ] kj Distributivityin C = n X k =1 [ A ] ik [ B ] kj Commutativityin C = n X k =1 [ A ] ik [ B ] kj DenitionMSM[183] =[ A B ] ij TheoremEMP[198] Sothematrices AB and A B areequal,entry-by-entry,andbythedenitionofmatrixequality DenitionME[182]wecansaytheyareequalmatrices. T40 ContributedbyRobertBeezerStatement[207] Toprovethatonesetisasubsetofanother,westartwithanelementofthesmallersetandseeifwecan determinethatitisamemberofthelargersetDenitionSSET[683].Suppose x 2N B .Thenwe knowthat B x = 0 byDenitionNSM[64].Consider AB x = A B x TheoremMMA[202] Version2.02

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SubsectionMM.SOLSolutions211 = A 0 Hypothesis = 0 TheoremMMZM[200] Thisestablishesthat x 2N AB ,so N B N AB Toshowthattheinclusiondoesnotholdintheoppositedirection,choose B tobeanynonsingular matrixofsize n .Then N B = f 0 g byTheoremNMTNS[74].Let A bethesquarezeromatrix, O ,of thesamesize.Then AB = O B = O byTheoremMMZM[200]andtherefore N AB = C n ,andis not a subsetof N B = f 0 g T41 ContributedbyDavidBraithwaiteStatement[207] FromthesolutiontoExerciseMM.T40[207]weknowthat N B N AB .Sotoestablishtheset equalityDenitionSE[684]weneedtoshowthat N AB N B Suppose x 2N AB .Thenweknowthat AB x = 0 byDenitionNSM[64].Consider 0 = AB x DenitionNSM[64] = A B x TheoremMMA[202] So, B x 2N A .Because A isnonsingular,ithasatrivialnullspaceTheoremNMTNS[74]andwe concludethat B x = 0 .Thisestablishesthat x 2N B ,so N AB N B andcombinedwiththe solutiontoExerciseMM.T40[207]wehave N B = N AB when A isnonsingular. T51 ContributedbyRobertBeezerStatement[207] Wewillworkwiththevectorequalityrepresentationsoftherelevantsystemsofequations,asdescribedby TheoremSLEMM[195]. Suppose y = w + z and z 2N A .Then A y = A w + z Substitution = A w + A z TheoremMMDAA[201] = b + 0z 2N A = b PropertyZC[86] demonstratingthat y isasolution. Suppose y isasolutionto LS A;b .Then A y )]TJ/F36 10.9091 Tf 10.91 0 Td [(w = A y )]TJ/F21 10.9091 Tf 10.909 0 Td [(A w TheoremMMDAA[201] = b )]TJ/F36 10.9091 Tf 10.909 0 Td [(by ; w solutionsto A x = b = 0 PropertyAIC[86] whichsaysthat y )]TJ/F36 10.9091 Tf 11.017 0 Td [(w 2N A .Inotherwords, y )]TJ/F36 10.9091 Tf 11.018 0 Td [(w = z forsomevector z 2N A .Rewritten,thisis y = w + z ,asdesired. T52 ContributedbyRobertBeezerStatement[207] LS A; b mustbehomogeneous.Toseethisconsiderthat b = A x TheoremSLEMM[195] = A x + 0 PropertyZC[86] = A x + A y )]TJ/F21 10.9091 Tf 10.909 0 Td [(A y PropertyAIC[86] = A x + y )]TJ/F21 10.9091 Tf 10.909 0 Td [(A y TheoremMMDAA[201] Version2.02

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SubsectionMM.SOLSolutions212 = b )]TJ/F36 10.9091 Tf 10.909 0 Td [(b TheoremSLEMM[195] = 0 PropertyAIC[86] ByDenitionHS[62]weseethat LS A; b ishomogeneous. Version2.02

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SectionMISLEMatrixInversesandSystemsofLinearEquations213 SectionMISLE MatrixInversesandSystemsofLinearEquations Webeginwithafamiliarexample,performedinanovelway. ExampleSABMI SolutionstoArchetypeBwithamatrixinverse ArchetypeB[707]isthesystemof m =3linearequationsin n =3variables, )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(12 x 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(33 5 x 1 +5 x 2 +7 x 3 =24 x 1 +4 x 3 =5 ByTheoremSLEMM[195]wecanrepresentthissystemofequationsas A x = b where A = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 557 104 3 5 x = 2 4 x 1 x 2 x 3 3 5 b = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(33 24 5 3 5 We'llpullarabbitoutofourhatandpresentthe3 3matrix B B = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 13 2 8 11 2 5 2 3 5 2 3 5 andnotethat BA = 2 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 13 2 8 11 2 5 2 3 5 2 3 5 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 557 104 3 5 = 2 4 100 010 001 3 5 Nowapplythiscomputationtotheproblemofsolvingthesystemofequations, x = I 3 x TheoremMMIM[200] = BA x Substitution = B A x TheoremMMA[202] = B b Substitution Sowehave x = B b = 2 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 13 2 8 11 2 5 2 3 5 2 3 5 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(33 24 5 3 5 = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 5 2 3 5 Sowiththehelpandassistanceof B wehavebeenabletodetermineasolutiontothesystemrepresented by A x = b throughjudicioususeofmatrixmultiplication.WeknowbyTheoremNMUS[74]thatsince thecoecientmatrixinthisexampleisnonsingular,therewouldbeauniquesolution,nomatterwhatthe choiceof b .Thederivationaboveampliesthisresult,sincewewere forced toconcludethat x = B b and Version2.02

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SubsectionMISLE.IMInverseofaMatrix214 thesolutioncouldn'tbeanythingelse.Youshouldnoticethatthisargumentwouldholdforanyparticular valueof b Thematrix B ofthepreviousexampleiscalledtheinverseof A .When A and B arecombinedvia matrixmultiplication,theresultistheidentitymatrix,whichcanbeinsertedinfront"of x astherst stepinndingthesolution.Thisisentirelyanalogoustohowwemightsolveasinglelinearequationlike 3 x =12. x =1 x = 1 3 x = 1 3 x = 1 3 =4 Herewehaveobtainedasolutionbyemployingthemultiplicativeinverse"of3,3 )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 = 1 3 .Thisworks neforanyscalarmultipleof x ,exceptforzero,sincezerodoesnothaveamultiplicativeinverse.For matrices,itismorecomplicated.Somematriceshaveinverses,somedonot.Andwhenamatrixdoes haveaninverse,justhowwouldwecomputeit?Inotherwords,justwheredidthatmatrix B inthelast examplecomefrom?Arethereothermatricesthatmighthaveworkedjustaswell? SubsectionIM InverseofaMatrix DenitionMI MatrixInverse Suppose A and B aresquarematricesofsize n suchthat AB = I n and BA = I n .Then A is invertible and B isthe inverse of A .Inthissituation,wewrite B = A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 ThisdenitioncontainsNotationMI. 4 Noticethatif B istheinverseof A ,thenwecanjustaseasilysay A istheinverseof B ,or A and B areinversesofeachother. Noteverysquarematrixhasaninverse.InExampleSABMI[212]thematrix B istheinversethe coecientmatrixofArchetypeB[707].Toseethisitonlyremainstocheckthat AB = I 3 .Whatabout ArchetypeA[702]?Itisanexampleofasquarematrixwithoutaninverse. ExampleMWIAA Amatrixwithoutaninverse,ArchetypeA ConsiderthecoecientmatrixfromArchetypeA[702], A = 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 211 110 3 5 Supposethat A isinvertibleanddoeshaveaninverse,say B .Choosethevectorofconstants b = 2 4 1 3 2 3 5 andconsiderthesystemofequations LS A; b .JustasinExampleSABMI[212],thisvectorequation wouldhavetheuniquesolution x = B b However,thesystem LS A; b isinconsistent.Formtheaugmentedmatrix[ A j b ]androw-reduceto 2 4 1 010 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 000 1 3 5 Version2.02

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SubsectionMISLE.CIMComputingtheInverseofaMatrix215 whichallowstorecognizetheinconsistencybyTheoremRCLS[53]. Sotheassumptionof A 'sinverseleadstoalogicalinconsistencythesystemcan'tbebothconsistent andinconsistent,soourassumptionisfalse. A isnotinvertible. Itspossiblethisexampleislessthansatisfying.Justwheredidthatparticularchoiceofthevector b comefromanyway?StaytunedforanapplicationofthefutureTheoremCSCS[237]inExampleCSAA [241]. Let'slookatonemorematrixinversebeforeweembarkonamoresystematicstudy. ExampleMI Matrixinverse Considerthematrices, A = 2 6 6 6 6 4 12121 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(30 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 11021 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(31 3 7 7 7 7 5 B = 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(336 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 1241 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 10110 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(201 3 7 7 7 7 5 Then AB = 2 6 6 6 6 4 12121 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(30 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 11021 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(31 3 7 7 7 7 5 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(336 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 1241 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 10110 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(201 3 7 7 7 7 5 = 2 6 6 6 6 4 10000 01000 00100 00010 00001 3 7 7 7 7 5 and BA = 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(336 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 1241 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 10110 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(201 3 7 7 7 7 5 2 6 6 6 6 4 12121 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(30 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 11021 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(31 3 7 7 7 7 5 = 2 6 6 6 6 4 10000 01000 00100 00010 00001 3 7 7 7 7 5 sobyDenitionMI[213],wecansaythat A isinvertibleandwrite B = A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 Wewillnowconcernourselveslesswithwhetherornotaninverseofamatrixexists,butinsteadwith howyoucanndonewhenitdoesexist.InSectionMINM[226]wewillhavesometheoremsthatallow ustomorequicklyandeasilydeterminejustwhenamatrixisinvertible. SubsectionCIM ComputingtheInverseofaMatrix We'veseenthatthematricesfromArchetypeB[707]andArchetypeK[746]bothhaveinverses,butthese inversematriceshavejustdroppedfromthesky.Howwouldwecomputeaninverse?Andjustwhenis amatrixinvertible,andwhenisitnot?Writingaputativeinversewith n 2 unknownsandsolvingthe resultant n 2 equationsisoneapproach.Applyingthisapproachto2 2matricescangetussomewhere, sojustforfun,let'sdoit. TheoremTTMI Two-by-TwoMatrixInverse Suppose A = ab cd Version2.02

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SubsectionMISLE.CIMComputingtheInverseofaMatrix216 Then A isinvertibleifandonlyif ad )]TJ/F21 10.9091 Tf 10.909 0 Td [(bc 6 =0.When A isinvertible,then A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 = 1 ad )]TJ/F21 10.9091 Tf 10.909 0 Td [(bc d )]TJ/F21 10.9091 Tf 8.485 0 Td [(b )]TJ/F21 10.9091 Tf 8.485 0 Td [(ca Proof Assumethat ad )]TJ/F21 10.9091 Tf 11.141 0 Td [(bc 6 =0.Wewillusethedenitionoftheinverseofamatrixtoestablish that A hasinverseDenitionMI[213].Notethatif ad )]TJ/F21 10.9091 Tf 11.178 0 Td [(bc 6 =0thenthedisplayedformulafor A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 is legitimatesincewearenotdividingbyzero.Usingthisproposedformulafortheinverseof A ,wecompute AA )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 = ab cd 1 ad )]TJ/F21 10.9091 Tf 10.909 0 Td [(bc d )]TJ/F21 10.9091 Tf 8.485 0 Td [(b )]TJ/F21 10.9091 Tf 8.484 0 Td [(ca = 1 ad )]TJ/F21 10.9091 Tf 10.909 0 Td [(bc ad )]TJ/F21 10.9091 Tf 10.909 0 Td [(bc 0 0 ad )]TJ/F21 10.9091 Tf 10.909 0 Td [(bc = 10 01 and A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 A = 1 ad )]TJ/F21 10.9091 Tf 10.909 0 Td [(bc d )]TJ/F21 10.9091 Tf 8.485 0 Td [(b )]TJ/F21 10.9091 Tf 8.485 0 Td [(ca ab cd = 1 ad )]TJ/F21 10.9091 Tf 10.909 0 Td [(bc ad )]TJ/F21 10.9091 Tf 10.909 0 Td [(bc 0 0 ad )]TJ/F21 10.9091 Tf 10.909 0 Td [(bc = 10 01 ByDenitionMI[213]thisissucienttoestablishthat A isinvertible,andthattheexpressionfor A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 is correct. Assumethat A isinvertible,andproceedwithaproofbycontradictionTechniqueCD[692], byassumingalsothat ad )]TJ/F21 10.9091 Tf 10.909 0 Td [(bc =0.Thistranslatesto ad = bc .Let B = ef gh beaputativeinverseof A .Thismeansthat I 2 = AB = ab cd ef gh = ae + bgaf + bh ce + dgcf + dh Workingonthematricesonbothendsofthisequation,wewillmultiplythetoprowby c andthebottom rowby a c 0 0 a = ace + bcgacf + bch ace + adgacf + adh Weareassumingthat ad = bc ,sowecanreplacetwooccurrencesof ad by bc inthebottomrowofthe rightmatrix. c 0 0 a = ace + bcgacf + bch ace + bcgacf + bch Thematrixontherightnowhastworowsthatareidentical,andthereforethesamemustbetrueofthe matrixontheleft.Giventheformofthematrixontheleft,identicalrowsimpliesthat a =0and c =0. Withthisinformation,theproduct AB becomes 10 01 = I 2 = AB = ae + bgaf + bh ce + dgcf + dh = bgbh dgdh So bg = dh =1andthus b;g;d;h areallnonzero.Butthen bh and dg theothercorners"mustalso benonzero,sothisisnallyacontradiction.Soourassumptionwasfalseandweseethat ad )]TJ/F21 10.9091 Tf 11.133 0 Td [(bc 6 =0 whenever A hasaninverse. Thereareseveralwaysonecouldtrytoprovethistheorem,butthereisacontinualtemptationtodivide byoneoftheeightentriesinvolved a through f ,butwecanneverbesureifthesenumbersarezeroor Version2.02

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SubsectionMISLE.CIMComputingtheInverseofaMatrix217 not.Thiscouldleadtoananalysisbycases,whichismessy,messy,messy.Notehowtheaboveproof neverdivides,butalwaysmultiplies,andhowzero/nonzeroconsiderationsarehandled.Payattentionto theexpression ad )]TJ/F21 10.9091 Tf 10.909 0 Td [(bc ,aswewillseeitagaininawhileChapterD[370]. Thistheoremiscute,anditisnicetohaveaformulafortheinverse,andaconditionthattellsuswhen wecanuseit.However,thisapproachbecomesimpracticalforlargermatrices,eventhoughitispossible todemonstratethat,intheory,thereisageneralformula.Thinkforaminuteaboutextendingthisresult tojust3 3matrices.Forstarters,weneed18letters!Instead,wewillworkcolumn-by-column.Let's rstworkanexamplethatwillmotivatethemaintheoremandremovesomeofthepreviousmystery. ExampleCMI Computingamatrixinverse ConsiderthematrixdenedinExampleMI[214]as, A = 2 6 6 6 6 4 12121 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(30 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 11021 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(31 3 7 7 7 7 5 Foritsinverse,wedesireamatrix B sothat AB = I 5 .Emphasizingthestructureofthecolumnsand employingthedenitionofmatrixmultiplicationDenitionMM[197], AB = I 5 A [ B 1 j B 2 j B 3 j B 4 j B 5 ]=[ e 1 j e 2 j e 3 j e 4 j e 5 ] [ A B 1 j A B 2 j A B 3 j A B 4 j A B 5 ]=[ e 1 j e 2 j e 3 j e 4 j e 5 ] : Equatingthematricescolumn-by-columnwehave A B 1 = e 1 A B 2 = e 2 A B 3 = e 3 A B 4 = e 4 A B 5 = e 5 : Sincethematrix B iswhatwearetryingtocompute,wecanvieweachcolumn, B i ,asacolumnvectorof unknowns.Thenwehavevesystemsofequationstosolve,eachwith5equationsin5variables.Notice thatall5ofthesesystemshavethesamecoecientmatrix.We'llnowsolveeachsysteminturn, Row-reducetheaugmentedmatrixofthelinearsystem LS A; e 1 2 6 6 6 6 4 121211 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(30 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 110210 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(310 3 7 7 7 7 5 RREF )443()223()222()443(! 2 6 6 6 6 6 4 1 0000 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 1 0000 00 1 001 000 1 01 0000 1 1 3 7 7 7 7 7 5 so B 1 = 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 1 1 1 3 7 7 7 7 5 Row-reducetheaugmentedmatrixofthelinearsystem LS A; e 2 2 6 6 6 6 4 121210 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(30 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 )]TJ/F15 10.9091 Tf 8.484 0 Td [(11 110210 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(20 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(310 3 7 7 7 7 5 RREF )443()223()222()443(! 2 6 6 6 6 6 4 1 00003 0 1 000 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 00 1 002 000 1 00 0000 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 7 7 7 5 so B 2 = 2 6 6 6 6 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 7 7 5 Version2.02

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SubsectionMISLE.CIMComputingtheInverseofaMatrix218 Row-reducetheaugmentedmatrixofthelinearsystem LS A; e 3 2 6 6 6 6 4 121210 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(30 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 )]TJ/F15 10.9091 Tf 8.484 0 Td [(10 110211 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(20 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(310 3 7 7 7 7 5 RREF )443()223()222()443(! 2 6 6 6 6 6 4 1 00006 0 1 000 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 00 1 004 000 1 01 0000 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 7 7 7 5 so B 3 = 2 6 6 6 6 4 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 7 7 5 Row-reducetheaugmentedmatrixofthelinearsystem LS A; e 4 2 6 6 6 6 4 121210 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(30 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 )]TJ/F15 10.9091 Tf 8.484 0 Td [(10 110210 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(310 3 7 7 7 7 5 RREF )443()223()222()443(! 2 6 6 6 6 6 4 1 0000 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 1 000 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 00 1 001 000 1 01 0000 1 0 3 7 7 7 7 7 5 so B 4 = 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 1 0 3 7 7 7 7 5 Row-reducetheaugmentedmatrixofthelinearsystem LS A; e 5 2 6 6 6 6 4 121210 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(30 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 )]TJ/F15 10.9091 Tf 8.484 0 Td [(10 110210 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(20 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(311 3 7 7 7 7 5 RREF )443()223()222()443(! 2 6 6 6 6 6 4 1 0000 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 1 0001 00 1 00 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 000 1 00 0000 1 1 3 7 7 7 7 7 5 so B 5 = 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 1 3 7 7 7 7 5 Wecannowcollectour5solutionvectorsintothematrix B B =[ B 1 j B 2 j B 3 j B 4 j B 5 ] = 2 6 6 6 6 4 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 1 1 1 3 7 7 7 7 5 2 6 6 6 6 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 7 7 5 2 6 6 6 6 4 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 7 7 5 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 1 0 3 7 7 7 7 5 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 1 3 7 7 7 7 5 3 7 7 7 7 5 = 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(336 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 1241 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 10110 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(201 3 7 7 7 7 5 Bythismethod,weknowthat AB = I 5 .Checkthat BA = I 5 ,andthenwewillknowthatwehavethe inverseof A Noticehowthevesystemsofequationsintheprecedingexamplewereallsolvedby exactly thesame sequenceofrowoperations.Wouldn'titbenicetoavoidthisobviousduplicationofeort?Ourmain theoremforthissectionfollows,anditmimicsthispreviousexample,whilealsoavoidingalltheoverhead. TheoremCINM ComputingtheInverseofaNonsingularMatrix Suppose A isanonsingularsquarematrixofsize n .Createthe n 2 n matrix M byplacingthe n n identitymatrix I n totherightofthematrix A .Let N beamatrixthatisrow-equivalentto M and Version2.02

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SubsectionMISLE.CIMComputingtheInverseofaMatrix219 inreducedrow-echelonform.Finally,let J bethematrixformedfromthenal n columnsof N .Then AJ = I n Proof A isnonsingular,sobyTheoremNMRRI[72]thereisasequenceofrowoperationsthatwill convert A into I n .Itisthissamesequenceofrowoperationsthatwillconvert M into N ,sincehaving theidentitymatrixintherst n columnsof N issucienttoguaranteethat N isinreducedrow-echelon form. Ifweconsiderthesystemsoflinearequations, LS A; e i ,1 i n ,weseethattheaforementioned sequenceofrowoperationswillalsobringtheaugmentedmatrixofeachofthesesystemsintoreducedrowechelonform.Furthermore,theuniquesolutionto LS A; e i appearsincolumn n +1oftherow-reduced augmentedmatrixofthesystemandisidenticaltocolumn n + i of N .Let N 1 ; N 2 ; N 3 ;:::; N 2 n denote thecolumnsof N .Sowend, AJ = A [ N n +1 j N n +2 j N n +3 j ::: j N n + n ] =[ A N n +1 j A N n +2 j A N n +3 j ::: j A N n + n ]DenitionMM[197] =[ e 1 j e 2 j e 3 j ::: j e n ] = I n DenitionIM[72] asdesired. Wehavetobejustabitcarefulhereaboutbothwhatthistheoremsaysandwhatitdoesn'tsay.If A isanonsingularmatrix,thenweareguaranteedamatrix B suchthat AB = I n ,andtheproofgivesusa processforconstructing B .However,thedenitionoftheinverseofamatrixDenitionMI[213]requires that BA = I n also.Soatthisjuncturewemustcomputethematrixproductintheopposite"orderbefore weclaim B astheinverseof A .However,we'llsoonseethatthisis always thecase,inTheoremOSIS [227],sothetitleofthistheoremisnotinaccurate. Whatif A issingular?AtthispointweonlyknowthatTheoremCINM[217]cannotbeapplied. Thequestionof A 'sinverseisstillopen.ButseeTheoremNI[228]inthenextsection.We'llnishby computingtheinverseforthecoecientmatrixofArchetypeB[707],theonewejustpulledfromahatin ExampleSABMI[212].TherearemoreexamplesintheArchetypesAppendixA[698]topracticewith, thoughnoticethatitissillytoaskfortheinverseofarectangularmatrixthesizesaren'trightandnot everysquarematrixhasaninverserememberExampleMWIAA[213]?. ExampleCMIAB Computingamatrixinverse,ArchetypeB ArchetypeB[707]hasacoecientmatrixgivenas B = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 557 104 3 5 ExercisingTheoremCINM[217]weset M = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12100 557010 104001 3 5 : whichrowreducesto N = 2 4 100 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.484 0 Td [(9 010 13 2 8 11 2 001 5 2 3 5 2 3 5 : Version2.02

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SubsectionMISLE.PMIPropertiesofMatrixInverses220 So B )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 13 2 8 11 2 5 2 3 5 2 3 5 oncewecheckthat B )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 B = I 3 theproductintheoppositeorderisaconsequenceofthetheorem. Whilewecanusearow-reducingproceduretocomputeanyneededinverse,mostcomputationaldevices haveabuilt-inproceduretocomputetheinverseofamatrixstraightaway.See:ComputationMI.MMA [671]ComputationMI.SAGE[677]. SubsectionPMI PropertiesofMatrixInverses Theinverseofamatrixenjoyssomeniceproperties.Wecollectafewhere.First,amatrixcanhavebut oneinverse. TheoremMIU MatrixInverseisUnique Supposethesquarematrix A hasaninverse.Then A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 isunique. Proof AsdescribedinTechniqueU[693],wewillassumethat A hastwoinverses.Thehypothesistells thereisatleastone.Supposethenthat B and C arebothinversesfor A .Then,repeateduseofDenition MI[213]andTheoremMMIM[200]plusoneapplicationofTheoremMMA[202]gives B = BI n TheoremMMIM[200] = B AC DenitionMI[213] = BA C TheoremMMA[202] = I n C DenitionMI[213] = C TheoremMMIM[200] Soweconcludethat B and C arethesame,andcannotbedierent.Soanymatrixthatactslike an inverse,mustbe the inverse. Whenmostofusdressinthemorning,weputonoursocksrst,followedbyourshoes.Intheevening wemustthenrstremoveourshoes,followedbyoursocks.Trytoconnecttheconclusionofthefollowing theoremwiththiseverydayexample. TheoremSS SocksandShoes Suppose A and B areinvertiblematricesofsize n .Then AB )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 = B )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 and AB isaninvertible matrix. Proof Attheriskofcarryingoureverydayanalogiestoofar,theproofofthistheoremisquiteeasywhen wecompareittotheworkingsofadatingservice.Wehaveastatementabouttheinverseofthematrix AB ,whichforallweknowrightnowmightnotevenexist.Suppose AB wastosignupforadatingservice withtworequirementsforacompatibledate.Uponmultiplicationontheleft,andontheright,theresult shouldbetheidentitymatrix.Inotherwords, AB 'sidealdatewouldbeitsinverse. Nowalongcomesthematrix B )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 whichweknowexistsbecauseourhypothesissaysboth A and B areinvertibleandwecanformtheproductofthesetwomatrices,alsolookingforadate.Let'sseeif Version2.02

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SubsectionMISLE.PMIPropertiesofMatrixInverses221 B )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 isagoodmatchfor AB .Firsttheymeetatanon-committalneutrallocation,sayacoeeshop, forquietconversation: B )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 AB = B )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 A B TheoremMMA[202] = B )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 I n B DenitionMI[213] = B )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 B TheoremMMIM[200] = I n DenitionMI[213] Therstdatehavinggonesmoothly,asecond,moreserious,dateisarranged,saydinnerandashow: AB B )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 = A BB )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 TheoremMMA[202] = AI n A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 DenitionMI[213] = AA )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 TheoremMMIM[200] = I n DenitionMI[213] Sothematrix B )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 hasmetalloftherequirementstobe AB 'sinversedateandwiththeensuing marriageproposalwecanannouncethat AB )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 = B )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 TheoremMIMI MatrixInverseofaMatrixInverse Suppose A isaninvertiblematrix.Then A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 isinvertibleand A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 = A Proof AswiththeproofofTheoremSS[219],weexamineif A isasuitableinversefor A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 bydenition, theoppositeistrue. AA )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 = I n DenitionMI[213] and A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 A = I n DenitionMI[213] Thematrix A hasmetalltherequirementstobetheinverseof A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 ,andsoisinvertibleandwecanwrite A = A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 TheoremMIT MatrixInverseofaTranspose Suppose A isaninvertiblematrix.Then A t isinvertibleand A t )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 = A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 t Proof AswiththeproofofTheoremSS[219],weseeif A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 t isasuitableinversefor A t .ApplyTheorem MMT[203]toseethat A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 t A t = AA )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 t TheoremMMT[203] = I t n DenitionMI[213] = I n I n issymmetric and A t A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 t = A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 A t TheoremMMT[203] = I t n DenitionMI[213] = I n I n issymmetric Version2.02

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SubsectionMISLE.READReadingQuestions222 Thematrix A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 t hasmetalltherequirementstobetheinverseof A t ,andsoisinvertibleandwecan write A t )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 = A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 t TheoremMISM MatrixInverseofaScalarMultiple Suppose A isaninvertiblematrixand isanonzeroscalar.Then A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 = 1 A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 and A isinvertible. Proof AswiththeproofofTheoremSS[219],weseeif 1 A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 isasuitableinversefor A 1 A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 A = 1 )]TJ/F21 10.9091 Tf 5 -8.836 Td [(AA )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 TheoremMMSMM[201] =1 I n Scalarmultiplicativeinverses = I n PropertyOM[184] and A 1 A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 = 1 )]TJ/F21 10.9091 Tf 5 -8.837 Td [(A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 A TheoremMMSMM[201] =1 I n Scalarmultiplicativeinverses = I n PropertyOM[184] Thematrix 1 A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 hasmetalltherequirementstobetheinverseof A ,sowecanwrite A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 = 1 A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 Noticethattherearesomelikelytheoremsthataremissinghere.Forexample,itwouldbetempting tothinkthat A + B )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 = A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 + B )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 ,butthisisfalse.Canyoundacounterexample?SeeExercise MISLE.T10[223]. SubsectionREAD ReadingQuestions 1.Computetheinverseofthematrixbelow. 410 26 2.Computetheinverseofthematrixbelow. 2 4 231 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(246 3 5 3.ExplainwhyTheoremSS[219]hasthetitleitdoes.Donotjuststatethetheorem,explainthe choiceofthetitlemakingreferencetothetheoremitself. Version2.02

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SubsectionMISLE.EXCExercises223 SubsectionEXC Exercises C21 Verifythat B istheinverseof A A = 2 6 6 4 11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1102 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1202 3 7 7 5 B = 2 6 6 4 420 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 84 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1010 )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 )]TJ/F15 10.9091 Tf 8.484 0 Td [(311 3 7 7 5 ContributedbyRobertBeezerSolution[224] C22 Recyclethematrices A and B fromExerciseMISLE.C21[222]andset c = 2 6 6 4 2 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 2 3 7 7 5 d = 2 6 6 4 1 1 1 1 3 7 7 5 Employthematrix B tosolvethetwolinearsystems LS A; c and LS A; d ContributedbyRobertBeezerSolution[224] C23 Ifitexists,ndtheinverseofthe2 2matrix A = 73 52 andcheckyouranswer.SeeTheoremTTMI[214]. ContributedbyRobertBeezer C24 Ifitexists,ndtheinverseofthe2 2matrix A = 63 42 andcheckyouranswer.SeeTheoremTTMI[214]. ContributedbyRobertBeezer C25 AttheconclusionofExampleCMI[216],verifythat BA = I 5 bycomputingthematrixproduct. ContributedbyRobertBeezer C26 Let D = 2 6 6 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(530 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(22 )]TJ/F15 10.9091 Tf 8.485 0 Td [(14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(104 105 )]TJ/F15 10.9091 Tf 8.485 0 Td [(25 3 7 7 7 7 5 Computetheinverseof D D )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 ,byformingthe5 10matrix[ D j I 5 ]androw-reducingTheoremCINM [217].Thenuseacalculatortocompute D )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 directly. ContributedbyRobertBeezerSolution[224] C27 Let E = 2 6 6 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(53 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(22 )]TJ/F15 10.9091 Tf 8.485 0 Td [(14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(102 105 )]TJ/F15 10.9091 Tf 8.485 0 Td [(24 3 7 7 7 7 5 Version2.02

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SubsectionMISLE.EXCExercises224 Computetheinverseof E E )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 ,byformingthe5 10matrix[ E j I 5 ]androw-reducingTheoremCINM [217].Thenuseacalculatortocompute E )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 directly. ContributedbyRobertBeezerSolution[224] C28 Let C = 2 6 6 4 1131 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 14102 )]TJ/F15 10.9091 Tf 8.484 0 Td [(20 )]TJ/F15 10.9091 Tf 8.485 0 Td [(45 3 7 7 5 Computetheinverseof C C )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 ,byformingthe4 8matrix[ C j I 4 ]androw-reducingTheoremCINM [217].Thenuseacalculatortocompute C )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 directly. ContributedbyRobertBeezerSolution[224] C40 Findallsolutionstothesystemofequationsbelow,makinguseofthematrixinversefoundin ExerciseMISLE.C28[223]. x 1 + x 2 +3 x 3 + x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 4 =4 x 1 +4 x 2 +10 x 3 +2 x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 3 +5 x 4 =9 ContributedbyRobertBeezerSolution[224] C41 Usetheinverseofamatrixtondallthesolutionstothefollowingsystemofequations. x 1 +2 x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 2 x 1 +5 x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 2 =2 ContributedbyRobertBeezerSolution[225] C42 Useamatrixinversetosolvethelinearsystemofequations. x 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 2 +2 x 3 =5 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 2 x 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 2 )]TJ/F21 10.9091 Tf 10.91 0 Td [(x 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 ContributedbyRobertBeezerSolution[225] T10 Constructanexampletodemonstratethat A + B )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 = A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 + B )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 isnottrueforallsquarematrices A and B ofthesamesize. ContributedbyRobertBeezerSolution[225] Version2.02

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SubsectionMISLE.SOLSolutions225 SubsectionSOL Solutions C21 ContributedbyRobertBeezerStatement[222] Checkthat both matrixproductsDenitionMM[197] AB and BA equalthe4 4identitymatrix I 4 DenitionIM[72]. C22 ContributedbyRobertBeezerStatement[222] Representeachofthetwosystemsbyavectorequality, A x = c and A y = d .TheninthespiritofExample SABMI[212],solutionsaregivenby x = B c = 2 6 6 4 8 21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(16 3 7 7 5 y = B d = 2 6 6 4 5 10 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 3 7 7 5 Noticehowwecouldsolvemanymoresystemshaving A asthecoecientmatrix,andhoweachsuchsystem hasauniquesolution.Youmightcheckyourworkbysubstitutingthesolutionsbackintothesystemsof equations,orformingthelinearcombinationsofthecolumnsof A suggestedbyTheoremSLSLC[93]. C26 ContributedbyRobertBeezerStatement[222] Theinverseof D is D )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 = 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(321 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(422 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(231 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3110 42 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 3 7 7 7 7 5 C27 ContributedbyRobertBeezerStatement[222] Thematrix E hasnoinverse,thoughwedonotyethaveatheoremthatallowsustoreachthisconclusion. However,whenrow-reducingthematrix[ E j I 5 ],therst5columnswillnotrow-reducetothe5 5identity matrix,soweareatalossonhowwemightcomputetheinverse.Whenrequestingthatyourcalculator compute E )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 ,itshouldgivesomeindicationthat E doesnothaveaninverse. C28 ContributedbyRobertBeezerStatement[223] EmployTheoremCINM[217], 2 6 6 4 11311000 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10100 141020010 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 )]TJ/F15 10.9091 Tf 8.484 0 Td [(450001 3 7 7 5 RREF )443()223()222()443(! 2 6 6 6 4 1 0003818 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 1 009647 )]TJ/F15 10.9091 Tf 8.484 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 00 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(39 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1952 000 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(16 )]TJ/F15 10.9091 Tf 8.485 0 Td [(821 3 7 7 7 5 Andthereforeweseethat C isnonsingular C row-reducestotheidentitymatrix,TheoremNMRRI[72] andbyTheoremCINM[217], C )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 = 2 6 6 4 3818 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 9647 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(39 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1952 )]TJ/F15 10.9091 Tf 8.485 0 Td [(16 )]TJ/F15 10.9091 Tf 8.485 0 Td [(821 3 7 7 5 C40 ContributedbyRobertBeezerStatement[223] Viewthissystemas LS C; b ,where C isthe4 4matrixfromExerciseMISLE.C28[223]and b = 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 9 3 7 7 5 Version2.02

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SubsectionMISLE.SOLSolutions226 Since C wasseentobenonsingularinExerciseMISLE.C28[223]TheoremSNCM[229]saysthesolution, whichisuniquebyTheoremNMUS[74],isgivenby C )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 b = 2 6 6 4 3818 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 9647 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(39 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1952 )]TJ/F15 10.9091 Tf 8.485 0 Td [(16 )]TJ/F15 10.9091 Tf 8.485 0 Td [(821 3 7 7 5 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 9 3 7 7 5 = 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 3 7 7 5 Noticethatthissolutioncanbeeasilycheckedintheoriginalsystemofequations. C41 ContributedbyRobertBeezerStatement[223] Thecoecientmatrixofthissystemofequationsis A = 2 4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 25 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(40 3 5 andthevectorofconstantsis b = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 2 3 5 .SobyTheoremSLEMM[195]wecanconvertthesystemtothe form A x = b .Row-reducingthismatrixyieldstheidentitymatrixsobyTheoremNMRRI[72]weknow A isnonsingular.ThisallowsustoapplyTheoremSNCM[229]tondtheuniquesolutionas x = A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 b = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(443 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(321 3 5 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 2 3 5 = 2 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 3 5 Remember,youcancheckthissolutioneasilybyevaluatingthematrix-vectorproduct A x DenitionMVP [194]. C42 ContributedbyRobertBeezerStatement[223] Wecanreformulatethelinearsystemasavectorequalitywithamatrix-vectorproductviaTheorem SLEMM[195].Thesystemisthenrepresentedby A x = b where A = 2 4 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(12 10 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 3 5 b = 2 4 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 3 5 AccordingtoTheoremSNCM[229],if A isnonsingularthentheuniquesolutionwillbegivenby A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 b Weattemptthecomputationof A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 throughTheoremCINM[217],orwithourfavoritecomputational deviceandobtain, A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 = 2 4 23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 35 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 5 SobyTheoremNI[228],weknow A isnonsingular,andsotheuniquesolutionis A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 b = 2 4 23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 35 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 5 2 4 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 3 5 = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 3 5 T10 ContributedbyRobertBeezerStatement[223] Let D beany2 2matrixthathasaninverseTheoremTTMI[214]canhelpyouconstructsuchamatrix, I 2 isasimplechoice.Set A = D and B = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 D .While A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 and B )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 bothexist,whatis A + B )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 ? Cantheproposedstatementbeatheorem? Version2.02

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SectionMINMMatrixInversesandNonsingularMatrices227 SectionMINM MatrixInversesandNonsingularMatrices WesawinTheoremCINM[217]thatifasquarematrix A isnonsingular,thenthereisamatrix B so that AB = I n .Inotherwords, B ishalfwaytobeinganinverseof A .Wewillseeinthissectionthat B automaticallyfulllsthesecondcondition BA = I n .ExampleMWIAA[213]showedusthatthe coecientmatrixfromArchetypeA[702]hadnoinverse.Notcoincidentally,thiscoecientmatrixis singular.We'llmakealltheseconnectionsprecisenow.Notmanyexamplesordenitionsinthissection, justtheorems. SubsectionNMI NonsingularMatricesareInvertible Weneedacoupleoftechnicalresultsforstarters.Somebookswouldcalltheseminor,butessential,results lemmas."We'lljustcall'emtheorems.SeeTechniqueLC[696]formoreonthedistinction. Therstofthesetechnicalresultsisinterestinginthatthehypothesissayssomethingaboutaproduct oftwosquarematricesandtheconclusionthensaysthesamethingabouteachindividualmatrixinthe product.Thisresulthasananalogyinthealgebraofcomplexnumbers:suppose ; 2 C ,then 6 =0 ifandonlyif 6 =0and 6 =0.Wecanviewthisresultassuggestingthatthetermnonsingular"for matricesislikethetermnonzero"forscalars. TheoremNPNT NonsingularProducthasNonsingularTerms Supposethat A and B aresquarematricesofsize n .Theproduct AB isnonsingularifandonlyif A and B arebothnonsingular. Proof We'lldothisportionoftheproofintwoparts,eachasaproofbycontradictionTechnique CD[692].Assumethat AB isnonsingular.Establishingthat B isnonsingularistheeasierpart,sowewill doitrst,butinreality,wewill need toknowthat B isnonsingularwhenweprovethat A isnonsingular. Youcanalsothinkofthisproofasbeingastudyoffourpossibleconclusionsinthetablebelow.One ofthefourrows must happenthelistisexhaustive.Intheproofwelearnthattherstthreerowslead tocontradictions,andsoareimpossible.Thatleavesthefourthrowasacertainty,whichisourdesired conclusion. AB Case Singular Singular 1 Nonsingular Singular 1 Singular Nonsingular 2 Nonsingular Nonsingular Part1.Suppose B issingular.Thenthereisanonzerovector z thatisasolutionto LS B; 0 .So AB z = A B z TheoremMMA[202] = A 0 TheoremSLEMM[195] = 0 TheoremMMZM[200] Because z isanonzerosolutionto LS AB; 0 ,weconcludethat AB issingularDenitionNM[71].This isacontradiction,so B isnonsingular,asdesired. Version2.02

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SubsectionMINM.NMINonsingularMatricesareInvertible228 Part2.Suppose A issingular.Thenthereisanonzerovector y thatisasolutionto LS A; 0 .Now considerthelinearsystem LS B; y .Sinceweknow B isnonsingularfromCase1,thesystemhasaunique solutionTheoremNMUS[74],whichwewilldenoteas w .Werstclaim w isnotthezerovectoreither. Assumingtheopposite,supposethat w = 0 TechniqueCD[692].Then y = B w TheoremSLEMM[195] = B 0 Hypothesis = 0 TheoremMMZM[200] contraryto y beingnonzero.So w 6 = 0 .Thepiecesareinplace,soherewego, AB w = A B w TheoremMMA[202] = A y TheoremSLEMM[195] = 0 TheoremSLEMM[195] So w isanonzerosolutionto LS AB; 0 ,andthuswecansaythat AB issingularDenitionNM[71]. Thisisacontradiction,so A isnonsingular,asdesired. Nowassumethatboth A and B arenonsingular.Supposethat x 2 C n isasolutionto LS AB; 0 Then 0 = AB x TheoremSLEMM[195] = A B x TheoremMMA[202] ByTheoremSLEMM[195], B x isasolutionto LS A; 0 ,andbythedenitionofanonsingularmatrix DenitionNM[71],weconcludethat B x = 0 .Now,byanentirelysimilarargument,thenonsingularity of B forcesustoconcludethat x = 0 .Sotheonlysolutionto LS AB; 0 isthezerovectorandwe concludethat AB isnonsingularbyDenitionNM[71]. Thisisapowerfulresultintheforward"direction,becauseitallowsustobeginwithahypothesis thatsomethingcomplicatedthematrixproduct AB hasthepropertyofbeingnonsingular,andwecan thenconcludethatthesimplerconstituents A and B individuallythenalsohavethepropertyofbeing nonsingular.Ifwehadthoughtthatthematrixproductwasanarticialconstruction,resultslikethis wouldmakeusbegintothinktwice. Thecontrapositiveofthisresultisequallyinteresting.Itsaysthat A or B orbothisasingularmatrix ifandonlyiftheproduct AB issingular.Noticehowthenegationofthetheorem'sconclusion A and B bothnonsingularbecomesthestatementatleastoneof A and B issingular."SeeTechniqueCP[691]. TheoremOSIS One-SidedInverseisSucient Suppose A and B aresquarematricesofsize n suchthat AB = I n .Then BA = I n Proof Thematrix I n isnonsingularsinceitrow-reduceseasilyto I n ,TheoremNMRRI[72].So A and B arenonsingularbyTheoremNPNT[226],soinparticular B isnonsingular.Wecantherefore applyTheoremCINM[217]toasserttheexistenceofamatrix C sothat BC = I n .Thisapplicationof TheoremCINM[217]couldbeabitconfusing,mostlybecauseofthenamesofthematricesinvolved. B isnonsingular,sotheremustbearight-inverse"for B ,andwe'recallingit C Now BA = BA I n TheoremMMIM[200] = BA BC TheoremCINM[217] = B AB C TheoremMMA[202] Version2.02

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SubsectionMINM.NMINonsingularMatricesareInvertible229 = BI n C Hypothesis = BC TheoremMMIM[200] = I n TheoremCINM[217] whichisthedesiredconclusion. SoTheoremOSIS[227]tellsusthatif A isnonsingular,thenthematrix B guaranteedbyTheorem CINM[217]willbebotharight-inverse"andaleft-inverse"for A ,so A isinvertibleand A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 = B Soifyouhaveanonsingularmatrix, A ,youcanusetheproceduredescribedinTheoremCINM[217] tondaninversefor A .If A issingular,thentheprocedureinTheoremCINM[217]willfailastherst n columnsof M willnotrow-reducetotheidentitymatrix.However,wecansayabitmore.When A issingular,then A doesnothaveaninversewhichisverydierentfromsayingthattheprocedurein TheoremCINM[217]failstondaninverse.Thismayfeellikewearesplittinghairs,butitsimportant thatwedonotmakeunfoundedassumptions.Theseobservationsmotivatethenexttheorem. TheoremNI NonsingularityisInvertibility Supposethat A isasquarematrix.Then A isnonsingularifandonlyif A isinvertible. Proof Suppose A isinvertible,andsupposethat x isanysolutiontothehomogeneoussystem LS A; 0 .Then x = I n x TheoremMMIM[200] = )]TJ/F21 10.9091 Tf 5 -8.836 Td [(A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 A x DenitionMI[213] = A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 A x TheoremMMA[202] = A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 0 TheoremSLEMM[195] = 0 TheoremMMZM[200] Sothe only solutionto LS A; 0 isthezerovector,sobyDenitionNM[71], A isnonsingular. Supposenowthat A isnonsingular.ByTheoremCINM[217]wend B sothat AB = I n .Then TheoremOSIS[227]tellsusthat BA = I n .So B is A 'sinverse,andbyconstruction, A isinvertible. Soforasquarematrix,thepropertiesofhavinganinverseandofhavingatrivialnullspaceareone andthesame.Can'thaveonewithouttheother. TheoremNME3 NonsingularMatrixEquivalences,Round3 Supposethat A isasquarematrixofsize n .Thefollowingareequivalent. 1. A isnonsingular. 2. A row-reducestotheidentitymatrix. 3.Thenullspaceof A containsonlythezerovector, N A = f 0 g 4.Thelinearsystem LS A; b hasauniquesolutionforeverypossiblechoiceof b 5.Thecolumnsof A arealinearlyindependentset. 6. A isinvertible. Proof WecanupdateourlistofequivalencesfornonsingularmatricesTheoremNME2[138]withthe equivalentconditionfromTheoremNI[228]. Inthecasethat A isanonsingularcoecientmatrixofasystemofequations,theinverseallowsusto veryquicklycomputetheuniquesolution,foranyvectorofconstants. Version2.02

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SubsectionMINM.UMUnitaryMatrices230 TheoremSNCM SolutionwithNonsingularCoecientMatrix Supposethat A isnonsingular.Thentheuniquesolutionto LS A; b is A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 b Proof ByTheoremNMUS[74]weknowalreadythat LS A; b hasauniquesolutionforeverychoiceof b .Weneedtoshowthattheexpressionstatedisindeedasolution the solution.That'seasy,justplug itin"tothecorrespondingvectorequationrepresentationTheoremSLEMM[195], A )]TJ/F21 10.9091 Tf 5 -8.837 Td [(A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 b = )]TJ/F21 10.9091 Tf 5 -8.837 Td [(AA )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 b TheoremMMA[202] = I n b DenitionMI[213] = b TheoremMMIM[200] Since A x = b istruewhenwesubstitute A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 b for x A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 b isathe!solutionto LS A; b SubsectionUM UnitaryMatrices Recallthattheadjointofamatrixis A = )]TJETq1 0 0 1 266.702 487.434 cm[]0 d 0 J 0.436 w 0 0 m 8.182 0 l SQBT/F21 10.9091 Tf 266.702 478.452 Td [(A t DenitionA[189]. DenitionUM UnitaryMatrices Supposethat U isasquarematrixofsize n suchthat U U = I n .Thenwesay U is unitary 4 Thisconditionmayseemratherfar-fetchedatrstglance.Wouldtherebe any matrixthatbehaved thisway?Well,yes,here'sone. ExampleUM3 Unitarymatrixofsize3 U = 2 6 4 1+ i p 5 3+2 i p 55 2+2 i p 22 1 )]TJ/F22 7.9701 Tf 6.587 0 Td [(i p 5 2+2 i p 55 )]TJ/F19 7.9701 Tf 6.586 0 Td [(3+ i p 22 i p 5 3 )]TJ/F19 7.9701 Tf 6.586 0 Td [(5 i p 55 )]TJ/F19 7.9701 Tf 15.326 4.295 Td [(2 p 22 3 7 5 Thecomputationsgetabittiresome,butifyouworkyourwaythroughthecomputationof U U ,you will arriveatthe3 3identitymatrix I 3 Unitarymatricesdonothavetolookquitesogruesome.Here'salargeronethatisabitmorepleasing. ExampleUPM Unitarypermutationmatrix Thematrix P = 2 6 6 6 6 4 01000 00010 10000 00001 00100 3 7 7 7 7 5 isunitaryascanbeeasilychecked.Noticethatitisjustarearrangementofthecolumnsofthe5 5 identitymatrix, I 5 DenitionIM[72]. Aninterestingexerciseistobuildanother5 5unitarymatrix, R ,usingadierentrearrangementof thecolumnsof I 5 .Thenformtheproduct PR .ThiswillbeanotherunitarymatrixExerciseMINM.T10 [234].Ifyouweretobuildall5!=5 4 3 2 1=120matricesofthistypeyouwouldhaveaset thatremainsclosedundermatrixmultiplication.Itisanexampleofanotheralgebraicstructureknownas Version2.02

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SubsectionMINM.UMUnitaryMatrices231 a group sincetogetherthesetandtheoneoperationmatrixmultiplicationhereisclosed,associative, hasanidentity I 5 ,andinversesTheoremUMI[230].Noticethoughthattheoperationinthisgroupis notcommutative! Ifamatrix A hasonlyrealnumberentrieswesayitisa realmatrix thenthedeningpropertyof beingunitarysimpliesto A t A = I n .Inthiscasewe,andeverybodyelse,callsthematrix orthogonal soyoumayoftenencounterthisterminyourotherreadingwhenthecomplexnumbersarenotunder consideration. Unitarymatriceshaveeasilycomputedinverses.Theyalsohavecolumnsthatformorthonormalsets. Herearethetheoremsthatshowusthatunitarymatricesarenotasstrangeastheymightinitiallyappear. TheoremUMI UnitaryMatricesareInvertible Supposethat U isaunitarymatrixofsize n .Then U isnonsingular,and U )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 = U Proof ByDenitionUM[229],weknowthat U U = I n .Thematrix I n isnonsingularsinceitrowreduceseasilyto I n ,TheoremNMRRI[72].SobyTheoremNPNT[226], U and U arebothnonsingular matrices. Theequation U U = I n getsushalfwaytoaninverseof U ,andTheoremOSIS[227]tellsusthatthen UU = I n also.So U and U areinversesofeachotherDenitionMI[213]. TheoremCUMOS ColumnsofUnitaryMatricesareOrthonormalSets Supposethat A isasquarematrixofsize n withcolumns S = f A 1 ; A 2 ; A 3 ;:::; A n g .Then A isaunitary matrixifandonlyif S isanorthonormalset. Proof Theproofrevolvesaroundrecognizingthatatypicalentryoftheproduct A A isaninnerproduct ofcolumnsof A .Herearethedetailstosupportthisclaim. [ A A ] ij = n X k =1 [ A ] ik [ A ] kj TheoremEMP[198] = n X k =1 h )]TJETq1 0 0 1 219.487 324.066 cm[]0 d 0 J 0.436 w 0 0 m 8.182 0 l SQBT/F21 10.9091 Tf 219.487 315.084 Td [(A t i ik [ A ] kj TheoremEMP[198] = n X k =1 A ki [ A ] kj DenitionTM[185] = n X k =1 [ A ] ki [ A ] kj DenitionCCM[187] = n X k =1 [ A ] kj [ A ] ki PropertyCMCN[680] = n X k =1 [ A j ] k [ A i ] k = h A j ; A i i DenitionIP[168] Wenowemploythisequalityinachainofequivalences, S = f A 1 ; A 2 ; A 3 ;:::; A n g isanorthonormalset h A j ; A i i = 0if i 6 = j 1if i = j DenitionONS[177] Version2.02

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SubsectionMINM.UMUnitaryMatrices232 [ A A ] ij = 0if i 6 = j 1if i = j [ A A ] ij =[ I n ] ij ; 1 i n; 1 j n DenitionIM[72] A A = I n DenitionME[182] A isaunitarymatrixDenitionUM[229] ExampleOSMC Orthonormalsetfrommatrixcolumns Thematrix U = 2 6 4 1+ i p 5 3+2 i p 55 2+2 i p 22 1 )]TJ/F22 7.9701 Tf 6.587 0 Td [(i p 5 2+2 i p 55 )]TJ/F19 7.9701 Tf 6.586 0 Td [(3+ i p 22 i p 5 3 )]TJ/F19 7.9701 Tf 6.586 0 Td [(5 i p 55 )]TJ/F19 7.9701 Tf 15.326 4.295 Td [(2 p 22 3 7 5 fromExampleUM3[229]isaunitarymatrix.ByTheoremCUMOS[230],itscolumns 8 > < > : 2 6 4 1+ i p 5 1 )]TJ/F22 7.9701 Tf 6.587 0 Td [(i p 5 i p 5 3 7 5 ; 2 6 4 3+2 i p 55 2+2 i p 55 3 )]TJ/F19 7.9701 Tf 6.587 0 Td [(5 i p 55 3 7 5 ; 2 6 4 2+2 i p 22 )]TJ/F19 7.9701 Tf 6.586 0 Td [(3+ i p 22 )]TJ/F19 7.9701 Tf 15.326 4.295 Td [(2 p 22 3 7 5 9 > = > ; formanorthonormalset.Youmightndcheckingthesixinnerproductsofpairsofthesevectorseasier thandoingthematrixproduct U U .Or,becausetheinnerproductisanti-commutativeTheoremIPAC [170]youonlyneedcheckthreeinnerproductsseeExerciseMINM.T12[234]. Whenusingvectorsandmatricesthatonlyhaverealnumberentries,orthogonalmatricesarethose matriceswithinversesthatequaltheirtranspose.Similarly,theinnerproductisthefamiliardotproduct. Keepthisspecialcaseinmindasyoureadthenexttheorem. TheoremUMPIP UnitaryMatricesPreserveInnerProducts Supposethat U isaunitarymatrixofsize n and u and v aretwovectorsfrom C n .Then h U u ;U v i = h u ; v i and k U v k = k v k Proof h U u ;U v i = U u t U v TheoremMMIP[202] = u t U t U v TheoremMMT[203] = u t U t U v TheoremMMCC[203] = u t U t U v TheoremCCT[682] = u t )]TJETq1 0 0 1 223.775 153.045 cm[]0 d 0 J 0.436 w 0 0 m 8.638 0 l SQBT/F21 10.9091 Tf 223.775 144.064 Td [(U t U v TheoremMCT[189] = u t )]TJETq1 0 0 1 223.775 131.43 cm[]0 d 0 J 0.436 w 0 0 m 8.638 0 l SQBT/F21 10.9091 Tf 223.775 122.448 Td [(U t U v TheoremMMCC[203] = u t U U v DenitionA[189] = u t I n v DenitionUM[229] = u t I n v DenitionIM[72] = u t v TheoremMMIM[200] Version2.02

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SubsectionMINM.READReadingQuestions233 = h u ; v i TheoremMMIP[202] Thesecondconclusionisjustaspecializationoftherstconclusion. k U v k = q k U v k 2 = p h U v ;U v i TheoremIPN[171] = p h v ; v i = q k v k 2 TheoremIPN[171] = k v k Asidefromtheinherentinterestinthistheorem,itmakesabiggerstatementaboutunitarymatrices. Whenweviewvectorsgeometricallyasdirectionsorforces,thenthenormequatestoanotionoflength.If wetransformavectorbymultiplicationwithaunitarymatrix,thenthelengthnormofthatvectorstays thesame.Ifweconsidercolumnvectorswithtwoorthreeslotscontainingonlyrealnumbers,thentheinner productoftwosuchvectorsisjustthedotproduct,andthisquantitycanbeusedtocomputetheangle betweentwovectors.Whentwovectorsaremultipliedtransformedbythesameunitarymatrix,their dotproductisunchangedandtheirindividuallengthsareunchanged.Theresultsintheanglebetween thetwovectorsremainingunchanged. Aunitarytransformation"matrix-vectorproductswithunitarymatricesthuspreservegeometrical relationshipsamongvectorsrepresentingdirections,forces,orotherphysicalquantities.Inthecaseofatwoslotvectorwithrealentries,thisissimplyarotation.Thesesortsofcomputationsareexceedinglyimportant incomputergraphicssuchasgamesandreal-timesimulations,especiallywhenincreasedrealismisachieved byperformingmanysuchcomputationsquickly.Wewillseeunitarymatricesagaininsubsequentsections especiallyTheoremOD[607]andineachinstance,considertheinterpretationoftheunitarymatrix asasortofgeometry-preservingtransformation.Someauthorsusetheterm isometry tohighlightthis behavior.Wewillspeaklooselyofaunitarymatrixasbeingasortofgeneralizedrotation. Analreminder:thetermsdotproduct,"symmetricmatrix"andorthogonalmatrix"usedinreferencetovectorsormatriceswithrealnumberentriescorrespondtothetermsinnerproduct,"Hermitian matrix"andunitarymatrix"whenwegeneralizetoincludecomplexnumberentries,sokeepthatinmind asyoureadelsewhere. SubsectionREAD ReadingQuestions 1.Computetheinverseofthecoecientmatrixofthesystemofequationsbelowandusetheinverseto solvethesystem. 4 x 1 +10 x 2 =12 2 x 1 +6 x 2 =4 2.InthereadingquestionsforSectionMISLE[212]youwereaskedtondtheinverseofthe3 3matrix below. 2 4 231 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(246 3 5 Version2.02

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SubsectionMINM.READReadingQuestions234 Becausethematrixwasnotnonsingular,youhadnotheoremsatthatpointthatwouldallowyouto computetheinverse.Explainwhyyounowknowthattheinversedoesnotexistwhichisdierent thannotbeingabletocomputeitbyquotingtherelevanttheorem'sacronym. 3.Isthematrix A unitary?Why? A = 1 p 22 +2 i 1 p 374 +3 i 1 p 22 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 1 p 374 +14 i # Version2.02

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SubsectionMINM.EXCExercises235 SubsectionEXC Exercises C40 Solvethesystemofequationsbelowusingtheinverseofamatrix. x 1 + x 2 +3 x 3 + x 4 =5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 3 )]TJ/F21 10.9091 Tf 10.91 0 Td [(x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 x 1 +4 x 2 +10 x 3 +2 x 4 =9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 3 +5 x 4 =9 ContributedbyRobertBeezerSolution[235] M20 Constructanexampleofa4 4unitarymatrix. ContributedbyRobertBeezerSolution[235] M80 Matrixmultiplicationinteractsnicelywithmanyoperations.Butnotalwayswithtransforminga matrixtoreducedrow-echelonform.Supposethat A isan m n matrixand B isan n p matrix.Let P be amatrixthatisrow-equivalentto A andinreducedrow-echelonform, Q beamatrixthatisrow-equivalent to B andinreducedrow-echelonform,andlet R beamatrixthatisrow-equivalentto AB andinreduced row-echelonform.Is PQ = R ?Inotherwords,withnonstandardnotation,isrref A rref B =rref AB ? Constructacounterexampletoshowthat,ingeneral,thisstatementisfalse.Thenndalargeclassof matriceswhereif A and B areintheclass,thenthestatementistrue. ContributedbyMarkHamrickSolution[235] T10 Supposethat Q and P areunitarymatricesofsize n .Provethat QP isaunitarymatrix. ContributedbyRobertBeezer T11 ProvethatHermitianmatricesDenitionHM[205]haverealentriesonthediagonal.More precisely,supposethat A isaHermitianmatrixofsize n .Then[ A ] ii 2 R ,1 i n ContributedbyRobertBeezer T12 Supposethatwearecheckingifasquarematrixofsize n isunitary.Showthatastraightforward applicationofTheoremCUMOS[230]requiresthecomputationof n 2 innerproductswhenthematrixis unitary,andfewerwhenthematrixisnotorthogonal.Thenshowthatthismaximumnumberofinner productscanbereducedto 1 2 n n +1inlightofTheoremIPAC[170]. ContributedbyRobertBeezer Version2.02

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SubsectionMINM.SOLSolutions236 SubsectionSOL Solutions C40 ContributedbyRobertBeezerStatement[234] Thecoecientmatrixandvectorofconstantsforthesystemare 2 6 6 4 1131 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 14102 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 )]TJ/F15 10.9091 Tf 8.485 0 Td [(45 3 7 7 5 b = 2 6 6 4 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 9 9 3 7 7 5 A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 canbecomputedbyusingacalculator,orbythemethodofTheoremCINM[217].ThenTheorem SNCM[229]saystheuniquesolutionis A )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 b = 2 6 6 4 3818 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 9647 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.484 0 Td [(39 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1952 )]TJ/F15 10.9091 Tf 8.484 0 Td [(16 )]TJ/F15 10.9091 Tf 8.485 0 Td [(821 3 7 7 5 2 6 6 4 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 9 9 3 7 7 5 = 2 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 3 3 7 7 5 M20 ContributedbyRobertBeezerStatement[234] The4 4identitymatrix, I 4 ,wouldbeoneexampleDenitionIM[72].Anyofthe23otherrearrangements ofthecolumnsof I 4 wouldbeasimple,butlesstrivial,example.SeeExampleUPM[229]. M80 ContributedbyRobertBeezerStatement[234] Take A = 10 00 B = 00 10 Then A isalreadyinreducedrow-echelonform,andbyswappingrows, B row-reducesto A .Sotheproduct oftherow-echelonformsof A is AA = A 6 = O .However,theproduct AB isthe2 2zeromatrix,which isinreduced-echelonform,andnotequalto AA .Whenyougetthere,TheoremPEEF[262]orTheorem EMDRO[372]mightshedsomelightonwhywewouldnotexpectthisstatementtobetrueingeneral. If A and B arenonsingular,then AB isnonsingularTheoremNPNT[226],andallthreematrices A B and AB row-reducetotheidentitymatrixTheoremNMRRI[72].ByTheoremMMIM[200],the desiredrelationshipistrue. Version2.02

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SectionCRSColumnandRowSpaces237 SectionCRS ColumnandRowSpaces TheoremSLSLC[93]showedusthatthereisanaturalcorrespondencebetweensolutionstolinearsystemsandlinearcombinationsofthecolumnsofthecoecientmatrix.Thisideamotivatesthefollowing importantdenition. DenitionCSM ColumnSpaceofaMatrix Supposethat A isan m n matrixwithcolumns f A 1 ; A 2 ; A 3 ;:::; A n g .Thenthe columnspace of A written C A ,isthesubsetof C m containingalllinearcombinationsofthecolumnsof A C A = hf A 1 ; A 2 ; A 3 ;:::; A n gi ThisdenitioncontainsNotationCSM. 4 Someauthorsrefertothecolumnspaceofamatrixasthe range ,butwewillreservethistermforuse withlineartransformationsDenitionRLT[496]. SubsectionCSSE ColumnSpacesandSystemsofEquations Uponencounteringanynewset,therstquestionweaskiswhatobjectsareintheset,andwhichobjects arenot?Here'sanexampleofonewaytoanswerthisquestion,anditwillmotivateatheoremthatwill thenanswerthequestionprecisely. ExampleCSMCS Columnspaceofamatrixandconsistentsystems ArchetypeD[716]andArchetypeE[720]arelinearsystemsofequations,withanidentical3 4coecient matrix,whichwecall A here.However,ArchetypeD[716]isconsistent,whileArchetypeE[720]isnot. Wecanexplainthisdierencebyemployingthecolumnspaceofthematrix A Thecolumnvectorofconstants, b ,inArchetypeD[716]is b = 2 4 8 )]TJ/F15 10.9091 Tf 8.484 0 Td [(12 4 3 5 Onesolutionto LS A; b ,aslisted,is x = 2 6 6 4 7 8 1 3 3 7 7 5 ByTheoremSLSLC[93],wecansummarizethissolutionasalinearcombinationofthecolumnsof A that equals b 7 2 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 3 5 +8 2 4 1 4 1 3 5 +1 2 4 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 4 3 5 +3 2 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(7 )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 3 5 = 2 4 8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 4 3 5 = b : Thisequationsaysthat b isalinearcombinationofthecolumnsof A ,andthenbyDenitionCSM[236], wecansaythat b 2C A Version2.02

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SubsectionCRS.CSSEColumnSpacesandSystemsofEquations238 Ontheotherhand,ArchetypeE[720]isthelinearsystem LS A; c ,wherethevectorofconstantsis c = 2 4 2 3 2 3 5 andthissystemofequationsisinconsistent.Thismeans c 62C A ,forifitwere,thenitwouldequala linearcombinationofthecolumnsof A andTheoremSLSLC[93]wouldleadustoasolutionofthesystem LS A; c Soifwexthecoecientmatrix,andvarythevectorofconstants,wecansometimesndconsistent systems,andsometimesinconsistentsystems.Thevectorsofconstantsthatleadtoconsistentsystems areexactlytheelementsofthecolumnspace.Thisisthecontentofthenexttheorem,andsinceitisan equivalence,itprovidesanalternateviewofthecolumnspace. TheoremCSCS ColumnSpacesandConsistentSystems Suppose A isan m n matrixand b isavectorofsize m .Then b 2C A ifandonlyif LS A; b is consistent. Proof Suppose b 2C A .Thenwecanwrite b assomelinearcombinationofthecolumnsof A .By TheoremSLSLC[93]wecanusethescalarsfromthislinearcombinationtoformasolutionto LS A; b sothissystemisconsistent. If LS A; b isconsistent,thereisasolutionthatmaybeusedwithTheoremSLSLC[93]towrite b asalinearcombinationofthecolumnsof A .Thisqualies b formembershipin C A Thistheoremtellsusthataskingifthesystem LS A; b isconsistentisexactlythesamequestionas askingif b isinthecolumnspaceof A .Orequivalently,ittellsusthatthecolumnspaceofthematrix A ispreciselythosevectorsofconstants, b ,thatcanbepairedwith A tocreateasystemoflinearequations LS A; b thatisconsistent. EmployingTheoremSLEMM[195]wecanformthechainofequivalences b 2C A LS A; b isconsistent A x = b forsome x Thus,analternativeandpopulardenitionofthecolumnspaceofan m n matrix A is C A = f y 2 C m j y = A x forsome x 2 C n g = f A x j x 2 C n g C m Werecognizethisassayingcreate all thematrixvectorproductspossiblewiththematrix A byletting x rangeoverallofthepossibilities.ByDenitionMVP[194]weseethatthismeanstakeallpossiblelinear combinationsofthecolumnsof A |preciselythedenitionofthecolumnspaceDenitionCSM[236] wehavechosen. Noticehowthisformulationofthecolumnspacelooksverymuchlikethedenitionofthenullspaceof amatrixDenitionNSM[64],butforarectangularmatrixthecolumnvectorsof C A and N A have dierentsizes,sothesetsareverydierent. Givenavector b andamatrix A itisnowverymechanicaltotestif b 2C A .Formthelinearsystem LS A; b ,row-reducetheaugmentedmatrix,[ A j b ],andtestforconsistencywithTheoremRCLS[53]. Here'sanexampleofthisprocedure. ExampleMCSM Membershipinthecolumnspaceofamatrix Considerthecolumnspaceofthe3 4matrix A A = 2 4 321 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(46 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 3 5 Version2.02

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SubsectionCRS.CSSEColumnSpacesandSystemsofEquations239 Werstshowthat v = 2 4 18 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 12 3 5 isinthecolumnspaceof A v 2C A .TheoremCSCS[237]saysweneed onlychecktheconsistencyof LS A; v .Formtheaugmentedmatrixandrow-reduce, 2 4 321 )]TJ/F15 10.9091 Tf 8.485 0 Td [(418 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(46 )]TJ/F15 10.9091 Tf 8.485 0 Td [(812 3 5 RREF )443()223()222()443(! 2 4 1 01 )]TJ/F15 10.9091 Tf 8.484 0 Td [(26 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(110 00000 3 5 Withoutaleading1inthenalcolumn,TheoremRCLS[53]tellsusthesystemisconsistentandtherefore byTheoremCSCS[237], v 2C A Ifwewishedtodemonstrateexplicitlythat v isalinearcombinationofthecolumnsof A ,wecan ndasolutionanysolutionof LS A; v anduseTheoremSLSLC[93]toconstructthedesiredlinear combination.Forexample,setthefreevariablesto x 3 =2and x 4 =1.Thenasolutionhas x 2 =1and x 1 =6.ThenbyTheoremSLSLC[93], v = 2 4 18 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 12 3 5 =6 2 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 5 +1 2 4 2 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 5 +2 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 6 3 5 +1 2 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(8 3 5 Nowweshowthat w = 2 4 2 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 5 isnotinthecolumnspaceof A w 62C A .TheoremCSCS[237]sayswe needonlychecktheconsistencyof LS A; w .Formtheaugmentedmatrixandrow-reduce, 2 4 321 )]TJ/F15 10.9091 Tf 8.485 0 Td [(42 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(231 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(46 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 5 RREF )443()223()222()443(! 2 4 1 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(110 0000 1 3 5 Withaleading1inthenalcolumn,TheoremRCLS[53]tellsusthesystemisinconsistentandtherefore byTheoremCSCS[237], w 62C A TheoremCSCS[237]completesacollectionofthreetheorems,andonedenition,thatdeservecomment. Manyquestionsaboutspans,linearindependence,nullspace,columnspacesandsimilarobjectscanbe convertedtoquestionsaboutsystemsofequationshomogeneousornot,whichweunderstandwellfrom ourpreviousresults,especiallythoseinChapterSLE[2].Thesepreviousresultsincludetheoremslike TheoremRCLS[53]whichallowsustoquicklydecideconsistencyofasystem,andTheoremBNS[139] whichallowsustodescribesolutionsetsforhomogeneoussystemscompactlyasthespanofalinearly independentsetofcolumnvectors. Thetablebelowliststhesefordenitionsandtheoremsalongwithabriefreminderofthestatement andanexampleofhowthestatementisused. DenitionNSM[64] Synopsis Nullspaceissolutionsetofhomogeneoussystem Example GeneralsolutionsetsdescribedbyTheoremPSPHS[105] TheoremSLSLC[93] Synopsis Solutionsforlinearcombinationswithunknownscalars Example Decidingmembershipinspans TheoremSLEMM[195] Synopsis Systemofequationsrepresentedbymatrix-vectorproduct Example Solutionto LS A; b is A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 b when A isnonsingular TheoremCSCS[237] Synopsis Columnspacevectorscreateconsistentsystems Example Decidingmembershipincolumnspaces Version2.02

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SubsectionCRS.CSSOCColumnSpaceSpannedbyOriginalColumns240 SubsectionCSSOC ColumnSpaceSpannedbyOriginalColumns Sowehaveafoolproof,automatedprocedurefordeterminingmembershipin C A .Whilethisworksjust neavectoratatime,wewouldliketohaveamoreusefuldescriptionoftheset C A asawhole.The nextexamplewillpreviewtherstoftwofundamentalresultsaboutthecolumnspaceofamatrix. ExampleCSTW Columnspace,twoways Considerthe5 7matrix A 2 6 6 6 6 4 241 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1144 1210247 0014187 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12196 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(413 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 7 7 5 AccordingtothedenitionDenitionCSM[236],thecolumnspaceof A is C A = 8 > > > > < > > > > : 2 6 6 6 6 4 2 1 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 7 7 5 ; 2 6 6 6 6 4 4 2 0 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 7 7 7 7 5 ; 2 6 6 6 6 4 1 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 3 7 7 7 7 5 ; 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 4 2 3 3 7 7 7 7 5 ; 2 6 6 6 6 4 1 2 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 7 7 5 ; 2 6 6 6 6 4 4 4 8 9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 7 7 5 ; 2 6 6 6 6 4 4 7 7 6 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 3 7 7 7 7 5 9 > > > > = > > > > ; + Whilethisisaconcisedescriptionofaninniteset,wemightbeabletodescribethespanwithfewerthan sevenvectors.ThisisthesubstanceofTheoremBS[157].Sowetakethesesevenvectorsandmakethem thecolumnsofmatrix,whichissimplytheoriginalmatrix A again.Nowwerow-reduce, 2 6 6 6 6 4 241 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1144 1210247 0014187 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12196 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(413 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 7 7 5 RREF )443()223()222()443(! 2 6 6 6 6 6 4 1 200031 00 1 00 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 000 1 021 0000 1 13 0000000 3 7 7 7 7 7 5 Thepivotcolumnsare D = f 1 ; 3 ; 4 ; 5 g ,sowecancreatetheset T = 8 > > > > < > > > > : 2 6 6 6 6 4 2 1 0 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 3 7 7 7 7 5 ; 2 6 6 6 6 4 1 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 3 7 7 7 7 5 ; 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 4 2 3 3 7 7 7 7 5 ; 2 6 6 6 6 4 1 2 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 7 7 5 9 > > > > = > > > > ; andknowthat C A = h T i and T isalinearlyindependentsetofcolumnsfromthesetofcolumnsof A Wewillnowformalizethepreviousexample,whichwillmakeittrivialtodeterminealinearlyindependentsetofvectorsthatwillspanthecolumnspaceofamatrix,andisconstitutedofjustcolumnsof A TheoremBCS BasisoftheColumnSpace Supposethat A isan m n matrixwithcolumns A 1 ; A 2 ; A 3 ;:::; A n ,and B isarow-equivalentmatrixin reducedrow-echelonformwith r nonzerorows.Let D = f d 1 ;d 2 ;d 3 ;:::;d r g bethesetofcolumnindices where B hasleading1's.Let T = f A d 1 ; A d 2 ; A d 3 ;:::; A d r g .Then Version2.02

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SubsectionCRS.CSSOCColumnSpaceSpannedbyOriginalColumns241 1. T isalinearlyindependentset. 2. C A = h T i Proof DenitionCSM[236]describesthecolumnspaceasthespanofthesetofcolumnsof A .Theorem BS[157]tellsusthatwecanreducethesetofvectorsusedinaspan.IfweapplyTheoremBS[157]to C A ,wewouldcollectthecolumnsof A intoamatrixwhichwouldjustbe A againandbringthematrix toreducedrow-echelonform,whichisthematrix B inthestatementofthetheorem.Inthiscase,the conclusionsofTheoremBS[157]appliedto A B and C A areexactlytheconclusionswedesire. Thisisaniceresultsinceitgivesusahandfulofvectorsthatdescribetheentirecolumnspacethrough thespan,andwebelievethissetisassmallaspossiblebecausewecannotcreateanymorerelationsof lineardependencetotrimitdownfurther.Furthermore,wedenedthecolumnspaceDenitionCSM [236]asalllinearcombinationsofthecolumnsofthematrix,andtheelementsoftheset S arestillcolumns ofthematrixwewon'tbesoluckyinthenexttwoconstructionsofthecolumnspace. Procedurallythistheoremisextremelyeasytoapply.Row-reducetheoriginalmatrix,identify r columnswithleading1'sinthisreducedmatrix,andgrabthecorrespondingcolumnsoftheoriginal matrix.ButitisstillimportanttostudytheproofofTheoremBS[157]anditsmotivationinExample COV[154]whichlieattherootofthistheorem.We'lltrotthroughanexampleallthesame. ExampleCSOCD Columnspace,originalcolumns,ArchetypeD Let'sdetermineacompactexpressionfortheentirecolumnspaceofthecoecientmatrixofthesystem ofequationsthatisArchetypeD[716].NoticethatinExampleCSMCS[236]wewereonlydeterminingif individualvectorswereinthecolumnspaceornot,nowwearedescribingtheentirecolumnspace. TostartwiththeapplicationofTheoremBCS[239],callthecoecientmatrix A A = 2 4 217 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(34 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 114 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 5 : androw-reduceittoreducedrow-echelonform, B = 2 4 1 03 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0000 3 5 : Thereareleading1'sincolumns1and2,so D = f 1 ; 2 g .Toconstructasetthatspans C A ,justgrabthe columnsof A indicatedbytheset D ,so C A = 8 < : 2 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 3 5 ; 2 4 1 4 1 3 5 9 = ; + : That'sit. InExampleCSMCS[236]wedeterminedthatthevector c = 2 4 2 3 2 3 5 wasnot inthecolumnspaceof A .Trytowrite c asalinearcombinationofthersttwocolumnsof A Whathappens? Version2.02

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SubsectionCRS.CSNMColumnSpaceofaNonsingularMatrix242 AlsoinExampleCSMCS[236]wedeterminedthatthevector b = 2 4 8 )]TJ/F15 10.9091 Tf 8.484 0 Td [(12 4 3 5 was inthecolumnspaceof A .Trytowrite b asalinearcombinationofthersttwocolumnsof A .What happens?Didyoundauniquesolutiontothisquestion?Hmmmm. SubsectionCSNM ColumnSpaceofaNonsingularMatrix Let'sspecializetosquarematricesandcontrastthecolumnspacesofthecoecientmatricesinArchetype A[702]andArchetypeB[707]. ExampleCSAA ColumnspaceofArchetypeA ThecoecientmatrixinArchetypeA[702]is A = 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 211 110 3 5 whichrow-reducesto 2 4 1 01 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 000 3 5 : Columns1and2haveleading1's,sobyTheoremBCS[239]wecanwrite C A = hf A 1 ; A 2 gi = 8 < : 2 4 1 2 1 3 5 ; 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 1 3 5 9 = ; + : Wewanttoshowinthisexamplethat C A 6 = C 3 .Sotake,forexample,thevector b = 2 4 1 3 2 3 5 .Thenthere isnosolutiontothesystem LS A; b ,orequivalently,itisnotpossibletowrite b asalinearcombination of A 1 and A 2 .Tryoneofthesetwocomputationsyourself.Ortryboth!.Since b 62C A ,thecolumn spaceof A cannotbeallof C 3 .Sobyvaryingthevectorofconstants,itispossibletocreateinconsistent systemsofequationswiththiscoecientmatrixthevector b beingonesuchexample. InExampleMWIAA[213]wewishedtoshowthatthecoecientmatrixfromArchetypeA[702]was notinvertibleasarstexampleofamatrixwithoutaninverse.Ourdevicetherewastondaninconsistent linearsystemwith A asthecoecientmatrix.Thevectorofconstantsinthatexamplewas b ,deliberately chosenoutsidethecolumnspaceof A ExampleCSAB ColumnspaceofArchetypeB ThecoecientmatrixinArchetypeB[707],callit B here,isknowntobenonsingularseeExampleNM [72].ByTheoremNMUS[74],thelinearsystem LS B; b hasauniquesolutionforeverychoiceof b TheoremCSCS[237]thensaysthat b 2C B forall b 2 C 3 .Stateddierently,thereisnowaytobuild Version2.02

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SubsectionCRS.CSNMColumnSpaceofaNonsingularMatrix243 aninconsistentsystemwiththecoecientmatrix B ,butthenweknewthatalreadyfromTheoremNMUS [74]. ExampleCSAA[241]andExampleCSAB[241]togethermotivatethefollowingequivalence,whichsays thatnonsingularmatriceshavecolumnspacesthatareasbigaspossible. TheoremCSNM ColumnSpaceofaNonsingularMatrix Suppose A isasquarematrixofsize n .Then A isnonsingularifandonlyif C A = C n Proof Suppose A isnonsingular.Wewishtoestablishthesetequality C A = C n .ByDenition CSM[236], C A C n Toshowthat C n C A choose b 2 C n .ByTheoremNMUS[74],weknowthelinearsystem LS A; b hasauniquesolutionandthereforeisconsistent.TheoremCSCS[237]thensaysthat b 2C A .Soby DenitionSE[684], C A = C n If e i iscolumn i ofthe n n identitymatrixDenitionSUV[173]andbyhypothesis C A = C n then e i 2C A for1 i n .ByTheoremCSCS[237],thesystem LS A; e i isconsistentfor1 i n Let b i denoteanyoneparticularsolutionto LS A; e i ,1 i n Denethe n n matrix B =[ b 1 j b 2 j b 3 j ::: j b n ].Then AB = A [ b 1 j b 2 j b 3 j ::: j b n ] =[ A b 1 j A b 2 j A b 3 j ::: j A b n ]DenitionMM[197] =[ e 1 j e 2 j e 3 j ::: j e n ] = I n DenitionSUV[173] Sothematrix B isaright-inverse"for A .ByTheoremNMRRI[72], I n isanonsingularmatrix,so byTheoremNPNT[226]both A and B arenonsingular.Thus,inparticular, A isnonsingular.Travis Osbornecontributedtothisproof. Withthisequivalencefornonsingularmatriceswecanupdateourlist,TheoremNME3[228]. TheoremNME4 NonsingularMatrixEquivalences,Round4 Supposethat A isasquarematrixofsize n .Thefollowingareequivalent. 1. A isnonsingular. 2. A row-reducestotheidentitymatrix. 3.Thenullspaceof A containsonlythezerovector, N A = f 0 g 4.Thelinearsystem LS A; b hasauniquesolutionforeverypossiblechoiceof b 5.Thecolumnsof A arealinearlyindependentset. 6. A isinvertible. 7.Thecolumnspaceof A is C n C A = C n Proof SinceTheoremCSNM[242]isanequivalence,wecanaddittothelistinTheoremNME3[228]. Version2.02

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SubsectionCRS.RSMRowSpaceofaMatrix244 SubsectionRSM RowSpaceofaMatrix Therowsofamatrixcanbeviewedasvectors,sincetheyarejustlistsofnumbers,arrangedhorizontally. Sowewilltransposeamatrix,turningrowsintocolumns,sowecanthenmanipulaterowsascolumn vectors.Asaresultwewillbeabletomakesomenewconnectionsbetweenrowoperationsandsolutions tosystemsofequations.OK,hereisthesecondprimarydenitionofthissection. DenitionRSM RowSpaceofaMatrix Suppose A isan m n matrix.Thenthe rowspace of A R A ,isthecolumnspaceof A t ,i.e. R A = C )]TJ/F21 10.9091 Tf 5 -8.836 Td [(A t ThisdenitioncontainsNotationRSM. 4 Informally,therowspaceisthesetofalllinearcombinationsoftherowsof A .However,wewrite therowsascolumnvectors,thusthenecessityofusingthetransposetomaketherowsintocolumns. Additionally,withtherowspacedenedintermsofthecolumnspace,allofthepreviousresultsofthis sectioncanbeappliedtorowspaces. Noticethatif A isarectangular m n matrix,then C A C m ,while R A C n andthetwosets arenotcomparablesincetheydonotevenholdobjectsofthesametype.However,when A issquareof size n ,both C A and R A aresubsetsof C n ,thoughusuallythesetswillnotbeequalbutseeExercise CRS.M20[251]. ExampleRSAI RowspaceofArchetypeI ThecoecientmatrixinArchetypeI[737]is I = 2 6 6 4 140 )]TJ/F15 10.9091 Tf 8.485 0 Td [(107 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 28 )]TJ/F15 10.9091 Tf 8.485 0 Td [(139 )]TJ/F15 10.9091 Tf 8.485 0 Td [(137 002 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(412 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4248 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3137 3 7 7 5 : Tobuildtherowspace,wetransposethematrix, I t = 2 6 6 6 6 6 6 6 6 4 120 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 480 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(122 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(34 09 )]TJ/F15 10.9091 Tf 8.485 0 Td [(48 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1312 )]TJ/F15 10.9091 Tf 8.485 0 Td [(31 )]TJ/F15 10.9091 Tf 8.485 0 Td [(97 )]TJ/F15 10.9091 Tf 8.485 0 Td [(837 3 7 7 7 7 7 7 7 7 5 Thenthecolumnsofthismatrixareusedinaspantobuildtherowspace, R I = C )]TJ/F21 10.9091 Tf 5 -8.836 Td [(I t = 8 > > > > > > > > < > > > > > > > > : 2 6 6 6 6 6 6 6 6 4 1 4 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 3 7 7 7 7 7 7 7 7 5 ; 2 6 6 6 6 6 6 6 6 4 2 8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 7 3 7 7 7 7 7 7 7 7 5 ; 2 6 6 6 6 6 6 6 6 4 0 0 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 3 7 7 7 7 7 7 7 7 5 ; 2 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 2 4 8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(31 37 3 7 7 7 7 7 7 7 7 5 9 > > > > > > > > = > > > > > > > > ; + : Version2.02

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SubsectionCRS.RSMRowSpaceofaMatrix245 However,wecanuseTheoremBCS[239]togetaslightlybetterdescription.First,row-reduce I t 2 6 6 6 6 6 6 6 6 6 4 1 00 )]TJ/F19 7.9701 Tf 9.681 4.295 Td [(31 7 0 1 0 12 7 00 1 13 7 0000 0000 0000 0000 3 7 7 7 7 7 7 7 7 7 5 : Sincethereareleading1'sincolumnswithindices D = f 1 ; 2 ; 3 g ,thecolumnspaceof I t canbespanned byjusttherstthreecolumnsof I t R I = C )]TJ/F21 10.9091 Tf 5 -8.837 Td [(I t = 8 > > > > > > > > < > > > > > > > > : 2 6 6 6 6 6 6 6 6 4 1 4 0 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 0 7 )]TJ/F15 10.9091 Tf 8.484 0 Td [(9 3 7 7 7 7 7 7 7 7 5 ; 2 6 6 6 6 6 6 6 6 4 2 8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 7 3 7 7 7 7 7 7 7 7 5 ; 2 6 6 6 6 6 6 6 6 4 0 0 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 3 7 7 7 7 7 7 7 7 5 9 > > > > > > > > = > > > > > > > > ; + : Therowspacewouldnotbetoointerestingifitwassimplythecolumnspaceofthetranspose.However, whenwedorowoperationsonamatrixwehavenoeectonthemanylinearcombinationsthatcanbe formedwiththerowsofthematrix.Thisisstatedmorecarefullyinthefollowingtheorem. TheoremREMRS Row-EquivalentMatriceshaveequalRowSpaces Suppose A and B arerow-equivalentmatrices.Then R A = R B Proof Twomatricesarerow-equivalentDenitionREM[28]ifonecanbeobtainedfromanotherby asequenceofpossiblymanyrowoperations.Wewillprovethetheoremfortwomatricesthatdier byasinglerowoperation,andthenthisresultcanbeappliedrepeatedlytogetthefullstatementofthe theorem.Therowspacesof A and B arespansofthecolumnsoftheirtransposes.Foreachrowoperation weperformonamatrix,wecandeneananalogousoperationonthecolumns.Perhapsweshouldcall these columnoperations .Instead,wewillstillcallthemrowoperations,butwewillapplythemtothe columnsofthetransposes. Refertothecolumnsof A t and B t as A i and B i ,1 i m .Therowoperationthatswitchesrowswill justswitchcolumnsofthetransposedmatrices.Thiswillhavenoeectonthepossiblelinearcombinations formedbythecolumns. Supposethat B t isformedfrom A t bymultiplyingcolumn A t by 6 =0.Inotherwords, B t = A t and B i = A i forall i 6 = t .Weneedtoestablishthattwosetsareequal, C )]TJ/F21 10.9091 Tf 5 -8.837 Td [(A t = C )]TJ/F21 10.9091 Tf 5 -8.837 Td [(B t .Wewilltakea genericelementofoneandshowthatitiscontainedintheother. 1 B 1 + 2 B 2 + 3 B 3 + + t B t + + m B m = 1 A 1 + 2 A 2 + 3 A 3 + + t A t + + m A m = 1 A 1 + 2 A 2 + 3 A 3 + + t A t + + m A m saysthat C )]TJ/F21 10.9091 Tf 5 -8.836 Td [(B t C )]TJ/F21 10.9091 Tf 5 -8.836 Td [(A t .Similarly, 1 A 1 + 2 A 2 + 3 A 3 + + t A t + + m A m = 1 A 1 + 2 A 2 + 3 A 3 + + t A t + + m A m Version2.02

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SubsectionCRS.RSMRowSpaceofaMatrix246 = 1 A 1 + 2 A 2 + 3 A 3 + + t A t + + m A m = 1 B 1 + 2 B 2 + 3 B 3 + + t B t + + m B m saysthat C )]TJ/F21 10.9091 Tf 5 -8.837 Td [(A t C )]TJ/F21 10.9091 Tf 5 -8.837 Td [(B t .So R A = C )]TJ/F21 10.9091 Tf 5 -8.837 Td [(A t = C )]TJ/F21 10.9091 Tf 5 -8.837 Td [(B t = R B whenasinglerowoperationofthesecond typeisperformed. Supposenowthat B t isformedfrom A t byreplacing A t with A s + A t forsome 2 C and s 6 = t .In otherwords, B t = A s + A t ,and B i = A i for i 6 = t 1 B 1 + 2 B 2 + 3 B 3 + + s B s + + t B t + + m B m = 1 A 1 + 2 A 2 + 3 A 3 + + s A s + + t A s + A t + + m A m = 1 A 1 + 2 A 2 + 3 A 3 + + s A s + + t A s + t A t + + m A m = 1 A 1 + 2 A 2 + 3 A 3 + + s A s + t A s + + t A t + + m A m = 1 A 1 + 2 A 2 + 3 A 3 + + s + t A s + + t A t + + m A m saysthat C )]TJ/F21 10.9091 Tf 5 -8.837 Td [(B t C )]TJ/F21 10.9091 Tf 5 -8.837 Td [(A t .Similarly, 1 A 1 + 2 A 2 + 3 A 3 + + s A s + + t A t + + m A m = 1 A 1 + 2 A 2 + 3 A 3 + + s A s + + )]TJ/F21 10.9091 Tf 8.485 0 Td [( t A s + t A s + t A t + + m A m = 1 A 1 + 2 A 2 + 3 A 3 + + )]TJ/F21 10.9091 Tf 8.485 0 Td [( t A s + s A s + + t A s + t A t + + m A m = 1 A 1 + 2 A 2 + 3 A 3 + + )]TJ/F21 10.9091 Tf 8.485 0 Td [( t + s A s + + t A s + A t + + m A m = 1 B 1 + 2 B 2 + 3 B 3 + + )]TJ/F21 10.9091 Tf 8.485 0 Td [( t + s B s + + t B t + + m B m saysthat C )]TJ/F21 10.9091 Tf 5 -8.837 Td [(A t C )]TJ/F21 10.9091 Tf 5 -8.837 Td [(B t .So R A = C )]TJ/F21 10.9091 Tf 5 -8.837 Td [(A t = C )]TJ/F21 10.9091 Tf 5 -8.837 Td [(B t = R B whenasinglerowoperationofthethird typeisperformed. Sotherowspaceofamatrixispreservedbyeachrowoperation,andhencerowspacesofrow-equivalent matricesareequalsets. ExampleRSREM Rowspacesoftworow-equivalentmatrices InExampleTREM[28]wesawthatthematrices A = 2 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(134 52 )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 1106 3 5 B = 2 4 1106 30 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(134 3 5 arerow-equivalentbydemonstratingasequenceoftworowoperationsthatconverted A into B .Applying TheoremREMRS[244]wecansay R A = 8 > > < > > : 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 4 3 7 7 5 ; 2 6 6 4 5 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 3 7 7 5 ; 2 6 6 4 1 1 0 6 3 7 7 5 9 > > = > > ; + = 8 > > < > > : 2 6 6 4 1 1 0 6 3 7 7 5 ; 2 6 6 4 3 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 3 7 7 5 ; 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 4 3 7 7 5 9 > > = > > ; + = R B TheoremREMRS[244]isatitsbestwhenoneoftherow-equivalentmatricesisinreducedrow-echelon form.Thevectorsthatcorrespondtothezerorowscanbeignored.Whoneedsthezerovectorwhen buildingaspan?SeeExerciseLI.T10[144].Theechelonpatterninsuresthatthenonzerorowsyield vectorsthatarelinearlyindependent.Here'sthetheorem. TheoremBRS BasisfortheRowSpace Supposethat A isamatrixand B isarow-equivalentmatrixinreducedrow-echelonform.Let S bethe setofnonzerocolumnsof B t .Then Version2.02

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SubsectionCRS.RSMRowSpaceofaMatrix247 1. R A = h S i 2. S isalinearlyindependentset. Proof FromTheoremREMRS[244]weknowthat R A = R B .If B hasanyzerorows,these correspondtocolumnsof B t thatarethezerovector.Wecansafelytossoutthezerovectorinthespan construction,sinceitcanberecreatedfromthenonzerovectorsbyalinearcombinationwhereallthe scalarsarezero.So R A = h S i Suppose B has r nonzerorowsandlet D = f d 1 ;d 2 ;d 3 ;:::;d r g denotethecolumnindicesof B that havealeadingoneinthem.Denotethe r columnvectorsof B t ,thevectorsin S ,as B 1 ; B 2 ; B 3 ;:::; B r Toshowthat S islinearlyindependent,startwitharelationoflineardependence 1 B 1 + 2 B 2 + 3 B 3 + + r B r = 0 Nowconsiderthisvectorequalityinlocation d i .Since B isinreducedrow-echelonform,theentriesof column d i of B areallzero,exceptforaleading1inrow i .Thus,in B t ,row d i isallzeros,exceptinga 1incolumn i .So,for1 i r 0=[ 0 ] d i DenitionZCV[25] =[ 1 B 1 + 2 B 2 + 3 B 3 + + r B r ] d i DenitionRLDCV[132] =[ 1 B 1 ] d i +[ 2 B 2 ] d i +[ 3 B 3 ] d i + +[ r B r ] d i +DenitionMA[182] = 1 [ B 1 ] d i + 2 [ B 2 ] d i + 3 [ B 3 ] d i + + r [ B r ] d i +DenitionMSM[183] = 1 + 2 + 3 + + i + + r DenitionRREF[30] = i Soweconcludethat i =0forall1 i r ,establishingthelinearindependenceof S DenitionLICV [132]. ExampleIAS Improvingaspan Supposeinthecourseofanalyzingamatrixitscolumnspace,itsnullspace,its...weencounterthe followingsetofvectors,describedbyaspan X = 8 > > > > < > > > > : 2 6 6 6 6 4 1 2 1 6 6 3 7 7 7 7 5 ; 2 6 6 6 6 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 6 3 7 7 7 7 5 ; 2 6 6 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 7 7 5 ; 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 3 7 7 7 7 5 9 > > > > = > > > > ; + Let A bethematrixwhoserowsarethevectorsin X ,sobydesign X = R A A = 2 6 6 4 12166 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(16 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(32 )]TJ/F15 10.9091 Tf 8.485 0 Td [(36 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 3 7 7 5 Row-reduce A toformarow-equivalentmatrixinreducedrow-echelonform, B = 2 6 6 4 1 002 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 1 031 00 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(25 00000 3 7 7 5 Version2.02

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SubsectionCRS.RSMRowSpaceofaMatrix248 ThenTheoremBRS[245]sayswecangrabthenonzerocolumnsof B t andwrite X = R A = R B = 8 > > > > < > > > > : 2 6 6 6 6 4 1 0 0 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 7 7 5 ; 2 6 6 6 6 4 0 1 0 3 1 3 7 7 7 7 5 ; 2 6 6 6 6 4 0 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 5 3 7 7 7 7 5 9 > > > > = > > > > ; + Thesethreevectorsprovideamuch-improveddescriptionof X .Therearefewervectors,andthepattern ofzerosandonesintherstthreeentriesmakesiteasiertodeterminemembershipin X .Andallwe hadtodowasrow-reducetherightmatrixandtossoutazerorow.Nexttorowoperationsthemselves, thisisprobablythemostpowerfulcomputationaltechniqueatyourdisposal asitquicklyprovidesamuch improveddescriptionofaspan,anyspan. TheoremBRS[245]andthetechniquesofExampleIAS[246]willprovideyetanotherdescriptionof thecolumnspaceofamatrix.Firstwestateatrivialityasatheorem,sowecanreferenceitlater. TheoremCSRST ColumnSpace,RowSpace,Transpose Suppose A isamatrix.Then C A = R )]TJ/F21 10.9091 Tf 5 -8.836 Td [(A t Proof C A = C )]TJ/F21 10.9091 Tf 5 -8.836 Td [(A t t TheoremTT[187] = R )]TJ/F21 10.9091 Tf 5 -8.836 Td [(A t DenitionRSM[243] Sotondanotherexpressionforthecolumnspaceofamatrix,builditstranspose,row-reduceit,toss outthezerorows,andconvertthenonzerorowstocolumnvectorstoyieldanimprovedsetforthespan construction.We'lldoArchetypeI[737],thenyoudoArchetypeJ[741]. ExampleCSROI Columnspacefromrowoperations,ArchetypeI TondthecolumnspaceofthecoecientmatrixofArchetypeI[737],weproceedasfollows.Thematrix is I = 2 6 6 4 140 )]TJ/F15 10.9091 Tf 8.485 0 Td [(107 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 28 )]TJ/F15 10.9091 Tf 8.485 0 Td [(139 )]TJ/F15 10.9091 Tf 8.485 0 Td [(137 002 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(412 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4248 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3137 3 7 7 5 : Thetransposeis 2 6 6 6 6 6 6 6 6 4 120 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 480 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(122 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(34 09 )]TJ/F15 10.9091 Tf 8.485 0 Td [(48 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1312 )]TJ/F15 10.9091 Tf 8.485 0 Td [(31 )]TJ/F15 10.9091 Tf 8.485 0 Td [(97 )]TJ/F15 10.9091 Tf 8.485 0 Td [(837 3 7 7 7 7 7 7 7 7 5 : Row-reducedthisbecomes, 2 6 6 6 6 6 6 6 6 6 4 1 00 )]TJ/F19 7.9701 Tf 9.681 4.295 Td [(31 7 0 1 0 12 7 00 1 13 7 0000 0000 0000 0000 3 7 7 7 7 7 7 7 7 7 5 : Version2.02

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SubsectionCRS.READReadingQuestions249 Now,usingTheoremCSRST[247]andTheoremBRS[245] C I = R )]TJ/F21 10.9091 Tf 5 -8.836 Td [(I t = 8 > > < > > : 2 6 6 4 1 0 0 )]TJ/F19 7.9701 Tf 9.681 4.295 Td [(31 7 3 7 7 5 ; 2 6 6 4 0 1 0 12 7 3 7 7 5 ; 2 6 6 4 0 0 1 13 7 3 7 7 5 9 > > = > > ; + : Thisisaverynicedescriptionofthecolumnspace.Fewervectorsthanthe7involvedinthedenition, andthepatternofthezerosandonesintherst3slotscanbeusedtoadvantage.Forexample,Archetype I[737]ispresentedasaconsistentsystemofequationswithavectorofconstants b = 2 6 6 4 3 9 1 4 3 7 7 5 : Since LS I; b isconsistent,TheoremCSCS[237]tellsusthat b 2C I .Butwecouldseethisquickly withthefollowingcomputation,whichreallyonlyinvolvesanyworkinthe4thentryofthevectorsasthe scalarsinthelinearcombinationare dictated bytherstthreeentriesof b b = 2 6 6 4 3 9 1 4 3 7 7 5 =3 2 6 6 4 1 0 0 )]TJ/F19 7.9701 Tf 9.68 4.295 Td [(31 7 3 7 7 5 +9 2 6 6 4 0 1 0 12 7 3 7 7 5 +1 2 6 6 4 0 0 1 13 7 3 7 7 5 Canyounowrapidlyconstructseveralvectors, b ,sothat LS I; b isconsistent,andseveralmoresothat thesystemisinconsistent? SubsectionREAD ReadingQuestions 1.Writethecolumnspaceofthematrixbelowasthespanofasetofthreevectorsandexplainyour choiceofmethod. 2 4 1313 2011 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1210 3 5 2.Supposethat A isan n n nonsingularmatrix.Whatcanyousayaboutitscolumnspace? 3.Isthevector 2 6 6 4 0 5 2 3 3 7 7 5 intherowspaceofthefollowingmatrix?Whyorwhynot? 2 4 1313 2011 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1210 3 5 Version2.02

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SubsectionCRS.EXCExercises250 SubsectionEXC Exercises C30 ExampleCSOCD[240]expressesthecolumnspaceofthecoecientmatrixfromArchetypeD[716] callthematrix A hereasthespanofthersttwocolumnsof A .InExampleCSMCS[236]wedetermined thatthevector c = 2 4 2 3 2 3 5 wasnot inthecolumnspaceof A andthatthevector b = 2 4 8 )]TJ/F15 10.9091 Tf 8.484 0 Td [(12 4 3 5 was inthecolumnspaceof A .Attempttowrite c and b aslinearcombinationsofthetwovectorsinthe spanconstructionforthecolumnspaceinExampleCSOCD[240]andrecordyourobservations. ContributedbyRobertBeezerSolution[253] C31 Forthematrix A belowndasetofvectors T meetingthefollowingrequirements:thespanof T isthecolumnspaceof A ,thatis, h T i = C A T islinearlyindependent,andtheelementsof T arecolumnsof A A = 2 6 6 4 214 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1511 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(701 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(18 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 3 7 7 5 ContributedbyRobertBeezerSolution[253] C32 InExampleCSAA[241],verifythatthevector b isnotinthecolumnspaceofthecoecientmatrix. ContributedbyRobertBeezer C33 Findalinearlyindependentset S sothatthespanof S h S i ,isrowspaceofthematrix B ,and S islinearlyindependent. B = 2 4 2311 1101 )]TJ/F15 10.9091 Tf 8.485 0 Td [(123 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 5 ContributedbyRobertBeezerSolution[253] C34 Forthe3 4matrix A andthecolumnvector y 2 C 4 givenbelow,determineif y isintherow spaceof A .Inotherwords,answerthequestion: y 2R A ?points A = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(267 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(30 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 8076 3 5 y = 2 6 6 4 2 1 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 5 ContributedbyRobertBeezerSolution[253] C35 Forthematrix A below,ndtwodierentlinearlyindependentsetswhosespansequalthecolumn spaceof A C A ,suchthat atheelementsareeachcolumnsof A Version2.02

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SubsectionCRS.EXCExercises251 bthesetisobtainedbyaprocedurethatissubstantiallydierentfromtheprocedureyouuseinpart a. A = 2 4 351 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 1233 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4713 3 5 ContributedbyRobertBeezerSolution[254] C40 Thefollowingarchetypesaresystemsofequations.Foreachsystem,writethevectorofconstants asalinearcombinationofthevectorsinthespanconstructionforthecolumnspaceprovidedbyTheorem BCS[239]thesevectorsarelistedforeachofthesearchetypes. ArchetypeA[702] ArchetypeB[707] ArchetypeC[712] ArchetypeD[716] ArchetypeE[720] ArchetypeF[724] ArchetypeG[729] ArchetypeH[733] ArchetypeI[737] ArchetypeJ[741] ContributedbyRobertBeezer C42 Thefollowingarchetypesareeithermatricesorsystemsofequationswithcoecientmatrices.For eachmatrix,computeasetofcolumnvectorssuchthatthevectorsarecolumnsofthematrix,the setislinearlyindependent,andthespanofthesetisthecolumnspaceofthematrix.SeeTheorem BCS[239]. ArchetypeA[702] ArchetypeB[707] ArchetypeC[712] ArchetypeD[716]/ArchetypeE[720] ArchetypeF[724] ArchetypeG[729]/ArchetypeH[733] ArchetypeI[737] ArchetypeJ[741] ArchetypeK[746] ArchetypeL[750] ContributedbyRobertBeezer C50 Thefollowingarchetypesareeithermatricesorsystemsofequationswithcoecientmatrices.For eachmatrix,computeasetofcolumnvectorssuchthatthesetislinearlyindependent,andthe spanofthesetistherowspaceofthematrix.SeeTheoremBRS[245]. ArchetypeA[702] ArchetypeB[707] ArchetypeC[712] ArchetypeD[716]/ArchetypeE[720] ArchetypeF[724] ArchetypeG[729]/ArchetypeH[733] Version2.02

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SubsectionCRS.EXCExercises252 ArchetypeI[737] ArchetypeJ[741] ArchetypeK[746] ArchetypeL[750] ContributedbyRobertBeezer C51 Thefollowingarchetypesareeithermatricesorsystemsofequationswithcoecientmatrices.For eachmatrix,computethecolumnspaceasthespanofalinearlyindependentsetasfollows:transposethe matrix,row-reduce,tossoutzerorows,convertrowsintocolumnvectors.SeeExampleCSROI[247]. ArchetypeA[702] ArchetypeB[707] ArchetypeC[712] ArchetypeD[716]/ArchetypeE[720] ArchetypeF[724] ArchetypeG[729]/ArchetypeH[733] ArchetypeI[737] ArchetypeJ[741] ArchetypeK[746] ArchetypeL[750] ContributedbyRobertBeezer C52 Thefollowingarchetypesaresystemsofequations.Foreachdierentcoecientmatrixbuildtwo newvectorsofconstants.Therstshouldleadtoaconsistentsystemandthesecondshouldleadtoan inconsistentsystem.Descriptionsofthecolumnspaceasspansoflinearlyindependentsetsofvectorswith nicepatterns"ofzerosandonesmightbemostusefulandinstructiveinconnectionwiththisexercise. SeetheendofExampleCSROI[247]. ArchetypeA[702] ArchetypeB[707] ArchetypeC[712] ArchetypeD[716]/ArchetypeE[720] ArchetypeF[724] ArchetypeG[729]/ArchetypeH[733] ArchetypeI[737] ArchetypeJ[741] ContributedbyRobertBeezer M10 Forthematrix E below,ndvectors b and c sothatthesystem LS E; b isconsistentand LS E; c isinconsistent. E = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2110 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(102 4116 3 5 ContributedbyRobertBeezerSolution[254] M20 Usuallythecolumnspaceandnullspaceofamatrixcontainvectorsofdierentsizes.Forasquare matrix,though,thevectorsinthesetwosetsarethesamesize.Usuallythetwosetswillbedierent. Constructanexampleofasquarematrixwherethecolumnspaceandnullspaceareequal. Version2.02

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SubsectionCRS.EXCExercises253 ContributedbyRobertBeezerSolution[255] M21 Wehaveavarietyoftheoremsabouthowtocreatecolumnspacesandrowspacesandtheyfrequently involverow-reducingamatrix.Hereisaprocedurethatsometrytousetogetacolumnspace.Begin withan m n matrix A androw-reducetoamatrix B withcolumns B 1 ; B 2 ; B 3 ;:::; B n .Thenformthe columnspaceof A as C A = hf B 1 ; B 2 ; B 3 ;:::; B n gi = C B Thisis not notalegitimateprocedure,andthereforeis not atheorem.Constructanexampletoshowthat theprocedurewillnotingeneralcreatethecolumnspaceof A ContributedbyRobertBeezerSolution[255] T40 Supposethat A isan m n matrixand B isan n p matrix.Provethatthecolumnspaceof AB is asubsetofthecolumnspaceof A ,thatis C AB C A .Provideanexamplewheretheoppositeisfalse, inotherwordsgiveanexamplewhere C A 6C AB .ComparewithExerciseMM.T40[207]. ContributedbyRobertBeezerSolution[255] T41 Supposethat A isan m n matrixand B isan n n nonsingularmatrix.Provethatthecolumn spaceof A isequaltothecolumnspaceof AB ,thatis C A = C AB .ComparewithExerciseMM.T41 [207]andExerciseCRS.T40[252]. ContributedbyRobertBeezerSolution[255] T45 Supposethat A isan m n matrixand B isan n m matrixwhere AB isanonsingularmatrix. Provethat N B = f 0 g C B N A = f 0 g Discussthecasewhen m = n inconnectionwithTheoremNPNT[226]. ContributedbyRobertBeezerSolution[255] Version2.02

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SubsectionCRS.SOLSolutions254 SubsectionSOL Solutions C30 ContributedbyRobertBeezerStatement[249] Ineachcase,beginwithavectorequationwhereonesidecontainsalinearcombinationofthetwovectors fromthespanconstructionthatgivesthecolumnspaceof A withunknownsforscalars,andthenuse TheoremSLSLC[93]tosetupasystemofequations.For c ,thecorrespondingsystemhasnosolution,as wewouldexpect. For b thereisasolution,aswewouldexpect.Whatisinterestingisthatthesolutionisunique.This isaconsequenceofthelinearindependenceofthesetoftwovectorsinthespanconstruction.Ifwewrote b asalinearcombinationofallfourcolumnsof A ,thentherewouldbeinnitelymanywaystodothis. C31 ContributedbyRobertBeezerStatement[249] TheoremBCS[239]istherighttoolforthisproblem.Row-reducethismatrix,identifythepivotcolumns andthengrabthecorrespondingcolumnsof A fortheset T .Thematrix A row-reducesto 2 6 6 6 4 1 0300 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(200 000 1 0 0000 1 3 7 7 7 5 So D = f 1 ; 2 ; 4 ; 5 g andthen T = f A 1 ; A 2 ; A 4 ; A 5 g = 8 > > < > > : 2 6 6 4 2 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 7 7 5 ; 2 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 5 ; 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 5 ; 2 6 6 4 2 1 1 2 3 7 7 5 9 > > = > > ; hastherequestedproperties. C33 ContributedbyRobertBeezerStatement[249] TheoremBRS[245]isthemostdirectroutetoasetwiththeseproperties.Row-reduce,tosszerorows, keeptheothers.Youcouldalsotransposethematrix,thenlookforthecolumnspacebyrow-reducingthe transposeandapplyingTheoremBCS[239].We'lldotheformer, B RREF )443()223()222()443(! 2 4 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 0 1 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 0000 3 5 Sotheset S is S = 8 > > < > > : 2 6 6 4 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 7 7 5 ; 2 6 6 4 0 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 5 9 > > = > > ; C34 ContributedbyRobertBeezerStatement[249] y 2R A y 2C )]TJ/F21 10.9091 Tf 5 -8.836 Td [(A t DenitionRSM[243] LS )]TJ/F21 10.9091 Tf 5 -8.836 Td [(A t ; y isconsistentTheoremCSCS[237] Version2.02

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SubsectionCRS.SOLSolutions255 Theaugmentedmatrix A t y rowreducesto 2 6 6 6 4 1 000 0 1 00 00 1 0 000 1 3 7 7 7 5 andwithaleading1inthenalcolumnTheoremRCLS[53]tellsusthelinearsystemisinconsistentand so y 62R A C35 ContributedbyRobertBeezerStatement[249] aByTheoremBCS[239]wecanrow-reduce A ,identifypivotcolumnswiththeset D ,andkeep"those columnsof A andwewillhaveasetwiththedesiredproperties. A RREF )443()223()222()443(! 2 4 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(19 0 1 811 0000 3 5 Sowehavethesetofpivotcolumns D = f 1 ; 2 g andwekeep"thersttwocolumnsof A 8 < : 2 4 3 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 3 5 ; 2 4 5 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 5 9 = ; bWecanviewthecolumnspaceastherowspaceofthetransposeTheoremCSRST[247].Wecan getabasisoftherowspaceofamatrixquicklybybringingthematrixtoreducedrow-echelonformand keepingthenonzerorowsascolumnvectorsTheoremBRS[245].Heregoes. A t RREF )443()223()222()443(! 2 6 6 4 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 1 3 000 000 3 7 7 5 Takingthenonzerorowsandtiltingthemupascolumnsgivesus 8 < : 2 4 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 5 ; 2 4 0 1 3 3 5 9 = ; Anapproachbasedonthematrix L fromextendedechelonformDenitionEEF[261]andTheoremFS [263]willworkaswellasanalternativeapproach. M10 ContributedbyRobertBeezerStatement[251] Anyvectorfrom C 3 willleadtoaconsistentsystem,andthereforethereisnovectorthatwillleadtoan inconsistentsystem. Howdoweconvinceourselvesofthis?First,row-reduce E E RREF )443()223()222()443(! 2 4 1 001 0 1 01 00 1 1 3 5 Ifweaugment E withanyvectorofconstants,androw-reducetheaugmentedmatrix,wewillnevernd aleading1inthenalcolumn,sobyTheoremRCLS[53]thesystemwillalwaysbeconsistent. Saidanotherway,thecolumnspaceof E isallof C 3 C E = C 3 .SobyTheoremCSCS[237]any vectorofconstantswillcreateaconsistentsystemandnonewillcreateaninconsistentsystem. Version2.02

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SubsectionCRS.SOLSolutions256 M20 ContributedbyRobertBeezerStatement[251] The2 2matrix 11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 has C A = N A = 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 M21 ContributedbyRobertBeezerStatement[252] Beginwithamatrix A ofanysizethatdoesnothaveanyzerorows,butwhichwhenrow-reducedto B yieldsatleastonerowofzeros.Suchamatrixshouldbeeasytoconstructornd,likesayfromArchetype A[702]. C A willcontainsomevectorswhosenalslotentry m isnon-zero,however,everycolumnvector fromthematrix B willhaveazeroinslot m andsoeveryvectorin C B willalsocontainazerointhe nalslot.Thismeansthat C A 6 = C B ,sincewehavevectorsin C A thatcannotbeelementsof C B T40 ContributedbyRobertBeezerStatement[252] Choose x 2C AB .ThenbyTheoremCSCS[237]thereisavector w thatisasolutionto LS AB; x Denethevector y by y = B w .We'reset, A y = A B w Denitionof y = AB w TheoremMMA[202] = xw solutionto LS AB; x Thissaysthat LS A; x isaconsistentsystem,andbyTheoremCSCS[237],weseethat x 2C A and therefore C AB C A Foranexamplewhere C A 6C AB choose A tobeanynonzeromatrixandchoose B tobeazero matrix.Then C A 6 = f 0 g and C AB = C O = f 0 g T41 ContributedbyRobertBeezerStatement[252] FromthesolutiontoExerciseCRS.T40[252]weknowthat C AB C A .Sotoestablishthesetequality DenitionSE[684]weneedtoshowthat C A C AB Choose x 2C A .ByTheoremCSCS[237]thelinearsystem LS A; x isconsistent,solet y beone suchsolution.Because B isnonsingular,andlinearsystemusing B asacoecientmatrixwillhavea solutionTheoremNMUS[74].Let w betheuniquesolutiontothelinearsystem LS B; y .Allset,here wego, AB w = A B w TheoremMMA[202] = A yw solutionto LS B; y = xy solutionto LS A; x Thissaysthatthelinearsystem LS AB; x isconsistent,sobyTheoremCSCS[237], x 2C AB .So C A C AB T45 ContributedbyRobertBeezerStatement[252] First, 0 2N B trivially.Nowsupposethat x 2N B .Then AB x = A B x TheoremMMA[202] = A 0x 2N B = 0 TheoremMMZM[200] Version2.02

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SubsectionCRS.SOLSolutions257 Sincewehaveassumed AB isnonsingular,DenitionNM[71]impliesthat x = 0 Second, 0 2C B and 0 2N A trivially,andsothezerovectorisintheintersectionaswellDenition SI[685].Nowsupposethat y 2C B N A .Because y 2C B ,TheoremCSCS[237]saysthesystem LS B; y isconsistent.Let x 2 C n beonesolutiontothissystem.Then AB x = A B x TheoremMMA[202] = A yx solutionto LS B; y = 0y 2N A Sincewehaveassumed AB isnonsingular,DenitionNM[71]impliesthat x = 0 .Then y = B x = B 0 = 0 When AB isnonsingularand m = n weknowthattherstcondition, N B = f 0 g ,meansthat B isnonsingularTheoremNMTNS[74].Because B isnonsingularTheoremCSNM[242]impliesthat C B = C m .Inordertohavethesecondconditionfullled, C B N A = f 0 g ,wemustrealizethat N A = f 0 g .However,asecondapplicationofTheoremNMTNS[74]showsthat A mustbenonsingular. ThisreproducesTheoremNPNT[226]. Version2.02

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SectionFSFourSubsets258 SectionFS FourSubsets Therearefournaturalsubsetsassociatedwithamatrix.Wehavemetthreealready:thenullspace, thecolumnspaceandtherowspace.Inthissectionwewillintroduceafourth,theleftnullspace.The objectiveofthissectionistodescribeoneprocedurethatwillallowustondlinearlyindependentsets thatspaneachofthesefoursetsofcolumnvectors.Alongtheway,wewillmakeaconnectionwiththe inverseofamatrix,soTheoremFS[263]willtietogethermostallofthischapterandtheentirecourse sofar. SubsectionLNS LeftNullSpace DenitionLNS LeftNullSpace Suppose A isan m n matrix.Thenthe leftnullspace isdenedas L A = N )]TJ/F21 10.9091 Tf 5 -8.836 Td [(A t C m ThisdenitioncontainsNotationLNS. 4 Theleftnullspacewillnotfeatureprominentlyinthesequel,butwecanexplainitsnameandconnect ittorowoperations.Suppose y 2L A .ThenbyDenitionLNS[257], A t y = 0 .Wecanthenwrite 0 t = )]TJ/F21 10.9091 Tf 5 -8.837 Td [(A t y t DenitionLNS[257] = y t )]TJ/F21 10.9091 Tf 5 -8.837 Td [(A t t TheoremMMT[203] = y t A TheoremTT[187] Theproduct y t A canbeviewedasthecomponentsof y actingasthescalarsinalinearcombinationof the rows of A .Andtheresultisarowvector", 0 t thatistotallyzeros.Whenweapplyasequence ofrowoperationstoamatrix,eachrowoftheresultingmatrixissomelinearcombinationoftherows. Theseobservationstellusthatthevectorsintheleftnullspacearescalarsthatrecordasequenceofrow operationsthatresultinarowofzerosintherow-reducedversionofthematrix.Wewillseethisidea moreexplicitlyinthecourseofprovingTheoremFS[263]. ExampleLNS Leftnullspace Wewillndtheleftnullspaceof A = 2 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(31 )]TJ/F15 10.9091 Tf 8.485 0 Td [(211 151 9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(40 3 7 7 5 Wetranspose A androw-reduce, A t = 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(219 )]TJ/F15 10.9091 Tf 8.485 0 Td [(315 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 1110 3 5 RREF )443()223()222()443(! 2 4 1 002 0 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 00 1 1 3 5 Version2.02

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SubsectionFS.CRSComputingColumnSpaces259 ApplyingDenitionLNS[257]andTheoremBNS[139]wehave L A = N )]TJ/F21 10.9091 Tf 5 -8.836 Td [(A t = 8 > > < > > : 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 3 7 7 5 9 > > = > > ; + Ifyourow-reduce A youwilldiscoveronezerorowinthereducedrow-echelonform.Thiszerorowis createdbyasequenceofrowoperations,whichintotalamountstoalinearcombination,withscalars a 1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2, a 2 =3, a 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1and a 4 =1,ontherowsof A andwhichresultsinthezerovectorcheckthis!. Sothecomponentsofthevectordescribingtheleftnullspaceof A providearelationoflineardependence ontherowsof A SubsectionCRS ComputingColumnSpaces Wehavethreewaystobuildthecolumnspaceofamatrix.First,wecanusejustthedenition,Denition CSM[236],andexpressthecolumnspaceasaspanofthecolumnsofthematrix.Asecondapproachgives usthecolumnspaceasthespanof some ofthecolumnsofthematrix,butthissetislinearlyindependent TheoremBCS[239].Finally,wecantransposethematrix,row-reducethetranspose,kickoutzerorows, andtransposetheremainingrowsbackintocolumnvectors.TheoremCSRST[247]andTheoremBRS [245]tellusthattheresultingvectorsarelinearlyindependentandtheirspanisthecolumnspaceofthe originalmatrix. Wewillnowdemonstrateafourthmethodbywayofarathercomplicatedexample.Studythisexample carefully,butrealizethatitsmainpurposeistomotivateatheoremthatsimpliesmuchoftheapparent complexity.Sootherthananinstructiveexerciseortwo,theprocedureweareabouttodescribewillnot beausualapproachtocomputingacolumnspace. ExampleCSANS Columnspaceasnullspace Letsndthecolumnspaceofthematrix A belowwithanewapproach. A = 2 6 6 6 6 6 6 4 100387 )]TJ/F15 10.9091 Tf 8.485 0 Td [(16 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(61 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 02 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 30123 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1110 3 7 7 7 7 7 7 5 ByTheoremCSCS[237]weknowthatthecolumnvector b isinthecolumnspaceof A ifandonlyifthe linearsystem LS A; b isconsistent.Solet'strytosolvethissysteminfullgenerality,usingavectorof variablesforthevectorofconstants.Inotherwords,whichvectors b leadtoconsistentsystems?Begin byformingtheaugmentedmatrix[ A j b ]withageneralversionof b [ A j b ]= 2 6 6 6 6 6 6 4 100387 b 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(16 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 b 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(61 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 b 3 02 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 b 4 30123 b 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1110 b 6 3 7 7 7 7 7 7 5 Version2.02

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SubsectionFS.CRSComputingColumnSpaces260 Toidentifysolutionswewillrow-reducethismatrixandbringittoreducedrow-echelonform.Despitethe presenceofvariablesinthelastcolumn,thereisnothingtostopusfromdoingthis.Exceptournumerical routinesoncalculatorscan'tbeused,andevensomeofthesymbolicalgebraroutinesdosomeunexpected maneuverswiththiscomputation.Sodoitbyhand.Yes,itisabitofwork.Butworthit.We'llstill beherewhenyougetback.Noticealongthewaythattherowoperationsare exactly thesameonesyou woulddoifyouwerejustrow-reducingthecoecientmatrixalone,sayinconnectionwithahomogeneous systemofequations.Thecolumnwiththe b i actsasasortofbookkeepingdevice.Therearemanydierent possibilitiesfortheresult,dependingonwhatorderyouchoosetoperformtherowoperations,butshortly we'llallbeonthesamepage.Here'sonepossibilityyoucanndthissameresultbydoingadditionalrow operationswiththefthandsixthrowstoremoveanyoccurrencesof b 1 and b 2 fromtherstfourrowsof yourresult: 2 6 6 6 6 6 6 6 4 1 0002 b 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 4 +2 b 5 )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 6 0 1 00 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 b 3 +3 b 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 b 5 +3 b 6 00 1 01 b 3 + b 4 +3 b 5 +3 b 6 000 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 b 3 + b 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 b 5 00000 b 1 +3 b 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 4 +3 b 5 + b 6 00000 b 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b 3 + b 4 + b 5 )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 6 3 7 7 7 7 7 7 7 5 Ourgoalistoidentifythosevectors b whichmake LS A; b consistent.ByTheoremRCLS[53]weknow thattheconsistentsystemsarepreciselythosewithoutaleading1inthelastcolumn.Aretheexpressions inthelastcolumnofrows5and6equaltozero,oraretheyleading1's?Theansweris:maybe.Itdepends on b .Withanonzerovalueforeitheroftheseexpressions,wewouldscaletherowandproducealeading 1.Sowegetaconsistentsystem,and b isinthecolumnspace,ifandonlyifthesetwoexpressionsare bothsimultaneouslyzero.Inotherwords,membersofthecolumnspaceof A areexactlythosevectors b thatsatisfy b 1 +3 b 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 4 +3 b 5 + b 6 =0 b 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b 3 + b 4 + b 5 )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 6 =0 Hmmm.Lookssuspiciouslylikeahomogeneoussystemoftwoequationswithsixvariables.Ifyou've beenplayingalongandwehopeyouhavethenyoumayhaveaslightlydierentsystem,butyoushould havejusttwoequations.Formthecoecientmatrixandrow-reducenoticethatthesystemabovehasa coecientmatrixthatisalreadyinreducedrow-echelonform.Weshouldallbetogethernowwiththe samematrix, L = 1 03 )]TJ/F15 10.9091 Tf 8.485 0 Td [(131 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(211 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 So, C A = N L andwecanapplyTheoremBNS[139]toobtainalinearlyindependentsettouseina spanconstruction, C A = N L = 8 > > > > > > < > > > > > > : 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 2 1 0 0 0 3 7 7 7 7 7 7 5 ; 2 6 6 6 6 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 1 0 0 3 7 7 7 7 7 7 5 ; 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 0 1 0 3 7 7 7 7 7 7 5 ; 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 1 0 0 0 1 3 7 7 7 7 7 7 5 9 > > > > > > = > > > > > > ; + Whew!Asapostscripttothiscentralexample,youmaywishtoconvinceyourselfthatthefourvectors abovereallyareelementsofthecolumnspace?Dotheycreateconsistentsystemswith A ascoecient matrix?Canyourecognizetheconstantvectorinyourdescriptionofthesesolutionsets? OK,thatwassomuchfun,let'sdoitagain.Butsimplerthistime.Andwe'llallgetthesameresults allthewaythrough.Doingrowoperationsbyhandwithvariablescanbeabiterrorprone,solet'sseeif Version2.02

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SubsectionFS.CRSComputingColumnSpaces261 wecanimprovetheprocesssome.Ratherthanrow-reduceacolumnvector b fullofvariables,let'swrite b = I 6 b andwewillrow-reducethematrix I 6 andwhenwenishrow-reducing, then wewillcomputethe matrix-vectorproduct.Youshouldrstconvinceyourselfthatwecanoperatelikethisthisisthesubject ofafuturehomeworkexercise.Ratherthanaugmenting A with b ,wewillinsteadaugmentitwith I 6 doesthisfeelfamiliar?, M = 2 6 6 6 6 6 6 4 100387100000 )]TJ/F15 10.9091 Tf 8.485 0 Td [(16 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13010000 )]TJ/F15 10.9091 Tf 8.485 0 Td [(61 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6001000 02 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2000100 30123000010 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1110000001 3 7 7 7 7 7 7 5 Wewanttorow-reducetheleft-handsideofthismatrix,butwewillapplythesamerowoperationstothe right-handsideaswell.Andoncewegettheleft-handsideinreducedrow-echelonform,wewillcontinueon toputleading1'sinthenaltworows,aswellasclearingoutthecolumnscontainingthosetwoadditional leading1's.Itistheseadditionalrowoperationsthatwillensurethatweallgettothesameplace,since thereducedrow-echelonformisuniqueTheoremRREFU[32], N = 2 6 6 6 6 6 6 4 10002001 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0100 )]TJ/F15 10.9091 Tf 8.485 0 Td [(300 )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(33 00101001133 0001 )]TJ/F15 10.9091 Tf 8.485 0 Td [(200 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(40 00000103 )]TJ/F15 10.9091 Tf 8.485 0 Td [(131 0000001 )]TJ/F15 10.9091 Tf 8.485 0 Td [(211 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 7 7 7 7 5 Weareafterthenalsixcolumnsofthismatrix,whichwewillmultiplyby b J = 2 6 6 6 6 6 6 4 001 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 00 )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(33 001133 00 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(40 103 )]TJ/F15 10.9091 Tf 8.485 0 Td [(131 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(211 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 7 7 7 7 5 so J b = 2 6 6 6 6 6 6 4 001 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 00 )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(33 001133 00 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(40 103 )]TJ/F15 10.9091 Tf 8.485 0 Td [(131 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(211 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 7 7 7 7 5 2 6 6 6 6 6 6 4 b 1 b 2 b 3 b 4 b 5 b 6 3 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 4 b 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 4 +2 b 5 )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 b 3 +3 b 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 b 5 +3 b 6 b 3 + b 4 +3 b 5 +3 b 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 b 3 + b 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 b 5 b 1 +3 b 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 4 +3 b 5 + b 6 b 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b 3 + b 4 + b 5 )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 6 3 7 7 7 7 7 7 5 Sobyapplyingthesamerowoperationsthatrow-reduce A totheidentitymatrixwhichwecoulddowith acalculatoronce I 6 isplacedalongsideof A ,wecanthenarriveattheresultofrow-reducingacolumn ofsymbolswherethevectorofconstantsusuallyresides.Sincetherow-reducedversionof A hastwozero rows,foraconsistentsystemwerequirethat b 1 +3 b 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 4 +3 b 5 + b 6 =0 b 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b 3 + b 4 + b 5 )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 6 =0 Nowweareexactlybackwherewewereontherstgo-round.Noticethatweobtainthematrix L as simplythelasttworowsandlastsixcolumnsof N Thisexamplemotivatestheremainderofthissection,soitisworthcarefulstudy.Youmightattempt tomimicthesecondapproachwiththecoecientmatricesofArchetypeI[737]andArchetypeJ[741].We willseeshortlythatthematrix L containsmoreinformationabout A thanjustthecolumnspace. Version2.02

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SubsectionFS.EEFExtendedechelonform262 SubsectionEEF Extendedechelonform Thenalmatrixthatwerow-reducedinExampleCSANS[258]shouldlookfamiliarinmostrespectsto theprocedureweusedtocomputetheinverseofanonsingularmatrix,TheoremCINM[217].Wewill nowgeneralizethatproceduretomatricesthatarenotnecessarilynonsingular,orevensquare.Firsta denition. DenitionEEF ExtendedEchelonForm Suppose A isan m n matrix.Add m newcolumnsto A thattogetherequalan m m identitymatrix toforman m n + m matrix M .Userowoperationstobring M toreducedrow-echelonformandcall theresult N N isthe extendedreducedrow-echelonform of A ,andwewillstandardizeonnames forvesubmatrices B C J K L of N Let B denotethe m n matrixformedfromtherst n columnsof N andlet J denotethe m m matrixformedfromthelast m columnsof N .Supposethat B has r nonzerorows.Furtherpartition N by letting C denotethe r n matrixformedfromallofthenon-zerorowsof B .Let K bethe r m matrix formedfromtherst r rowsof J ,while L willbethe m )]TJ/F21 10.9091 Tf 10.882 0 Td [(r m matrixformedfromthebottom m )]TJ/F21 10.9091 Tf 10.882 0 Td [(r rowsof J .Pictorially, M =[ A j I m ] RREF )443()223()222()443(! N =[ B j J ]= C K 0 L 4 ExampleSEEF Submatricesofextendedechelonform WeillustrateDenitionEEF[261]withthematrix A A = 2 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2716 )]TJ/F15 10.9091 Tf 8.485 0 Td [(62 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(18 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(26 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1410217 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(129112 3 7 7 5 Augmentingwiththe4 4identitymatrix,M= 2 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(27161000 )]TJ/F15 10.9091 Tf 8.485 0 Td [(62 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(18 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(260100 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(14102170010 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1291120001 3 7 7 5 androw-reducing,weobtain N = 2 6 6 6 4 1 021030111 0 1 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(60 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10230 0000 1 20 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 000000 1 221 3 7 7 7 5 Sowethenobtain B = 2 6 6 4 1 02103 0 1 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(60 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0000 1 2 000000 3 7 7 5 Version2.02

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SubsectionFS.EEFExtendedechelonform263 C = 2 4 1 02103 0 1 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(60 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0000 1 2 3 5 J = 2 6 6 4 0111 0230 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 221 3 7 7 5 K = 2 4 0111 0230 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 5 L = 1 221 Youcanobserveorverifythepropertiesofthefollowingtheoremwiththisexample. TheoremPEEF PropertiesofExtendedEchelonForm Supposethat A isan m n matrixandthat N isitsextendedechelonform.Then 1. J isnonsingular. 2. B = JA 3.If x 2 C n and y 2 C m ,then A x = y ifandonlyif B x = J y 4. C isinreducedrow-echelonform,hasnozerorowsandhas r pivotcolumns. 5. L isinreducedrow-echelonform,hasnozerorowsandhas m )]TJ/F21 10.9091 Tf 10.909 0 Td [(r pivotcolumns. Proof J istheresultofapplyingasequenceofrowoperationsto I m ,assuch J and I m arerow-equivalent. LS I m ; 0 hasonlythezerosolution,since I m isnonsingularTheoremNMRRI[72].Thus, LS J; 0 also hasonlythezerosolutionTheoremREMES[28],DenitionESYS[11]and J isthereforenonsingular DenitionNSM[64]. Toprovethesecondpartofthisconclusion,rstconvinceyourselfthatrowoperationsandthematrixvectorarecommutativeoperations.Bythiswemeanthefollowing.Supposethat F isan m n matrixthat isrow-equivalenttothematrix G .Applytothecolumnvector F w thesamesequenceofrowoperations thatconverts F to G .Thentheresultis G w .Sowecandorowoperationsonthematrix,thendoa matrix-vectorproduct, or doamatrix-vectorproductandthendorowoperationsonacolumnvector,and theresultwillbethesameeitherway.SincematrixmultiplicationisdenedbyacollectionofmatrixvectorproductsDenitionMM[197],ifweapplytothematrixproduct FH thesamesequenceofrow operationsthatconverts F to G thentheresultwillequal GH .Nowapplytheseobservationsto A Write AI n = I m A andapplytherowoperationsthatconvert M to N A isconvertedto B ,while I m isconvertedto J ,sowehave BI n = JA .Simplifyingtheleftsidegivesthedesiredconclusion. Forthethirdconclusion,wenowestablishthetwoequivalences A x = y JA x = J y B x = J y Theforwarddirectionoftherstequivalenceisaccomplishedbymultiplyingbothsidesofthematrix equalityby J ,whilethebackwarddirectionisaccomplishedbymultiplyingbytheinverseof J whichwe knowexistsbyTheoremNI[228]since J isnonsingular.Thesecondequivalenceisobtainedsimplyby thesubstitutionsgivenby JA = B Version2.02

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SubsectionFS.FSFourSubsets264 Therst r rowsof N areinreducedrow-echelonform,sinceanycontiguouscollectionofrowstaken fromamatrixinreducedrow-echelonformwillformamatrixthatisagaininreducedrow-echelonform. Sincethematrix C isformedbyremovingthelast n entriesofeachtheserows,theremainderisstillin reducedrow-echelonform.Byitsconstruction, C hasnozerorows. C has r rowsandeachcontainsa leading1,sothereare r pivotcolumnsin C Thenal m )]TJ/F21 10.9091 Tf 11.417 0 Td [(r rowsof N areinreducedrow-echelonform,sinceanycontiguouscollectionofrows takenfromamatrixinreducedrow-echelonformwillformamatrixthatisagaininreducedrow-echelon form.Sincethematrix L isformedbyremovingtherst n entriesofeachtheserows,andtheseentries areallzerotheyformthezerorowsof B ,theremainderisstillinreducedrow-echelonform. L isthe nal m )]TJ/F21 10.9091 Tf 11.242 0 Td [(r rowsofthenonsingularmatrix J ,sononeoftheserowscanbetotallyzero,or J wouldnot row-reducetotheidentitymatrix. L has m )]TJ/F21 10.9091 Tf 11.311 0 Td [(r rowsandeachcontainsaleading1,sothereare m )]TJ/F21 10.9091 Tf 11.311 0 Td [(r pivotcolumnsin L Noticethatinthecasewhere A isanonsingularmatrixweknowthatthereducedrow-echelonform of A istheidentitymatrixTheoremNMRRI[72],so B = I n .Thenthesecondconclusionabovesays JA = B = I n ,so J istheinverseof A .ThusthistheoremgeneralizesTheoremCINM[217],thoughthe resultisaleft-inverse"of A ratherthanaright-inverse." ThethirdconclusionofTheoremPEEF[262]isthemosttelling.Itsaysthat x isasolutiontothelinear system LS A; y ifandonlyif x isasolutiontothelinearsystem LS B;J y .Orsaiddierently,ifwe row-reducetheaugmentedmatrix[ A j y ]wewillgettheaugmentedmatrix[ B j J y ].Thematrix J tracks thecumulativeeectoftherowoperationsthatconverts A toreducedrow-echelonform,hereeectively applyingthemtothevectorofconstantsinasystemofequationshaving A asacoecientmatrix.When A row-reducestoamatrixwithzerorows,then J y shouldalsohavezeroentriesinthesamerowsifthe systemistobeconsistent. SubsectionFS FourSubsets Withallthepreliminariesinplacewecanstateourmainresultforthissection.Inessencethisresultwill allowustosaythatwecanndlinearlyindependentsetstouseinspanconstructionsforallfoursubsets nullspace,columnspace,rowspace,leftnullspacebyanalyzingonlytheextendedechelonformofthe matrix,andspecically,justthetwosubmatrices C and L ,whichwillberipeforanalysissincetheyare alreadyinreducedrow-echelonformTheoremPEEF[262]. TheoremFS FourSubsets Suppose A isan m n matrixwithextendedechelonform N .Supposethereducedrow-echelonformof A has r nonzerorows.Then C isthesubmatrixof N formedfromtherst r rowsandtherst n columns and L isthesubmatrixof N formedfromthelast m columnsandthelast m )]TJ/F21 10.9091 Tf 10.909 0 Td [(r rows.Then 1.Thenullspaceof A isthenullspaceof C N A = N C 2.Therowspaceof A istherowspaceof C R A = R C 3.Thecolumnspaceof A isthenullspaceof L C A = N L 4.Theleftnullspaceof A istherowspaceof L L A = R L Proof First, N A = N B since B isrow-equivalentto A TheoremREMES[28].Thezerorowsof B representequationsthatarealwaystrueinthehomogeneoussystem LS B; 0 ,sotheremovalofthese equationswillnotchangethesolutionset.Thus,inturn, N B = N C Version2.02

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SubsectionFS.FSFourSubsets265 Second, R A = R B since B isrow-equivalentto A TheoremREMRS[244].Thezerorowsof B contributenothingtothespanthatistherowspaceof B ,sotheremovaloftheserowswillnotchangethe rowspace.Thus,inturn, R B = R C Third,weprovethesetequality C A = N L withDenitionSE[684].Beginbyshowingthat C A N L .Choose y 2C A C m .Thenthereexistsavector x 2 C n suchthat A x = y Theorem CSCS[237].Thenfor1 k m )]TJ/F21 10.9091 Tf 10.909 0 Td [(r [ L y ] k =[ J y ] r + k L asubmatrixof J =[ B x ] r + k TheoremPEEF[262] =[ O x ] k Zeromatrixasubmatrixof B =[ 0 ] k TheoremMMZM[200] So,forall1 k m )]TJ/F21 10.9091 Tf 10.909 0 Td [(r ,[ L y ] k =[ 0 ] k .SobyDenitionCVE[84]wehave L y = 0 andthus y 2N L Now,showthat N L C A .Choose y 2N L C m .Formthevector K y 2 C r .Thelinearsystem LS C;K y isconsistentsince C isinreducedrow-echelonformandhasnozerorowsTheoremPEEF [262].Let x 2 C n denoteasolutionto LS C;K y Thenfor1 j r [ B x ] j =[ C x ] j C asubmatrixof B =[ K y ] j x asolutionto LS C;K y =[ J y ] j K asubmatrixof J Andfor r +1 k m [ B x ] k =[ O x ] k )]TJ/F22 7.9701 Tf 6.587 0 Td [(r Zeromatrixasubmatrixof B =[ 0 ] k )]TJ/F22 7.9701 Tf 6.586 0 Td [(r TheoremMMZM[200] =[ L y ] k )]TJ/F22 7.9701 Tf 6.586 0 Td [(r y in N L =[ J y ] k L asubmatrixof J Soforall1 i m ,[ B x ] i =[ J y ] i andbyDenitionCVE[84]wehave B x = J y .FromTheoremPEEF [262]weknowthenthat A x = y ,andtherefore y 2C A TheoremCSCS[237].ByDenitionSE[684] wenowhave C A = N L Fourth,weprovethesetequality L A = R L withDenitionSE[684].Beginbyshowingthat R L L A .Choose y 2R L C m .Thenthereexistsavector w 2 C m )]TJ/F22 7.9701 Tf 6.587 0 Td [(r suchthat y = L t w DenitionRSM[243],TheoremCSCS[237].Thenfor1 i n A t y i = m X k =1 A t ik [ y ] k TheoremEMP[198] = m X k =1 A t ik L t w k Denitionof w = m X k =1 A t ik m )]TJ/F22 7.9701 Tf 6.587 0 Td [(r X ` =1 L t k` [ w ] ` TheoremEMP[198] = m X k =1 m )]TJ/F22 7.9701 Tf 6.587 0 Td [(r X ` =1 A t ik L t k` [ w ] ` PropertyDCN[681] Version2.02

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SubsectionFS.FSFourSubsets266 = m )]TJ/F22 7.9701 Tf 6.587 0 Td [(r X ` =1 m X k =1 A t ik L t k` [ w ] ` PropertyCACN[680] = m )]TJ/F22 7.9701 Tf 6.587 0 Td [(r X ` =1 m X k =1 A t ik L t k` [ w ] ` PropertyDCN[681] = m )]TJ/F22 7.9701 Tf 6.587 0 Td [(r X ` =1 m X k =1 A t ik J t k;r + ` [ w ] ` L asubmatrixof J = m )]TJ/F22 7.9701 Tf 6.587 0 Td [(r X ` =1 A t J t i;r + ` [ w ] ` TheoremEMP[198] = m )]TJ/F22 7.9701 Tf 6.586 0 Td [(r X ` =1 JA t i;r + ` [ w ] ` TheoremMMT[203] = m )]TJ/F22 7.9701 Tf 6.586 0 Td [(r X ` =1 B t i;r + ` [ w ] ` TheoremPEEF[262] = m )]TJ/F22 7.9701 Tf 6.586 0 Td [(r X ` =1 0[ w ] ` Zerorowsin B =0PropertyZCN[681] =[ 0 ] i DenitionZCV[25] Since A t y i =[ 0 ] i for1 i n ,DenitionCVE[84]impliesthat A t y = 0 .Thismeansthat y 2N )]TJ/F21 10.9091 Tf 5 -8.837 Td [(A t Now,showthat L A R L .Choose y 2L A C m .Thematrix J isnonsingularTheoremPEEF [262],so J t isalsononsingularTheoremMIT[220]andthereforethelinearsystem LS )]TJ/F21 10.9091 Tf 5 -8.836 Td [(J t ; y hasa uniquesolution.Denotethissolutionas x 2 C m .Wewillneedtoworkwithtwohalves"of x ,whichwe willdenoteas z and w withformaldenitionsgivenby [ z ] j =[ x ] i 1 j r; [ w ] k =[ x ] r + k 1 k m )]TJ/F21 10.9091 Tf 10.909 0 Td [(r Now,for1 j r C t z j = r X k =1 C t jk [ z ] k TheoremEMP[198] = r X k =1 C t jk [ z ] k + m )]TJ/F22 7.9701 Tf 6.586 0 Td [(r X ` =1 [ O ] j` [ w ] ` DenitionZM[185] = r X k =1 B t jk [ z ] k + m )]TJ/F22 7.9701 Tf 6.586 0 Td [(r X ` =1 B t j;r + ` [ w ] ` C O submatricesof B = r X k =1 B t jk [ x ] k + m )]TJ/F22 7.9701 Tf 6.586 0 Td [(r X ` =1 B t j;r + ` [ x ] r + ` Denitionsof z and w = r X k =1 B t jk [ x ] k + m X k = r +1 B t jk [ x ] k Re-indexsecondsum = m X k =1 B t jk [ x ] k Combinesums = m X k =1 JA t jk [ x ] k TheoremPEEF[262] Version2.02

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SubsectionFS.FSFourSubsets267 = m X k =1 A t J t jk [ x ] k TheoremMMT[203] = m X k =1 m X ` =1 A t j` J t `k [ x ] k TheoremEMP[198] = m X ` =1 m X k =1 A t j` J t `k [ x ] k PropertyCACN[680] = m X ` =1 A t j` m X k =1 J t `k [ x ] k PropertyDCN[681] = m X ` =1 A t j` J t x ` TheoremEMP[198] = m X ` =1 A t j` [ y ] ` Denitionof x = A t y j TheoremEMP[198] =[ 0 ] j y 2L A So,byDenitionCVE[84], C t z = 0 andthevector z givesusalinearcombinationofthecolumnsof C t thatequalsthezerovector.Inotherwords, z givesarelationoflineardependenceonthetherowsof C However,therowsof C arealinearlyindependentsetbyTheoremBRS[245].AccordingtoDenition LICV[132]wemustconcludethattheentriesof z areallzero,i.e. z = 0 Now,for1 i m ,wehave [ y ] i = J t x i Denitionof x = m X k =1 J t ik [ x ] k TheoremEMP[198] = r X k =1 J t ik [ x ] k + m X k = r +1 J t ik [ x ] k Breakapartsum = r X k =1 J t ik [ z ] k + m X k = r +1 J t ik [ w ] k )]TJ/F22 7.9701 Tf 6.587 0 Td [(r Denitionof z and w = r X k =1 J t ik 0+ m )]TJ/F22 7.9701 Tf 6.587 0 Td [(r X ` =1 J t i;r + ` [ w ] ` z = 0 ,re-index =0+ m )]TJ/F22 7.9701 Tf 6.586 0 Td [(r X ` =1 L t i;` [ w ] ` L asubmatrixof J = L t w i TheoremEMP[198] SobyDenitionCVE[84], y = L t w .Theexistenceof w impliesthat y 2R L ,andtherefore L A R L .SobyDenitionSE[684]wehave L A = R L Thersttwoconclusionsofthistheoremarenearlytrivial.Buttheysetupapatternofresultsfor C thatisreectedinthelattertwoconclusionsabout L .Intotal,theytellusthatwecancomputeallfour subsetsjustbyndingnullspacesandrowspaces.Thistheoremdoesnottellusexactlyhowtocompute thesesubsets,butinsteadsimplyexpressesthemasnullspacesandrowspacesofmatricesinreduced row-echelonformwithoutanyzerorows C and L .Alinearlyindependentsetthatspansthenullspace Version2.02

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SubsectionFS.FSFourSubsets268 ofamatrixinreducedrow-echelonformcanbefoundeasilywithTheoremBNS[139].Itisaneveneasier mattertondalinearlyindependentsetthatspanstherowspaceofamatrixinreducedrow-echelonform withTheoremBRS[245],especiallywhentherearenozerorowspresent.SoanapplicationofTheorem FS[263]istypicallyfollowedbytwoapplicationseachofTheoremBNS[139]andTheoremBRS[245]. Thesituationwhen r = m deservescomment,sincenowthematrix L hasnorows.Whatis C A when wetrytoapplyTheoremFS[263]andencounter N L ?Oneinterpretationofthissituationisthat L is thecoecientmatrixofahomogeneoussystemthathasnoequations.Howhardisittondasolution vectortothissystem?Somethoughtwillconvinceyouthat any proposedvectorwillqualifyasasolution, sinceitmakes all oftheequationstrue.Soeverypossiblevectorisinthenullspaceof L andtherefore C A = N L = C m .OK,perhapsthissoundslikesometwistedargumentfrom AliceinWonderland .Let ustryanotherargumentthatmightsolidlyconvinceyouofthislogic. If r = m ,whenwerow-reducetheaugmentedmatrixof LS A; b theresultwillhavenozerorows,and alltheleading1'swilloccurinrst n columns,sobyTheoremRCLS[53]thesystemwillbeconsistent. ByTheoremCSCS[237], b 2C A .Since b wasarbitrary,everypossiblevectorisinthecolumnspaceof A ,soweagainhave C A = C m .Thesituationwhenamatrixhas r = m isknownbytheterm fullrank andinthecaseofasquarematrixcoincideswithnonsingularityseeExerciseFS.M50[273]. Thepropertiesofthematrix L describedbythistheoremcanbeexplainedinformallyasfollows.A columnvector y 2 C m isinthecolumnspaceof A ifthelinearsystem LS A; y isconsistentTheorem CSCS[237].ByTheoremRCLS[53],thereducedrow-echelonformoftheaugmentedmatrix[ A j y ]ofa consistentsystemwillhavezerosinthebottom m )]TJ/F21 10.9091 Tf 11.235 0 Td [(r locationsofthelastcolumn.ByTheoremPEEF [262]thisnalcolumnisthevector J y andsoshouldthenhavezerosinthenal m )]TJ/F21 10.9091 Tf 11.268 0 Td [(r locations.But since L comprisesthenal m )]TJ/F21 10.9091 Tf 10.909 0 Td [(r rowsof J ,thisconditionisexpressedbysaying y 2N L Additionally,therowsof J arethescalarsinlinearcombinationsoftherowsof A thatcreatethe rowsof B .Thatis,therowsof J recordtheneteectofthesequenceofrowoperationsthattakes A to itsreducedrow-echelonform, B .Thiscanbeseenintheequation JA = B TheoremPEEF[262].As such,therowsof L arescalarsforlinearcombinationsoftherowsof A thatyieldzerorows.Butsuch linearcombinationsarepreciselytheelementsoftheleftnullspace.Soanyelementoftherowspaceof L isalsoanelementoftheleftnullspaceof A .WewillnowillustrateTheoremFS[263]withafewexamples. ExampleFS1 Foursubsets,#1 InExampleSEEF[261]wefoundtheverelevantsubmatricesofthematrix A = 2 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2716 )]TJ/F15 10.9091 Tf 8.485 0 Td [(62 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(18 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(26 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1410217 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(129112 3 7 7 5 ToapplyTheoremFS[263]weonlyneed C and L C = 2 4 1 02103 0 1 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(60 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0000 1 2 3 5 L = 1 221 ThenweuseTheoremFS[263]toobtain N A = N C = 8 > > > > > > < > > > > > > : 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 1 0 0 0 3 7 7 7 7 7 7 5 ; 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 6 0 1 0 0 3 7 7 7 7 7 7 5 ; 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 0 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 3 7 7 7 7 7 7 5 9 > > > > > > = > > > > > > ; + TheoremBNS[139] Version2.02

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SubsectionFS.FSFourSubsets269 R A = R C = 8 > > > > > > < > > > > > > : 2 6 6 6 6 6 6 4 1 0 2 1 0 3 3 7 7 7 7 7 7 5 ; 2 6 6 6 6 6 6 4 0 1 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 0 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 3 7 7 7 7 7 7 5 ; 2 6 6 6 6 6 6 4 0 0 0 0 1 2 3 7 7 7 7 7 7 5 9 > > > > > > = > > > > > > ; + TheoremBRS[245] C A = N L = 8 > > < > > : 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 0 0 3 7 7 5 ; 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 1 0 3 7 7 5 ; 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 0 1 3 7 7 5 9 > > = > > ; + TheoremBNS[139] L A = R L = 8 > > < > > : 2 6 6 4 1 2 2 1 3 7 7 5 9 > > = > > ; + TheoremBRS[245] Boom! ExampleFS2 Foursubsets,#2 Nowletsreturntothematrix A thatweusedtomotivatethissectioninExampleCSANS[258], A = 2 6 6 6 6 6 6 4 100387 )]TJ/F15 10.9091 Tf 8.485 0 Td [(16 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(61 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 02 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 30123 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1110 3 7 7 7 7 7 7 5 Weformthematrix M byadjoiningthe6 6identitymatrix I 6 M = 2 6 6 6 6 6 6 4 100387100000 )]TJ/F15 10.9091 Tf 8.485 0 Td [(16 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13010000 )]TJ/F15 10.9091 Tf 8.485 0 Td [(61 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6001000 02 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2000100 30123000010 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1110000001 3 7 7 7 7 7 7 5 androw-reducetoobtain N N = 2 6 6 6 6 6 6 6 4 1 0002001 )]TJ/F15 10.9091 Tf 8.484 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 1 00 )]TJ/F15 10.9091 Tf 8.485 0 Td [(300 )]TJ/F15 10.9091 Tf 8.484 0 Td [(23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(33 00 1 01001133 000 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(200 )]TJ/F15 10.9091 Tf 8.484 0 Td [(21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(40 00000 1 03 )]TJ/F15 10.9091 Tf 8.484 0 Td [(131 000000 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(211 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 7 7 7 7 7 5 Tondthefoursubsetsfor A ,weonlyneedidentifythe4 5matrix C andthe2 6matrix L C = 2 6 6 6 4 1 0002 0 1 00 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 00 1 01 000 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 3 7 7 7 5 L = 1 03 )]TJ/F15 10.9091 Tf 8.485 0 Td [(131 0 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(211 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 Version2.02

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SubsectionFS.FSFourSubsets270 ThenweapplyTheoremFS[263], N A = N C = 8 > > > > < > > > > : 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 1 3 7 7 7 7 5 9 > > > > = > > > > ; + TheoremBNS[139] R A = R C = 8 > > > > < > > > > : 2 6 6 6 6 4 1 0 0 0 2 3 7 7 7 7 5 ; 2 6 6 6 6 4 0 1 0 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 7 7 7 7 5 ; 2 6 6 6 6 4 0 0 1 0 1 3 7 7 7 7 5 ; 2 6 6 6 6 4 0 0 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 7 7 5 9 > > > > = > > > > ; + TheoremBRS[245] C A = N L = 8 > > > > > > < > > > > > > : 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 2 1 0 0 0 3 7 7 7 7 7 7 5 ; 2 6 6 6 6 6 6 4 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 0 1 0 0 3 7 7 7 7 7 7 5 ; 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 0 1 0 3 7 7 7 7 7 7 5 ; 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 0 0 0 1 3 7 7 7 7 7 7 5 9 > > > > > > = > > > > > > ; + TheoremBNS[139] L A = R L = 8 > > > > > > < > > > > > > : 2 6 6 6 6 6 6 4 1 0 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 1 3 7 7 7 7 7 7 5 ; 2 6 6 6 6 6 6 4 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 7 7 7 7 5 9 > > > > > > = > > > > > > ; + TheoremBRS[245] Thenextexampleisjustabitdierentsincethematrixhasmorerowsthancolumns,andatrivial nullspace. ExampleFSAG Foursubsets,ArchetypeG ArchetypeG[729]andArchetypeH[733]arebothsystemsof m =5equationsin n =2variables.They haveidenticalcoecientmatrices,whichwewilldenotehereasthematrix G G = 2 6 6 6 6 4 23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(14 310 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 69 3 7 7 7 7 5 Adjointhe5 5identitymatrix, I 5 ,toform M = 2 6 6 6 6 4 2310000 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1401000 31000100 3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(100010 6900001 3 7 7 7 7 5 Thisrow-reducesto N = 2 6 6 6 6 6 4 1 0000 3 11 1 33 0 1 000 )]TJ/F19 7.9701 Tf 11.797 4.296 Td [(2 11 1 11 00 1 000 )]TJ/F19 7.9701 Tf 9.68 4.295 Td [(1 3 000 1 01 )]TJ/F19 7.9701 Tf 9.68 4.295 Td [(1 3 0000 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 7 7 7 5 Version2.02

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SubsectionFS.FSFourSubsets271 Therst n =2columnscontain r =2leading1's,soweobtain C asthe2 2identitymatrixandextract L fromthenal m )]TJ/F21 10.9091 Tf 10.909 0 Td [(r =3rowsinthenal m =5columns. C = 1 0 0 1 L = 2 4 1 000 )]TJ/F19 7.9701 Tf 9.68 4.295 Td [(1 3 0 1 01 )]TJ/F19 7.9701 Tf 9.68 4.296 Td [(1 3 00 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 5 ThenweapplyTheoremFS[263], N G = N C = h;i = f 0 g TheoremBNS[139] R G = R C = 1 0 ; 0 1 = C 2 TheoremBRS[245] C G = N L = 8 > > > > < > > > > : 2 6 6 6 6 4 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 0 3 7 7 7 7 5 ; 2 6 6 6 6 4 1 3 1 3 1 0 1 3 7 7 7 7 5 9 > > > > = > > > > ; + TheoremBNS[139] = 8 > > > > < > > > > : 2 6 6 6 6 4 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 0 3 7 7 7 7 5 ; 2 6 6 6 6 4 1 1 3 0 3 3 7 7 7 7 5 9 > > > > = > > > > ; + L G = R L = 8 > > > > < > > > > : 2 6 6 6 6 4 1 0 0 0 )]TJ/F19 7.9701 Tf 9.68 4.296 Td [(1 3 3 7 7 7 7 5 ; 2 6 6 6 6 4 0 1 0 1 )]TJ/F19 7.9701 Tf 9.681 4.296 Td [(1 3 3 7 7 7 7 5 ; 2 6 6 6 6 4 0 0 1 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 3 7 7 7 7 5 9 > > > > = > > > > ; + TheoremBRS[245] = 8 > > > > < > > > > : 2 6 6 6 6 4 3 0 0 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 7 7 5 ; 2 6 6 6 6 4 0 3 0 3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 3 7 7 7 7 5 ; 2 6 6 6 6 4 0 0 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 7 7 5 9 > > > > = > > > > ; + Asmentionedearlier,ArchetypeG[729]isconsistent,whileArchetypeH[733]isinconsistent.Seeifyou canwritethetwodierentvectorsofconstantsfromthesetwoarchetypesaslinearcombinationsofthetwo vectorsin C G .Howaboutthetwocolumnsof G ,canyouwriteeachindividuallyasalinearcombination ofthetwovectorsin C G ?Theymustbeinthecolumnspaceof G also.Areyouranswersunique?Do younoticeanythingaboutthescalarsthatappearinthelinearcombinationsyouareforming? ExampleCOV[154]andExampleCSROI[247]eachdescribesthecolumnspaceofthecoecientmatrix fromArchetypeI[737]asthespanofasetof r =3linearlyindependentvectors.Itisnoaccidentthat thesetwodierentsetsbothhavethesamesize.Ifweyou?weretocalculatethecolumnspaceofthis matrixusingthenullspaceofthematrix L fromTheoremFS[263]thenwewouldagainndasetof3 linearlyindependentvectorsthatspantherange.Moreonthislater. Sowehavethreedierentmethodstoobtainadescriptionofthecolumnspaceofamatrixasthe spanofalinearlyindependentset.TheoremBCS[239]issometimesusefulsincethevectorsitspecies areequaltoactualcolumnsofthematrix.TheoremBRS[245]andTheoremCSRST[247]combineto createvectorswithlotsofzeros,andstrategicallyplaced1'snearthetopofthevector.TheoremFS[263] andthematrix L fromtheextendedechelonformgivesusathirdmethod,whichtendstocreatevectors withlotsofzeros,andstrategicallyplaced1'snearthebottomofthevector.Ifwedon'tcareaboutlinear Version2.02

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SubsectionFS.READReadingQuestions272 independencewecanalsoappealtoDenitionCSM[236]andsimplyexpressthecolumnspaceasthespan ofallthecolumnsofthematrix,givingusafourthdescription. WithTheoremCSRST[247]andDenitionRSM[243],wecancomputecolumnspaceswiththeorems aboutrowspaces,andwecancomputerowspaceswiththeoremsaboutrowspaces,butineachcase wemusttransposethematrixrst.Atthispointyoumaybeoverwhelmedbyallthepossibilitiesfor computingcolumnandrowspaces.DiagramCSRST[271]ismeanttohelp.Forboththecolumnspace androwspace,itsuggestsfourtechniques.Oneistoappealtothedenition,anotheryieldsaspanofa linearlyindependentset,andathirdusesTheoremFS[263].Afourthsuggeststransposingthematrix andthedashedlineimpliesthatthenthecompanionsetoftechniquescanbeapplied.Thiscanleadto abitofsilliness,sinceifyouweretofollowthedashedlines twice youwouldtransposethematrixtwice, andbyTheoremTT[187]wouldaccomplishnothingproductive. DiagramCSRST.ColumnSpaceandRowSpaceTechniques Althoughwehavemanywaystodescribeacolumnspace,noticethatonetemptingstrategywillusually fail.Itisnotpossibletosimplyrow-reduceamatrixdirectlyandthenusethecolumnsoftherow-reduced matrixasasetwhosespanequalsthecolumnspace.Inotherwords,rowoperations donot preservecolumn spaceshoweverrowoperationsdopreserverowspaces,TheoremREMRS[244].SeeExerciseCRS.M21 [252]. SubsectionREAD ReadingQuestions 1.Findanontrivialelementoftheleftnullspaceof A A = 2 4 21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(34 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(112 3 5 2.Findthematrices C and L intheextendedechelonformof A A = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(95 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(53 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 5 3.WhyisTheoremFS[263]agreatconclusiontoChapterM[182]? Version2.02

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SubsectionFS.EXCExercises273 SubsectionEXC Exercises C20 ExampleFSAG[269]concludeswithseveralquestions.Performtheanalysissuggestedbythese questions. ContributedbyRobertBeezer C25 Giventhematrix A below,usetheextendedechelonformof A toanswereachpartofthisproblem. Ineachpart,ndalinearlyindependentsetofvectors, S ,sothatthespanof S h S i ,equalsthespecied setofvectors. A = 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(53 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(111 )]TJ/F15 10.9091 Tf 8.485 0 Td [(85 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 3 7 7 5 aTherowspaceof A R A bThecolumnspaceof A C A cThenullspaceof A N A dTheleftnullspaceof A L A ContributedbyRobertBeezerSolution[274] C26 Forthematrix D belowusetheextendedechelonformtond aalinearlyindependentsetwhosespanisthecolumnspaceof D balinearlyindependentsetwhosespanistheleftnullspaceof D D = 2 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(7 )]TJ/F15 10.9091 Tf 8.484 0 Td [(11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(19 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 6101814 3597 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 7 7 5 ContributedbyRobertBeezerSolution[274] C41 Thefollowingarchetypesaresystemsofequations.Foreachsystem,writethevectorofconstants asalinearcombinationofthevectorsinthespanconstructionforthecolumnspaceprovidedbyTheorem FS[263]andTheoremBNS[139]thesevectorsarelistedforeachofthesearchetypes. ArchetypeA[702] ArchetypeB[707] ArchetypeC[712] ArchetypeD[716] ArchetypeE[720] ArchetypeF[724] ArchetypeG[729] ArchetypeH[733] ArchetypeI[737] ArchetypeJ[741] ContributedbyRobertBeezer C43 Thefollowingarchetypesareeithermatricesorsystemsofequationswithcoecientmatrices.For eachmatrix,computetheextendedechelonform N andidentifythematrices C and L .UsingTheorem Version2.02

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SubsectionFS.EXCExercises274 FS[263],TheoremBNS[139]andTheoremBRS[245]expressthenullspace,therowspace,thecolumn spaceandleftnullspaceofeachcoecientmatrixasaspanofalinearlyindependentset. ArchetypeA[702] ArchetypeB[707] ArchetypeC[712] ArchetypeD[716]/ArchetypeE[720] ArchetypeF[724] ArchetypeG[729]/ArchetypeH[733] ArchetypeI[737] ArchetypeJ[741] ArchetypeK[746] ArchetypeL[750] ContributedbyRobertBeezer C60 Forthematrix B below,ndsetsofvectorswhosespanequalsthecolumnspaceof B C B and whichindividuallymeetthefollowingextrarequirements. aThesetillustratesthedenitionofthecolumnspace. bThesetislinearlyindependentandthemembersofthesetarecolumnsof B cThesetislinearlyindependentwithanicepatternofzerosandones"atthe top ofeachvector. dThesetislinearlyindependentwithanicepatternofzerosandones"atthebottomofeachvector. B = 2 4 2311 1101 )]TJ/F15 10.9091 Tf 8.485 0 Td [(123 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 5 ContributedbyRobertBeezerSolution[275] C61 Let A bethematrixbelow,andndtheindicatedsetswiththerequestedproperties. A = 2 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(53 )]TJ/F15 10.9091 Tf 8.485 0 Td [(127 114 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 5 aAlinearlyindependentset S sothat C A = h S i and S iscomposedofcolumnsof A bAlinearlyindependentset S sothat C A = h S i andthevectorsin S haveanicepatternofzeros andonesatthetopofthevectors. cAlinearlyindependentset S sothat C A = h S i andthevectorsin S haveanicepatternofzeros andonesatthebottomofthevectors. dAlinearlyindependentset S sothat R A = h S i ContributedbyRobertBeezerSolution[276] M50 Supposethat A isanonsingularmatrix.ExtendthefourconclusionsofTheoremFS[263]inthis specialcaseanddiscussconnectionswithpreviousresultssuchasTheoremNME4[242]. ContributedbyRobertBeezer M51 Supposethat A isasingularmatrix.ExtendthefourconclusionsofTheoremFS[263]inthis specialcaseanddiscussconnectionswithpreviousresultssuchasTheoremNME4[242]. ContributedbyRobertBeezer Version2.02

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SubsectionFS.SOLSolutions275 SubsectionSOL Solutions C25 ContributedbyRobertBeezerStatement[272] Adda4 4identitymatrixtotherightof A toformthematrix M andthenrow-reducetothematrix N M = 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(53 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11000 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1110100 )]TJ/F15 10.9091 Tf 8.485 0 Td [(85 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10010 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(200001 3 7 7 5 RREF )443()223()222()443(! 2 6 6 6 4 1 0200 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 0 1 300 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 000 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0000 1 13 3 7 7 7 5 = N ToapplyTheoremFS[263]ineachofthesefourparts,weneedthetwomatrices, C = 1 02 0 1 3 L = 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 0 1 13 a R A = R C TheoremFS[263] = 2 4 1 0 2 3 5 ; 2 4 0 1 3 3 5 + TheoremBRS[245] b C A = N L TheoremFS[263] = 2 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 0 3 7 7 5 ; 2 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 1 3 7 7 5 + TheoremBNS[139] c N A = N C TheoremFS[263] = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 3 5 + TheoremBNS[139] d L A = R L TheoremFS[263] = 2 6 6 4 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 5 ; 2 6 6 4 0 1 1 3 3 7 7 5 + TheoremBRS[245] C26 ContributedbyRobertBeezerStatement[272] Forbothparts,weneedtheextendedechelonformofthematrix. 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F15 10.9091 Tf 8.484 0 Td [(11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(19 )]TJ/F15 10.9091 Tf 8.485 0 Td [(151000 61018140100 35970010 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(30001 3 7 7 5 RREF )443()222()223()443(! 2 6 6 6 4 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10025 0 1 3200 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0000 1 032 00000 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 3 7 7 7 5 Version2.02

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SubsectionFS.SOLSolutions276 Fromthismatrixweextractthelasttworows,inthelastfourcolumnstoformthematrix L L = 1 032 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 aByTheoremFS[263]andTheoremBNS[139]wehave C D = N L = 8 > > < > > : 2 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 2 1 0 3 7 7 5 ; 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 0 1 3 7 7 5 9 > > = > > ; + bByTheoremFS[263]andTheoremBRS[245]wehave L D = R L = 8 > > < > > : 2 6 6 4 1 0 3 2 3 7 7 5 ; 2 6 6 4 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 3 7 7 5 9 > > = > > ; + C60 ContributedbyRobertBeezerStatement[273] aThedenitionofthecolumnspaceisthespanofthesetofcolumnsDenitionCSM[236].Sothe desiredsetisjustthefourcolumnsof B S = 8 < : 2 4 2 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 3 5 ; 2 4 3 1 2 3 5 ; 2 4 1 0 3 3 5 ; 2 4 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 5 9 = ; bTheoremBCS[239]suggestsrow-reducingthematrixandusingthecolumnsof B thatcorrespondto thepivotcolumns. B RREF )443()223()222()443(! 2 4 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 0 1 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 0000 3 5 Sothepivotcolumnsarenumberedbyelementsof D = f 1 ; 2 g ,sotherequestedsetis S = 8 < : 2 4 2 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 5 ; 2 4 3 1 2 3 5 9 = ; cWecanndthissetbyrow-reducingthetransposeof B ,deletingthezerorows,andusingthenonzero rowsascolumnvectorsintheset.ThisisanapplicationofTheoremCSRST[247]followedbyTheorem BRS[245]. B t RREF )443()223()222()443(! 2 6 6 4 1 03 0 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(7 000 000 3 7 7 5 Sotherequestedsetis S = 8 < : 2 4 1 0 3 3 5 ; 2 4 0 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(7 3 5 9 = ; Version2.02

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SubsectionFS.SOLSolutions277 dWiththecolumnspaceexpressedasanullspace,thevectorsobtainedviaTheoremBNS[139]will beofthedesiredshape.SowerstproceedwithTheoremFS[263]andcreatetheextendedechelonform, [ B j I 3 ]= 2 4 2311100 1101010 )]TJ/F15 10.9091 Tf 8.485 0 Td [(123 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4001 3 5 RREF )443()223()222()443(! 2 4 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(120 2 3 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 3 0 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 1 3 1 3 0000 1 )]TJ/F19 7.9701 Tf 6.587 0 Td [(7 3 )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 3 3 5 So,employingTheoremFS[263],wehave C B = N L ,where L = 1 )]TJ/F19 7.9701 Tf 6.587 0 Td [(7 3 )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 3 WecanndthedesiredsetofvectorsfromTheoremBNS[139]as S = 8 < : 2 4 7 3 1 0 3 5 ; 2 4 1 3 0 1 3 5 9 = ; C61 ContributedbyRobertBeezerStatement[273] aFirstndamatrix B thatisrow-equivalentto A andinreducedrow-echelonform B = 2 4 1 03 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 0 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0000 3 5 ByTheoremBCS[239]wecanchoosethecolumnsof A thatcorrespondtodependentvariables D = f 1 ; 2 g astheelementsof S andobtainthedesiredproperties.So S = 8 < : 2 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 1 3 5 ; 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 1 3 5 9 = ; bWecanwritethecolumnspaceof A astherowspaceofthetransposeTheoremCSRST[247].So werow-reducethetransposeof A toobtaintherow-equivalentmatrix C inreducedrow-echelonform C = 2 6 6 4 108 013 000 000 3 7 7 5 Thenonzerorowswrittenascolumnswillbealinearlyindependentsetthatspanstherowspaceof A t byTheoremBRS[245],andthezerosandoneswillbeatthetopofthevectors, S = 8 < : 2 4 1 0 8 3 5 ; 2 4 0 1 3 3 5 9 = ; cInpreparationforTheoremFS[263],augment A withthe3 3identitymatrix I 3 androw-reduceto obtaintheextendedechelonform, 2 4 103 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 )]TJ/F19 7.9701 Tf 9.68 4.296 Td [(1 8 3 8 011 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 1 8 5 8 00001 3 8 )]TJ/F19 7.9701 Tf 9.68 4.295 Td [(1 8 3 5 Thensincetherstfourcolumnsofrow3areallzeros,weextract L = 1 3 8 )]TJ/F19 7.9701 Tf 9.68 4.295 Td [(1 8 Version2.02

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SubsectionFS.SOLSolutions278 TheoremFS[263]saysthat C A = N L .WecanthenuseTheoremBNS[139]toconstructthedesired set S ,basedonthefreevariableswithindicesin F = f 2 ; 3 g forthehomogeneoussystem LS L; 0 ,so S = 8 < : 2 4 )]TJ/F19 7.9701 Tf 9.68 4.295 Td [(3 8 1 0 3 5 ; 2 4 1 8 0 1 3 5 9 = ; Noticethatthezerosandonesareatthebottomofthevectors. dThisisastraightforwardapplicationofTheoremBRS[245].Usetherow-reducedmatrix B from parta,grabthenonzerorows,andwritethemascolumnvectors, S = 8 > > < > > : 2 6 6 4 1 0 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 5 ; 2 6 6 4 0 1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 5 9 > > = > > ; Version2.02

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AnnotatedAcronymsFS.MMatrices279 AnnotatedAcronymsM Matrices TheoremVSPM[184] Thesearethefundamentalrulesforworkingwiththeaddition,andscalarmultiplication,ofmatrices.We sawsomethingverysimilarinthepreviouschapterTheoremVSPCV[86].Together,thesetwodenitions willprovideourdenitionforthekeydenition,DenitionVS[279]. TheoremSLEMM[195] TheoremSLSLC[93]connectedlinearcombinationswithsystemsofequations.TheoremSLEMM[195] connectsthematrix-vectorproductDenitionMVP[194]andcolumnvectorequalityDenitionCVE [84]withsystemsofequations.We'llseethisoneregularly. TheoremEMP[198] ThistheoremisaworkhorseinSectionMM[194]andwillcontinuetomakeregularappearances.Ifyou wanttogetbetteratformulatingproofs,theapplicationofthistheoremcanbeakeystepingainingthat broaderunderstanding.WhileitmightbehardtoimagineTheoremEMP[198]asa denition ofmatrix multiplication,we'llseeinExerciseMR.T80[564]thatintheoryitisactuallya better denitionofmatrix multiplicationlong-term. TheoremCINM[217] Theinverseofamatrixiskey.Here'showyoucangetoneifyouknowhowtorow-reduce. TheoremNI[228] Nonsingularity"orinvertibility"?Pickyourfavorite,orshowyourversatilitybyusingoneortheother intherightcontext.Theymeanthesamething. TheoremCSCS[237] Givenacoecientmatrix,whichvectorsofconstantscreateconsistentsystems.Thistheoremtellsusthat theanswerisexactlythosecolumnvectorsinthecolumnspace.Conversely,wealsousethisteoremtotest formembershipinthecolumnspacebycheckingtheconsistencyoftheappropriatesystemofequations. TheoremBCS[239] Anothertheoremthatprovidesalinearlyindependentsetofvectorswhosespanequalssomesetofinterest acolumnspacethistime. TheoremBRS[245] Yetanothertheoremthatprovidesalinearlyindependentsetofvectorswhosespanequalssomesetof interestarowspace. TheoremCSRST[247] Columnspaces,rowspaces,transposes,rows,columns.Manyoftheconnectionsbetweentheseobjects arebasedonthesimpleobservationcapturedinthistheorem.Thisisnotadeepresult.Westateitasa theoremforconvenience,sowecanrefertoitasneeded. TheoremFS[263] Thistheoremisinherentlyinteresting,ifnotcomputationallysatisfying.Nullspace,rowspace,column space,leftnullspace|heretheyallare,simplybyrowreducingtheextendedmatrixandapplying TheoremBNS[139]andTheoremBCS[239]twiceeach.Nice. Version2.02

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ChapterVS VectorSpaces Wenowhaveacomputationaltoolkitinplaceandsowecanbeginourstudyoflinearalgebrainamore theoreticalstyle. Linearalgebraisthestudyoftwofundamentalobjects,vectorspacesandlineartransformationssee ChapterLT[452].Thischapterwillfocusontheformer.Thepowerofmathematicsisoftenderivedfrom generalizingmanydierentsituationsintooneabstractformulation,andthatisexactlywhatwewillbe doingthroughoutthischapter. SectionVS VectorSpaces Inthissectionwepresentaformaldenitionofavectorspace,whichwillleadtoanextraincrementof abstraction.Oncedened,westudyitsmostbasicproperties. SubsectionVS VectorSpaces Hereisoneofthetwomostimportantdenitionsintheentirecourse. DenitionVS VectorSpace Supposethat V isasetuponwhichwehavedenedtwooperations: vectoraddition ,whichcombines twoelementsof V andisdenotedby+",and scalarmultiplication ,whichcombinesacomplex numberwithanelementof V andisdenotedbyjuxtaposition.Then V ,alongwiththetwooperations,is a vectorspace ifthefollowingtenpropertieshold. ACAdditiveClosure If u ; v 2 V ,then u + v 2 V SCScalarClosure If 2 C and u 2 V ,then u 2 V CCommutativity If u ; v 2 V ,then u + v = v + u AAAdditiveAssociativity If u ; v ; w 2 V ,then u + v + w = u + v + w 280

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SubsectionVS.EVSExamplesofVectorSpaces281 ZZeroVector Thereisavector, 0 ,calledthe zerovector ,suchthat u + 0 = u forall u 2 V AIAdditiveInverses If u 2 V ,thenthereexistsavector )]TJ/F36 10.9091 Tf 8.484 0 Td [(u 2 V sothat u + )]TJ/F36 10.9091 Tf 8.485 0 Td [(u = 0 SMAScalarMultiplicationAssociativity If ; 2 C and u 2 V ,then u = u DVADistributivityacrossVectorAddition If 2 C and u ; v 2 V ,then u + v = u + v DSADistributivityacrossScalarAddition If ; 2 C and u 2 V ,then + u = u + u OOne If u 2 V ,then1 u = u Theobjectsin V arecalled vectors ,nomatterwhatelsetheymightreallybe,simplybyvirtueofbeing elementsofavectorspace. 4 Now,thereareseveralimportantobservationstomake.Manyofthesewillbeeasiertounderstand onasecondorthirdreading,andespeciallyaftercarefullystudyingtheexamplesinSubsectionVS.EVS [280]. An axiom isoftenaself-evident"truth.Somethingsofundamentalthatweallagreeitistrueand acceptitwithoutproof.Typically,itwouldbethelogicalunderpinningthatwewouldbegintobuild theoremsupon.SomemightrefertothetenpropertiesofDenitionVS[279]asaxioms,implyingthat avectorspaceisaverynaturalobjectandthetenpropertiesaretheessenceofavectorspace.Wewill insteademphasizethatwewillbeginwithadenitionofavectorspace.Afterstudyingtheremainderof thischapter,youmightreturnhereandremindyourselfhowallourforthcomingtheoremsanddenitions restonthisfoundation. Aswewillseeshortly,theobjectsin V canbe anything ,eventhoughwewillcallthemvectors.We havebeenworkingwithvectorsfrequently,butweshouldstressherethatthesehavesofarjustbeen column vectors|scalarsarrangedinacolumnarlistofxedlength.Inasimilarvein,youhaveusedthe symbol+"formanyyearstorepresenttheadditionofnumbersscalars.Wehaveextendeditsusetothe additionofcolumnvectorsandtotheadditionofmatrices,andnowwearegoingtorecycleitevenfurther andletitdenotevectoradditionin any possiblevectorspace.Sowhendescribinganewvectorspace,we willhaveto dene exactlywhat+"is.Similarcommentsapplytoscalarmultiplication.Conversely,we can dene ouroperationsanywaywelike,solongasthetenpropertiesarefullledseeExampleCVS [283]. Avectorspaceiscomposedofthreeobjects,asetandtwooperations.However,weusuallyusethe samesymbolforboththesetandthevectorspaceitself.Donotletthisconveniencefoolyouintothinking theoperationsaresecondary! Thisdiscussionhaseitherconvincedyouthatwearereallyembarkingonanewlevelofabstraction, ortheyhaveseemedcryptic,mysteriousornonsensical.Youmightwanttoreturntothissectioninafew daysandgiveitanotherreadthen.Inanycase,let'slookatsomeconcreteexamplesnow. SubsectionEVS ExamplesofVectorSpaces Ouraiminthissubsectionistogiveyouastorehouseofexamplestoworkwith,tobecomecomfortable withthetenvectorspacepropertiesandtoconvinceyouthatthemultitudeofexamplesjustiesatleast Version2.02

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SubsectionVS.EVSExamplesofVectorSpaces282 initiallymakingsuchabroaddenitionasDenitionVS[279].Someofourclaimswillbejustiedby referencetoprevioustheorems,wewillprovesomefactsfromscratch,andwewilldoonenon-trivial examplecompletely.Inotherplaces,ourusualthoroughnesswillbeneglected,sograbpaperandpencil andplayalong. ExampleVSCV Thevectorspace C m Set: C m ,allcolumnvectorsofsize m ,DenitionVSCV[83]. Equality:Entry-wise,DenitionCVE[84]. VectorAddition:Theusual"addition,giveninDenitionCVA[84]. ScalarMultiplication:Theusual"scalarmultiplication,giveninDenitionCVSM[85]. Doesthissetwiththeseoperationsfulllthetenproperties?Yes.Andbydesignallweneedtodois quoteTheoremVSPCV[86].Thatwaseasy. ExampleVSM Thevectorspaceofmatrices, M mn Set: M mn ,thesetofallmatricesofsize m n andentriesfrom C ,ExampleVSM[281]. Equality:Entry-wise,DenitionME[182]. VectorAddition:Theusual"addition,giveninDenitionMA[182]. ScalarMultiplication:Theusual"scalarmultiplication,giveninDenitionMSM[183]. Doesthissetwiththeseoperationsfulllthetenproperties?Yes.Andallweneedtodoisquote TheoremVSPM[184].Anothereasyonebydesign. So,thesetofallmatricesofaxedsizeformsavectorspace.Thatentitlesustocallamatrixavector, sinceamatrixisanelementofavectorspace.Forexample,if A;B 2 M 3 ; 4 thenwecall A and B vectors," andweevenuseourpreviousnotationforcolumnvectorstoreferto A and B .Sowecouldlegitimately writeexpressionslike u + v = A + B = B + A = v + u Thiscouldleadtosomeconfusion,butitisnottoogreatadanger.Butitisworthcomment. Theprevioustwoexamplesmaybelessthansatisfying.Wemadealltherelevantdenitionslong ago.Andtherequiredvericationswereallhandledbyquotingoldtheorems.However,itisimportantto considerthesetwoexamplesrst.WehavebeenstudyingvectorsandmatricescarefullyChapterV[83], ChapterM[182],andbothobjects,alongwiththeiroperations,havecertainpropertiesincommon,as youmayhavenoticedincomparingTheoremVSPCV[86]withTheoremVSPM[184].Indeed,itisthese twotheoremsthat motivate ustoformulatetheabstractdenitionofavectorspace,DenitionVS[279]. Now,shouldweprovesomegeneraltheoremsaboutvectorspacesaswewillshortlyinSubsectionVS.VSP [285],wecaninstantlyapplytheconclusionsto both C m and M mn .Noticetoohowwehavetakensix denitionsandtwotheoremsandreducedthemdowntotwo examples .Withgreatergeneralizationand abstractionouroldideasgetdowngradedinstature. Letuslookatsomemoreexamples,nowconsideringsomenewvectorspaces. ExampleVSP Thevectorspaceofpolynomials, P n Set: P n ,thesetofallpolynomialsofdegree n orlessinthevariable x withcoecientsfrom C Equality: a 0 + a 1 x + a 2 x 2 + + a n x n = b 0 + b 1 x + b 2 x 2 + + b n x n ifandonlyif a i = b i for0 i n VectorAddition: a 0 + a 1 x + a 2 x 2 + + a n x n + b 0 + b 1 x + b 2 x 2 + + b n x n = a 0 + b 0 + a 1 + b 1 x + a 2 + b 2 x 2 + + a n + b n x n Version2.02

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SubsectionVS.EVSExamplesofVectorSpaces283 ScalarMultiplication: a 0 + a 1 x + a 2 x 2 + + a n x n = a 0 + a 1 x + a 2 x 2 + + a n x n Thisset,withtheseoperations,willfulllthetenproperties,thoughwewillnotworkallthedetails here.However,wewillmakeafewcommentsandproveoneoftheproperties.First,thezerovector PropertyZ[280]iswhatyoumightexpect,andyoucancheckthatithastherequiredproperty. 0 =0+0 x +0 x 2 + +0 x n TheadditiveinversePropertyAI[280]isalsonosurprise,thoughconsiderhowwehavechosentowrite it. )]TJ/F27 10.9091 Tf 10.303 8.836 Td [()]TJ/F21 10.9091 Tf 5 -8.836 Td [(a 0 + a 1 x + a 2 x 2 + + a n x n = )]TJ/F21 10.9091 Tf 8.485 0 Td [(a 0 + )]TJ/F21 10.9091 Tf 8.485 0 Td [(a 1 x + )]TJ/F21 10.9091 Tf 8.485 0 Td [(a 2 x 2 + + )]TJ/F21 10.9091 Tf 8.485 0 Td [(a n x n Nowlet'sprovetheassociativityofvectoradditionPropertyAA[279].Thisisabittedious,though necessary.Throughout,theplussign+"doestriple-duty.Youmightaskyourselfwhateachplussign representsasyouworkthroughthisproof. u + v + w = a 0 + a 1 x + + a n x n + b 0 + b 1 x + + b n x n + c 0 + c 1 x + + c n x n = a 0 + a 1 x + + a n x n + b 0 + c 0 + b 1 + c 1 x + + b n + c n x n = a 0 + b 0 + c 0 + a 1 + b 1 + c 1 x + + a n + b n + c n x n = a 0 + b 0 + c 0 + a 1 + b 1 + c 1 x + + a n + b n + c n x n = a 0 + b 0 + a 1 + b 1 x + + a n + b n x n + c 0 + c 1 x + + c n x n = a 0 + a 1 x + + a n x n + b 0 + b 1 x + + b n x n + c 0 + c 1 x + + c n x n = u + v + w Noticehowitistheapplicationoftheassociativityoftheoldadditionofcomplexnumbersinthemiddle ofthischainofequalitiesthatmakesthewholeproofhappen.Theremainderissuccessiveapplicationsof ournewdenitionofvectorpolynomialaddition.Provingtheremainderofthetenpropertiesissimilar instyleandtedium.YoumighttryprovingthecommutativityofvectoradditionPropertyC[279],or oneofthedistributivitypropertiesPropertyDVA[280],PropertyDSA[280]. ExampleVSIS Thevectorspaceofinnitesequences Set: C 1 = f c 0 ;c 1 ;c 2 ;c 3 ;::: j c i 2 C ;i 2 N g Equality: c 0 ;c 1 ;c 2 ;::: = d 0 ;d 1 ;d 2 ;::: ifandonlyif c i = d i forall i 0 VectorAddition: c 0 ;c 1 ;c 2 ;::: + d 0 ;d 1 ;d 2 ;::: = c 0 + d 0 ;c 1 + d 1 ;c 2 + d 2 ;::: ScalarMultiplication: c 0 ;c 1 ;c 2 ;c 3 ;::: = c 0 ;c 1 ;c 2 ;c 3 ;::: Thisshouldremindyouofthevectorspace C m ,thoughnowourlistsofscalarsarewrittenhorizontally withcommasasdelimitersandtheyareallowedtobeinniteinlength.Whatdoesthezerovectorlook likePropertyZ[280]?AdditiveinversesPropertyAI[280]?Canyouprovetheassociativityofvector additionPropertyAA[279]? ExampleVSF Thevectorspaceoffunctions Set: F = f f j f : C C g Version2.02

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SubsectionVS.EVSExamplesofVectorSpaces284 Equality: f = g ifandonlyif f x = g x forall x 2 C VectorAddition: f + g isthefunctionwithoutputsdenedby f + g x = f x + g x ScalarMultiplication: f isthefunctionwithoutputsdenedby f x = f x Sothisisthesetofallfunctionsofonevariablethattakeacomplexnumbertoacomplexnumber. Youmighthavestudiedfunctionsofonevariablethattakearealnumbertoarealnumber,andthatmight beamorenaturalsettostudy.Butsinceweareallowingourscalarstobecomplexnumbers,weneedto expandthedomainandrangeofourfunctionsalso.Studycarefullyhowthedenitionsoftheoperation aremade,andthinkaboutthedierentusesof+"andjuxtaposition.Asanexampleofwhatisrequired whenverifyingthatthisisavectorspace,considerthatthezerovectorPropertyZ[280]isthefunction z whosedenitionis z x =0foreveryinput x Whilevectorspacesoffunctionsareveryimportantinmathematicsandphysics,wewillnotdevote themmuchmoreattention. Here'sauniqueexample. ExampleVSS Thesingletonvectorspace Set: Z = f z g Equality:Huh? VectorAddition: z + z = z ScalarMultiplication: z = z Thisshouldlookprettywild.First,justwhatis z ?Columnvector,matrix,polynomial,sequence, function?Mineral,plant,oranimal?Wearen'tsaying! z just is .Andwehavedenitionsofvector additionandscalarmultiplicationthataresucientforanoccurrenceofeitherthatmaycomealong. Ouronlyconcernisifthisset,alongwiththedenitionsoftwooperations,fulllsthetenpropertiesof DenitionVS[279].Let'scheckassociativityofvectoradditionPropertyAA[279].Forall u ; v ; w 2 Z u + v + w = z + z + z = z + z = z + z + z = u + v + w WhatisthezerovectorinthisvectorspacePropertyZ[280]?Withonlyoneelementintheset,wedo nothavemuchchoice.Is z = 0 ?Itappearsthat z behaveslikethezerovectorshould,soitgetsthetitle. Maybenowthedenitionofthisvectorspacedoesnotseemsobizarre.Itisasetwhoseonlyelementis theelementthatbehaveslikethezerovector,sothatloneelement is thezerovector. Perhapssomeoftheabovedenitionsandvericationsseemobviousorlikesplittinghairs,butthe nextexampleshouldconvinceyouthatthey are necessary.Wewillstudythisonecarefully.Ready?Check yourpreconceptionsatthedoor. ExampleCVS Thecrazyvectorspace Set: C = f x 1 ;x 2 j x 1 ;x 2 2 C g VectorAddition: x 1 ;x 2 + y 1 ;y 2 = x 1 + y 1 +1 ;x 2 + y 2 +1. ScalarMultiplication: x 1 ;x 2 = x 1 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ;x 2 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1. Now,therstthingIhearyousayisYoucan'tdothat!"Andmyresponseis,Ohyes,Ican!"Iam freetodenemysetandmyoperationsanywayIplease.Theymaynotlooknatural,orevenuseful,but wewillnowverifythattheyprovideuswithanotherexampleofavectorspace.Andthatisenough.If youareadventurous,youmighttryrstcheckingsomeofthepropertiesyourself.Whatisthezerovector? Additiveinverses?Canyouproveassociativity?Ready,herewego. Version2.02

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SubsectionVS.EVSExamplesofVectorSpaces285 PropertyAC[279],PropertySC[279]:Theresultofeachoperationisapairofcomplexnumbers,so thesetwoclosurepropertiesarefullled. PropertyC[279]: u + v = x 1 ;x 2 + y 1 ;y 2 = x 1 + y 1 +1 ;x 2 + y 2 +1 = y 1 + x 1 +1 ;y 2 + x 2 +1= y 1 ;y 2 + x 1 ;x 2 = v + u PropertyAA[279]: u + v + w = x 1 ;x 2 + y 1 ;y 2 + z 1 ;z 2 = x 1 ;x 2 + y 1 + z 1 +1 ;y 2 + z 2 +1 = x 1 + y 1 + z 1 +1+1 ;x 2 + y 2 + z 2 +1+1 = x 1 + y 1 + z 1 +2 ;x 2 + y 2 + z 2 +2 = x 1 + y 1 +1+ z 1 +1 ; x 2 + y 2 +1+ z 2 +1 = x 1 + y 1 +1 ;x 2 + y 2 +1+ z 1 ;z 2 = x 1 ;x 2 + y 1 ;y 2 + z 1 ;z 2 = u + v + w PropertyZ[280]:Thezerovectoris... 0 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.NowIhearyousay,No,no,thatcan'tbe,itmust be ; 0!"Indulgemeforamomentandletuscheckmyproposal. u + 0 = x 1 ;x 2 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1= x 1 + )]TJ/F15 10.9091 Tf 8.484 0 Td [(1+1 ;x 2 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1+1= x 1 ;x 2 = u Feelingbetter?Orworse? PropertyAI[280]:Foreachvector, u ,wemustlocateanadditiveinverse, )]TJ/F36 10.9091 Tf 8.485 0 Td [(u .Hereitis, )]TJ/F15 10.9091 Tf 8.485 0 Td [( x 1 ;x 2 = )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 ; )]TJ/F21 10.9091 Tf 8.484 0 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2.Asoddasitmaylook,Ihopeyouarewithholdingjudgment.Check: u + )]TJ/F36 10.9091 Tf 8.485 0 Td [(u = x 1 ;x 2 + )]TJ/F21 10.9091 Tf 8.484 0 Td [(x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 ; )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2= x 1 + )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2+1 ; )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 2 + x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2+1= )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1= 0 PropertySMA[280]: u = x 1 ;x 2 = x 1 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ;x 2 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = x 1 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+ )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ; x 2 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+ )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = x 1 + )]TJ/F21 10.9091 Tf 10.909 0 Td [( + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ; x 2 + )]TJ/F21 10.9091 Tf 10.909 0 Td [( + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = x 1 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ;x 2 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = x 1 ;x 2 = u PropertyDVA[280]:Ifyouhavehungonsofar,here'swhereitgetsevenwilder.Inthenexttwoproperties wemixandmashthetwooperations. u + v = x 1 ;x 2 + y 1 ;y 2 = x 1 + y 1 +1 ;x 2 + y 2 +1 = x 1 + y 1 +1+ )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ; x 2 + y 2 +1+ )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = x 1 + y 1 + + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ;x 2 + y 2 + + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = x 1 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+ y 1 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+1 ;x 2 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+ y 2 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+1 = x 1 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+ y 1 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+1 ; x 2 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+ y 2 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+1 Version2.02

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SubsectionVS.VSPVectorSpaceProperties286 = x 1 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ;x 2 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+ y 1 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ;y 2 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = x 1 ;x 2 + y 1 ;y 2 = u + v PropertyDSA[280]: + u = + x 1 ;x 2 = + x 1 + + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ; + x 2 + + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = x 1 + x 1 + + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ;x 2 + x 2 + + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = x 1 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+ x 1 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+1 ;x 2 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+ x 2 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+1 = x 1 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+ x 1 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+1 ; x 2 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+ x 2 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+1 = x 1 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ;x 2 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+ x 1 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ;x 2 + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = x 1 ;x 2 + x 1 ;x 2 = u + u PropertyO[280]:Afterallthat,thisoneiseasy,butnolesspleasing. 1 u =1 x 1 ;x 2 = x 1 +1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ;x 2 +1 )]TJ/F15 10.9091 Tf 10.91 0 Td [(1= x 1 ;x 2 = u That'sit, C isavectorspace,ascrazyasthatmayseem. Noticethatinthecaseofthezerovectorandadditiveinverses,weonlyhadtoproposepossibilitiesand thenverifythattheywerethecorrectchoices.Youmighttrytodiscoverhowyouwouldarriveatthese choices,thoughyoushouldunderstandwhytheprocessofdiscoveringthemisnotanecessarycomponent oftheproofitself. SubsectionVSP VectorSpaceProperties SubsectionVS.EVS[280]hasprovideduswithanabundanceofexamplesofvectorspaces,mostofthem containingusefulandinterestingmathematicalobjectsalongwithnaturaloperations.Inthissubsection wewillprovesomegeneralpropertiesofvectorspaces.Someoftheseresultswillagainseemobvious,but itisimportanttounderstandwhyitisnecessarytostateandprovethem.Atypicalhypothesiswillbe Let V beavectorspace."FromthiswemayassumethetenpropertiesofDenitionVS[279], andnothing more .Itslikestartingover,aswelearnaboutwhatcanhappeninthisnewalgebrawearelearning.But thepowerofthiscarefulapproachisthatwecanapplythesetheoremstoanyvectorspaceweencounter |thoseinthepreviousexamples,ornewoneswehavenotyetcontemplated.Orperhapsnewonesthat nobodyhasevercontemplated.Wewillillustratesomeoftheseresultswithexamplesfromthecrazyvector spaceExampleCVS[283],butmostlywearestatingtheoremsanddoingproofs.Theseproofsdonot gettooinvolved,butarenottrivialeither,sothesearegoodtheoremstotryprovingyourselfbeforeyou studytheproofgivenhere.SeeTechniqueP[695]. Firstweshowthatthereisjustonezerovector.Noticethatthepropertiesonlyrequiretheretobe at least one,andsaynothingabouttherepossiblybeingmore.Thatisbecausewecanusethetenproperties ofavectorspaceDenitionVS[279]tolearnthattherecan never bemorethanone.Torequirethat thisextraconditionbestatedasaneleventhpropertywouldmakethedenitionofavectorspacemore complicatedthanitneedstobe. TheoremZVU ZeroVectorisUnique Version2.02

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SubsectionVS.VSPVectorSpaceProperties287 Supposethat V isavectorspace.Thezerovector, 0 ,isunique. Proof Toproveuniqueness,astandardtechniqueistosupposetheexistenceoftwoobjectsTechnique U[693].Solet 0 1 and 0 2 betwozerovectorsin V .Then 0 1 = 0 1 + 0 2 PropertyZ[280]for 0 2 = 0 2 + 0 1 PropertyC[279] = 0 2 PropertyZ[280]for 0 1 Thisprovestheuniquenesssincethetwozerovectorsarereallythesame. TheoremAIU AdditiveInversesareUnique Supposethat V isavectorspace.Foreach u 2 V ,theadditiveinverse, )]TJ/F36 10.9091 Tf 8.485 0 Td [(u ,isunique. Proof Toproveuniqueness,astandardtechniqueistosupposetheexistenceoftwoobjectsTechnique U[693].Solet )]TJ/F36 10.9091 Tf 8.485 0 Td [(u 1 and )]TJ/F36 10.9091 Tf 8.485 0 Td [(u 2 betwoadditiveinversesfor u .Then )]TJ/F36 10.9091 Tf 8.485 0 Td [(u 1 = )]TJ/F36 10.9091 Tf 8.485 0 Td [(u 1 + 0 PropertyZ[280] = )]TJ/F36 10.9091 Tf 8.485 0 Td [(u 1 + u + )]TJ/F36 10.9091 Tf 8.485 0 Td [(u 2 PropertyAI[280] = )]TJ/F36 10.9091 Tf 8.485 0 Td [(u 1 + u + )]TJ/F36 10.9091 Tf 8.485 0 Td [(u 2 PropertyAA[279] = 0 + )]TJ/F36 10.9091 Tf 8.485 0 Td [(u 2 PropertyAI[280] = )]TJ/F36 10.9091 Tf 8.485 0 Td [(u 2 PropertyZ[280] Sothetwoadditiveinversesarereallythesame. Asobviousasthenextthreetheoremsappear,nowherehaveweguaranteedthatthezeroscalar,scalar multiplicationandthezerovectorallinteractthisway.Untilwehaveprovedit,anyway. TheoremZSSM ZeroScalarinScalarMultiplication Supposethat V isavectorspaceand u 2 V .Then0 u = 0 Proof Noticethat0isascalar, u isavector,soPropertySC[279]says0 u isagainavector.Assuch, 0 u hasanadditiveinverse, )]TJ/F15 10.9091 Tf 8.484 0 Td [( u byPropertyAI[280]. 0 u = 0 +0 u PropertyZ[280] = )]TJ/F15 10.9091 Tf 8.485 0 Td [( u +0 u +0 u PropertyAI[280] = )]TJ/F15 10.9091 Tf 8.485 0 Td [( u + u +0 u PropertyAA[279] = )]TJ/F15 10.9091 Tf 8.485 0 Td [( u ++0 u PropertyDSA[280] = )]TJ/F15 10.9091 Tf 8.485 0 Td [( u +0 u PropertyZCN[681] = 0 PropertyAI[280] Here'sanothertheoremthat looks likeitshouldbeobvious,butisstillinneedofaproof. TheoremZVSM ZeroVectorinScalarMultiplication Supposethat V isavectorspaceand 2 C .Then 0 = 0 Proof Noticethat isascalar, 0 isavector,soPropertySC[279]means 0 isagainavector.Assuch, 0 hasanadditiveinverse, )]TJ/F15 10.9091 Tf 8.485 0 Td [( 0 byPropertyAI[280]. 0 = 0 + 0 PropertyZ[280] Version2.02

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SubsectionVS.VSPVectorSpaceProperties288 = )]TJ/F15 10.9091 Tf 8.485 0 Td [( 0 + 0 + 0 PropertyAI[280] = )]TJ/F15 10.9091 Tf 8.485 0 Td [( 0 + 0 + 0 PropertyAA[279] = )]TJ/F15 10.9091 Tf 8.485 0 Td [( 0 + 0 + 0 PropertyDVA[280] = )]TJ/F15 10.9091 Tf 8.485 0 Td [( 0 + 0 PropertyZ[280] = 0 PropertyAI[280] Here'sanotheronethatsurelooksobvious.Butunderstandthatwehavechosentousecertainnotation becauseitmakesthetheorem'sconclusionlooksonice.Thetheoremisnottruebecausethenotationlooks sogood,itstillneedsaproof.Ifwehadreallywantedtomakethispoint,wemighthavedenedtheadditive inverseof u as u ] .Thenwewouldhavewrittenthedeningproperty,PropertyAI[280],as u + u ] = 0 Thistheoremwouldbecome u ] = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 u .Notreallyquiteaspretty,isit? TheoremAISM AdditiveInversesfromScalarMultiplication Supposethat V isavectorspaceand u 2 V .Then )]TJ/F36 10.9091 Tf 8.485 0 Td [(u = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 u Proof )]TJ/F36 10.9091 Tf 8.485 0 Td [(u = )]TJ/F36 10.9091 Tf 8.485 0 Td [(u + 0 PropertyZ[280] = )]TJ/F36 10.9091 Tf 8.485 0 Td [(u +0 u TheoremZSSM[286] = )]TJ/F36 10.9091 Tf 8.485 0 Td [(u ++ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 u = )]TJ/F36 10.9091 Tf 8.485 0 Td [(u + u + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 u PropertyDSA[280] = )]TJ/F36 10.9091 Tf 8.485 0 Td [(u + u + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 u PropertyO[280] = )]TJ/F36 10.9091 Tf 8.485 0 Td [(u + u + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 u PropertyAA[279] = 0 + )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 u PropertyAI[280] = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 u PropertyZ[280] Becauseofthistheorem,wecannowwritelinearcombinationslike6 u 1 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 u 2 as6 u 1 )]TJ/F15 10.9091 Tf 11.335 0 Td [(4 u 2 ,eventhoughwehavenotformallydenedanoperationcalled vectorsubtraction .Our nexttheoremisabitdierentfromseveraloftheothersinthelist.Ratherthanmakingadeclaration thezerovectorisunique"itisanimplicationif...,then..."andsocanbeusedinproofstoconvert avectorequalityintotwopossibilities,oneascalarequalityandtheotheravectorequality.Itshould remindyouofthesituationforcomplexnumbers.If ; 2 C and =0,then =0or =0.This criticalpropertyisthedrivingforcebehindusingafactorizationtosolveapolynomialequation. TheoremSMEZV ScalarMultiplicationEqualstheZeroVector Supposethat V isavectorspaceand 2 C .If u = 0 ,theneither =0or u = 0 Proof Weprovethistheorembybreakinguptheanalysisintotwocases.Therstseemstootrivial,and itis,butthelogicoftheargumentisstilllegitimate. Case1.Suppose =0.Inthiscaseourconclusionistruetherstpartoftheeither/oristrueand wearedone.Thatwaseasy. Case2.Suppose 6 =0. u =1 u PropertyO[280] = 1 u 6 =0 Version2.02

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SubsectionVS.RDRecyclingDenitions289 = 1 u PropertySMA[280] = 1 0 Hypothesis = 0 TheoremZVSM[286] Sointhiscase,theconclusionistruethesecondpartoftheeither/oristrueandwearedonesincethe conclusionwastrueineachofthetwocases. ExamplePCVS PropertiesfortheCrazyVectorSpace Severaloftheabovetheoremshaveinterestingdemonstrationswhenappliedtothecrazyvectorspace, C ExampleCVS[283].Wearenotprovinganythingnewhere,orlearninganythingwedidnotknow alreadyabout C .Itisjustplainfuntoseehowthesegeneraltheoremsapplyinaspecicinstance.For mostofourexamples,theapplicationsareobviousortrivial,butnotwith C Suppose u 2 C Then,asgivenbyTheoremZSSM[286], 0 u =0 x 1 ;x 2 = x 1 +0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ; 0 x 2 +0 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1= )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1= 0 AndasgivenbyTheoremZVSM[286], 0 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(1= )]TJ/F15 10.9091 Tf 8.484 0 Td [(1+ )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1+ )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = )]TJ/F21 10.9091 Tf 8.484 0 Td [( + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ; )]TJ/F21 10.9091 Tf 8.485 0 Td [( + )]TJ/F15 10.9091 Tf 10.909 0 Td [(1= )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1= 0 Finally,asgivenbyTheoremAISM[287], )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 u = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 x 1 ;x 2 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 x 1 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 x 2 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 ; )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2= )]TJ/F36 10.9091 Tf 8.485 0 Td [(u SubsectionRD RecyclingDenitions Whenwesaythat V isavectorspace,wethenknowwehaveasetofobjectsthevectors",butwealso knowwehavebeenprovidedwithtwooperationsvectoraddition"andscalarmultiplication"andthese operationsbehavewiththeseobjectsaccordingtothetenpropertiesofDenitionVS[279].Onecombines twovectorsandproducesavector,theothertakesascalarandavector,producingavectorastheresult. Soif u 1 ; u 2 ; u 3 2 V thenanexpressionlike 5 u 1 +7 u 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(13 u 3 wouldbeunambiguousin any ofthevectorspaceswehavediscussedinthissection.Andtheresulting objectwouldbeanothervectorinthevectorspace.Ifyouweretemptedtocalltheaboveexpressiona linearcombination,youwouldberight.Fourofthedenitionsthatwerecentraltoourdiscussionsin ChapterV[83]werestatedinthecontextofvectorsbeing columnvectors ,butwerepurposelykeptbroad enoughthattheycouldbeappliedinthecontextofanyvectorspace.Theyonlyrelyonthepresenceof scalars,vectors,vectoradditionandscalarmultiplicationtomakesense.Wewillrestatethemshortly, unchanged,exceptthattheirtitlesandacronymsnolongerrefertocolumnvectors,andthehypothesisof beinginavectorspacehasbeenadded.Takethetimenowtolookforwardandrevieweachone,andbegin Version2.02

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SubsectionVS.READReadingQuestions290 toformsomeconnectionstowhatwehavedoneearlierandwhatwewillbedoinginsubsequentsections andchapters.Specically,comparethefollowingpairsofdenitions: DenitionLCCV[90]andDenitionLC[297] DenitionSSCV[112]andDenitionSS[298] DenitionRLDCV[132]andDenitionRLD[308] DenitionLICV[132]andDenitionLI[308] SubsectionREAD ReadingQuestions 1.Commentonhowthevectorspace C m wentfromatheoremTheoremVSPCV[86]toanexample ExampleVSCV[281]. 2.Inthecrazyvectorspace, C ,ExampleCVS[283]computethelinearcombination 2 ; 4+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 ; 2 : 3.Supposethat isascalarand 0 isthezerovector.Whyshouldweproveanythingasobviousas 0 = 0 suchaswedidinTheoremZVSM[286]? Version2.02

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SubsectionVS.EXCExercises291 SubsectionEXC Exercises M10 Deneapossiblynewvectorspacebybeginningwiththesetandvectoradditionfrom C 2 Example VSCV[281]butchangethedenitionofscalarmultiplicationto x = 0 = 0 0 2 C ; x 2 C 2 Provethattherstninepropertiesrequiredforavectorspacehold,butPropertyO[280]doesnothold. ThisexampleshowsusthatwecannotexpecttobeabletoderivePropertyO[280]asaconsequenceof assumingtherstnineproperties.Inotherwords,wecannotslimdownourlistofpropertiesbyjettisoning thelastone,andstillhavethesamecollectionofobjectsqualifyasvectorspaces. ContributedbyRobertBeezer T10 ProveeachofthetenpropertiesofDenitionVS[279]foreachofthefollowingexamplesofavector space: ExampleVSP[281] ExampleVSIS[282] ExampleVSF[282] ExampleVSS[283] ContributedbyRobertBeezer Thenextthreeproblemssuggestthatundertherightsituationswecancancel."Inpractice,these techniquesshouldbeavoidedinotherproofs.Proveeachofthefollowingstatements. T21 Supposethat V isavectorspace,and u ; v ; w 2 V .If w + u = w + v ,then u = v ContributedbyRobertBeezerSolution[291] T22 Suppose V isavectorspace, u ; v 2 V and isanonzeroscalarfrom C .If u = v ,then u = v ContributedbyRobertBeezerSolution[291] T23 Suppose V isavectorspace, u 6 = 0 isavectorin V and ; 2 C .If u = u ,then = ContributedbyRobertBeezerSolution[291] Version2.02

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SubsectionVS.SOLSolutions292 SubsectionSOL Solutions T21 ContributedbyRobertBeezerStatement[290] u = 0 + u PropertyZ[280] = )]TJ/F36 10.9091 Tf 8.485 0 Td [(w + w + u PropertyAI[280] = )]TJ/F36 10.9091 Tf 8.485 0 Td [(w + w + u PropertyAA[279] = )]TJ/F36 10.9091 Tf 8.485 0 Td [(w + w + v Hypothesis = )]TJ/F36 10.9091 Tf 8.485 0 Td [(w + w + v PropertyAA[279] = 0 + v PropertyAI[280] = v PropertyZ[280] T22 ContributedbyRobertBeezerStatement[290] u =1 u PropertyO[280] = 1 u 6 =0 = 1 u PropertySMA[280] = 1 v Hypothesis = 1 v PropertySMA[280] =1 v = v PropertyO[280] T23 ContributedbyRobertBeezerStatement[290] 0 = u + )]TJ/F15 10.9091 Tf 10.303 0 Td [( u PropertyAI[280] = u + )]TJ/F15 10.9091 Tf 10.303 0 Td [( u Hypothesis = u + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 u TheoremAISM[287] = u + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 u PropertySMA[280] = u + )]TJ/F21 10.9091 Tf 8.485 0 Td [( u = )]TJ/F21 10.9091 Tf 10.909 0 Td [( u PropertyDSA[280] Byhypothesis, u 6 = 0 ,soTheoremSMEZV[287]implies 0= )]TJ/F21 10.9091 Tf 10.909 0 Td [( = Version2.02

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SectionSSubspaces293 SectionS Subspaces Asubspaceisavectorspacethatiscontainedwithinanothervectorspace.Soeverysubspaceisavector spaceinitsownright,butitisalsodenedrelativetosomeotherlargervectorspace.Wewilldiscover shortlythatwearealreadyfamiliarwithawidevarietyofsubspacesfromprevioussections.Here'sthe denition. DenitionS Subspace Supposethat V and W aretwovectorspacesthathaveidenticaldenitionsofvectoradditionandscalar multiplication,andthat W isasubsetof V W V .Then W isa subspace of V 4 Letslookatanexampleofavectorspaceinsideanothervectorspace. ExampleSC3 Asubspaceof C 3 Weknowthat C 3 isavectorspaceExampleVSCV[281].Considerthesubset, W = 8 < : 2 4 x 1 x 2 x 3 3 5 j 2 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 2 +7 x 3 =0 9 = ; Itisclearthat W C 3 ,sincetheobjectsin W arecolumnvectorsofsize3.Butis W avectorspace?Does itsatisfythetenpropertiesofDenitionVS[279]whenweusethesameoperations?Thatisthemain question.Suppose x = 2 4 x 1 x 2 x 3 3 5 and y = 2 4 y 1 y 2 y 3 3 5 arevectorsfrom W .Thenweknowthatthesevectorscannot betotallyarbitrary,theymusthavegainedmembershipin W byvirtueofmeetingthemembershiptest. Forexample,weknowthat x mustsatisfy2 x 1 )]TJ/F15 10.9091 Tf 10.995 0 Td [(5 x 2 +7 x 3 =0while y mustsatisfy2 y 1 )]TJ/F15 10.9091 Tf 10.995 0 Td [(5 y 2 +7 y 3 =0. OurrstpropertyPropertyAC[279]asksthequestion,is x + y 2 W ?Whenoursetofvectorswas C 3 thiswasaneasyquestiontoanswer.Nowitisnotsoobvious.Noticerstthat x + y = 2 4 x 1 x 2 x 3 3 5 + 2 4 y 1 y 2 y 3 3 5 = 2 4 x 1 + y 1 x 2 + y 2 x 3 + y 3 3 5 andwecantestthisvectorformembershipin W asfollows, 2 x 1 + y 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 2 + y 2 +7 x 3 + y 3 =2 x 1 +2 y 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 y 2 +7 x 3 +7 y 3 = x 1 )]TJ/F15 10.9091 Tf 10.91 0 Td [(5 x 2 +7 x 3 + y 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 y 2 +7 y 3 =0+0 x 2 W; y 2 W =0 andbythiscomputationweseethat x + y 2 W .Onepropertydown,ninetogo. If isascalarand x 2 W ,isitalwaystruethat x 2 W ?ThisiswhatweneedtoestablishProperty SC[279].Again,theanswerisnotasobviousasitwaswhenoursetofvectorswasallof C 3 .Let'ssee. x = 2 4 x 1 x 2 x 3 3 5 = 2 4 x 1 x 2 x 3 3 5 Version2.02

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SubsectionS.TSTestingSubspaces294 andwecantestthisvectorformembershipin W with 2 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 2 +7 x 3 = x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 2 +7 x 3 = 0 x 2 W =0 andweseethatindeed x 2 W .Always. If W hasazerovector,itwillbeuniqueTheoremZVU[285].Thezerovectorfor C 3 shouldalso performtherequireddutieswhenaddedtoelementsof W .Sothelikelycandidateforazerovectorin W isthesamezerovectorthatweknow C 3 has.Youcancheckthat 0 = 2 4 0 0 0 3 5 isazerovectorin W too PropertyZ[280]. Withazerovector,wecannowaskaboutadditiveinversesPropertyAI[280].Asyoumightsuspect, thenaturalcandidateforanadditiveinversein W isthesameastheadditiveinversefrom C 3 .However, wemustinsurethattheseadditiveinversesactuallyareelementsof W .Given x 2 W ,is )]TJ/F36 10.9091 Tf 8.485 0 Td [(x 2 W ? )]TJ/F36 10.9091 Tf 8.484 0 Td [(x = 2 4 )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 1 )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 2 )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 3 3 5 andwecantestthisvectorformembershipin W with 2 )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 2 +7 )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 3 = )]TJ/F15 10.9091 Tf 8.484 0 Td [( x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 2 +7 x 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(0 x 2 W =0 andwenowbelievethat )]TJ/F36 10.9091 Tf 8.485 0 Td [(x 2 W Isthevectoradditionin W commutativePropertyC[279]?Is x + y = y + x ?Ofcourse!Nothing aboutrestrictingthescopeofoursetofvectorswillpreventtheoperationfromstillbeingcommutative. Indeed,theremainingvepropertiesareunaectedbythetransitiontoasmallersetofvectors,andso remaintrue.Thatwasconvenient. So W satisesalltenproperties,isthereforeavectorspace,andthusearnsthetitleofbeingasubspace of C 3 SubsectionTS TestingSubspaces InExampleSC3[292]weproceededthroughalltenofthevectorspacepropertiesbeforebelievingthat asubsetwasasubspace.Butsixofthepropertieswereeasytoprove,andwecanleanonsomeofthe propertiesofthevectorspacethesupersettomaketheotherfoureasier.Hereisatheoremthatwill makeiteasiertotestifasubsetisavectorspace.Ashortcutifthereeverwasone. TheoremTSS TestingSubsetsforSubspaces Supposethat V isavectorspaceand W isasubsetof V W V .Endow W withthesameoperationsas V .Then W isasubspaceifandonlyifthreeconditionsaremet 1. W isnon-empty, W 6 = ; 2.If x 2 W and y 2 W ,then x + y 2 W Version2.02

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SubsectionS.TSTestingSubspaces295 3.If 2 C and x 2 W ,then x 2 W Proof Wehavethehypothesisthat W isasubspace,sobyDenitionVS[279]weknowthat W containsazerovector.Thisisenoughtoshowthat W 6 = ; .Also,since W isavectorspaceitsatisesthe additiveandscalarmultiplicationclosureproperties,andsoexactlymeetsthesecondandthirdconditions. Ifthatwaseasy,thetheotherdirectionmightrequireabitmorework. Wehavethreepropertiesforourhypothesis,andfromthisweshouldconcludethat W hasthe tendeningpropertiesofavectorspace.Thesecondandthirdconditionsofourhypothesisareexactly PropertyAC[279]andPropertySC[279].Ourhypothesisthat V isavectorspaceimpliesthatProperty C[279],PropertyAA[279],PropertySMA[280],PropertyDVA[280],PropertyDSA[280]andProperty O[280]allhold.Theycontinuetobetrueforvectorsfrom W sincepassingtoasubset,andkeepingthe operationthesame,leavestheirstatementsunchanged.Eightdown,twotogo. Suppose x 2 W .Thenbythethirdpartofourhypothesisscalarclosure,weknowthat )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 x 2 W ByTheoremAISM[287] )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 x = )]TJ/F36 10.9091 Tf 8.485 0 Td [(x ,sotogetherthesestatementsshowusthat )]TJ/F36 10.9091 Tf 8.485 0 Td [(x 2 W )]TJ/F36 10.9091 Tf 8.485 0 Td [(x isthe additiveinverseof x in V ,butwillcontinueinthisrolewhenviewedaselementofthesubset W .Soevery elementof W hasanadditiveinversethatisanelementof W andPropertyAI[280]iscompleted.Just onepropertyleft. Whilewehaveimplicitlydiscussedthezerovectorinthepreviousparagraph,weneedtobecertain thatthezerovectorof V reallylivesin W .Since W isnon-empty,wecanchoosesomevector z 2 W Thenbytheargumentinthepreviousparagraph,weknow )]TJ/F36 10.9091 Tf 8.485 0 Td [(z 2 W .NowbyPropertyAI[280]for V and thenbythesecondpartofourhypothesisadditiveclosureweseethat 0 = z + )]TJ/F36 10.9091 Tf 8.484 0 Td [(z 2 W So W containthezerovectorfrom V .Sincethisvectorperformstherequireddutiesofazerovectorin V itwillcontinueinthatroleasanelementof W .Thisgivesus,PropertyZ[280],thenalpropertyofthe tenrequired.SarahFellezcontributedtothisproof. Sojustthreeconditions,plusbeingasubsetofaknownvectorspace,getsusalltenproperties. Fabulous!Thistheoremcanbeparaphrasedbysayingthatasubspaceisanon-emptysubsetofavector spacethatisclosedundervectoradditionandscalarmultiplication." YoumightwanttogobackandreworkExampleSC3[292]inlightofthisresult,perhapsseeingwhere wecannoweconomizeorwheretheworkdoneintheexamplemirroredtheproofandwhereitdidnot. Wewillpressonandapplythistheoreminaslightlymoreabstractsetting. ExampleSP4 Asubspaceof P 4 P 4 isthevectorspaceofpolynomialswithdegreeatmost4ExampleVSP[281].Deneasubset W as W = f p x j p 2 P 4 ;p =0 g so W isthecollectionofthosepolynomialswithdegree4orlesswhosegraphscrossthe x -axisat x =2. Wheneverweencounteranewsetitisagoodideatogainabetterunderstandingofthesetbyndinga fewelementsintheset,andafewoutsideit.Forexample x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 2 W ,while x 4 + x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 62 W Is W nonempty?Yes, x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 W Additiveclosure?Suppose p 2 W and q 2 W .Is p + q 2 W ? p and q arenottotallyarbitrary,we knowthat p =0and q =0.Thenwecancheck p + q formembershipin W p + q = p + q Additionin P 4 =0+0 p 2 W;q 2 W Version2.02

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SubsectionS.TSTestingSubspaces296 =0 soweseethat p + q qualiesformembershipin W Scalarmultiplicationclosure?Supposethat 2 C and p 2 W .Thenweknowthat p =0.Testing p formembership, p = p Scalarmultiplicationin P 4 = 0 p 2 W =0 so p 2 W Wehaveshownthat W meetsthethreeconditionsofTheoremTSS[293]andsoqualiesasasubspace of P 4 .NoticethatbyDenitionS[292]wenowknowthat W isalsoavectorspace.Soalltheproperties ofavectorspaceDenitionVS[279]andthetheoremsofSectionVS[279]applyinfull. MuchofthepowerofTheoremTSS[293]isthatwecaneasilyestablishnewvectorspacesifwecan locatethemassubsetsofothervectorspaces,suchastheonespresentedinSubsectionVS.EVS[280]. Itcanbeasinstructivetoconsidersomesubsetsthatare not subspaces.SinceTheoremTSS[293]isan equivalenceseeTechniqueE[690]wecanbeassuredthatasubsetisnotasubspaceifitviolatesoneof thethreeconditions,andinanyexampleofinterestthiswillnotbethenon-empty"condition.However, sinceasubspacehastobeavectorspaceinitsownright,wecanalsosearchforaviolationofanyoneof thetendeningpropertiesinDenitionVS[279]oranyinherentpropertyofavectorspace,suchasthose givenbythebasictheoremsofSubsectionVS.VSP[285].Noticealsothataviolationneedonlybefora specicvectororpairofvectors. ExampleNSC2Z Anon-subspacein C 2 ,zerovector Considerthesubset W belowasacandidateforbeingasubspaceof C 2 W = x 1 x 2 j 3 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 2 =12 Thezerovectorof C 2 0 = 0 0 willneedtobethezerovectorin W also.However, 0 62 W since 3 )]TJ/F15 10.9091 Tf 11.35 0 Td [(5=0 6 =12.So W hasnozerovectorandfailsPropertyZ[280]ofDenitionVS[279].This subspacealsofailstobeclosedunderadditionandscalarmultiplication.Canyoundexamplesofthis? ExampleNSC2A Anon-subspacein C 2 ,additiveclosure Considerthesubset X belowasacandidateforbeingasubspaceof C 2 X = x 1 x 2 j x 1 x 2 =0 Youcancheckthat 0 2 X ,sotheapproachofthelastexamplewillnotgetusanywhere.However,notice that x = 1 0 2 X and y = 0 1 2 X .Yet x + y = 1 0 + 0 1 = 1 1 62 X Version2.02

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SubsectionS.TSTestingSubspaces297 So X failstheadditiveclosurerequirementofeitherPropertyAC[279]orTheoremTSS[293],andis thereforenotasubspace. ExampleNSC2S Anon-subspacein C 2 ,scalarmultiplicationclosure Considerthesubset Y belowasacandidateforbeingasubspaceof C 2 Y = x 1 x 2 j x 1 2 Z ;x 2 2 Z Z isthesetofintegers,soweareonlyallowingwholenumbers"astheconstituentsofourvectors.Now, 0 2 Y ,andadditiveclosurealsoholdscanyouprovetheseclaims?.Sowewillhavetotrysomething dierent.Notethat = 1 2 2 C and 2 3 2 Y ,but x = 1 2 2 3 = 1 3 2 62 Y So Y failsthescalarmultiplicationclosurerequirementofeitherPropertySC[279]orTheoremTSS[293], andisthereforenotasubspace. Therearetwoexamplesofsubspacesthataretrivial.Supposethat V isanyvectorspace.Then V is asubsetofitselfandisavectorspace.ByDenitionS[292], V qualiesasasubspaceofitself.Theset containingjustthezerovector Z = f 0 g isalsoasubspaceascanbeseenbyapplyingTheoremTSS[293] orbysimplemodicationsofthetechniqueshintedatinExampleVSS[283].Sincethesesubspacesareso obviousandthereforenottoointerestingwewillrefertothemasbeingtrivial. DenitionTS TrivialSubspaces Giventhevectorspace V ,thesubspaces V and f 0 g areeachcalleda trivialsubspace 4 WecanalsouseTheoremTSS[293]toprovemoregeneralstatementsaboutsubspaces,asillustrated inthenexttheorem. TheoremNSMS NullSpaceofaMatrixisaSubspace Supposethat A isan m n matrix.Thenthenullspaceof A N A ,isasubspaceof C n Proof WewillexaminethethreerequirementsofTheoremTSS[293].Recallthat N A = f x 2 C n j A x = 0 g First, 0 2N A ,whichcanbeinferredasaconsequenceofTheoremHSC[62].So N A 6 = ; Second,checkadditiveclosurebysupposingthat x 2N A and y 2N A .Soweknowalittle somethingabout x and y : A x = 0 and A y = 0 ,andthatisallweknow.Question:Is x + y 2N A ? Let'scheck. A x + y = A x + A y TheoremMMDAA[201] = 0 + 0x 2N A ; y 2N A = 0 TheoremVSPCV[86] So,yes, x + y qualiesformembershipin N A Third,checkscalarmultiplicationclosurebysupposingthat 2 C and x 2N A .Soweknowalittle somethingabout x : A x = 0 ,andthatisallweknow.Question:Is x 2N A ?Let'scheck. A x = A x TheoremMMSMM[201] = 0x 2N A Version2.02

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SubsectionS.TSSTheSpanofaSet298 = 0 TheoremZVSM[286] So,yes, x qualiesformembershipin N A HavingmetthethreeconditionsinTheoremTSS[293]wecannowsaythatthenullspaceofamatrix isasubspaceandhenceavectorspaceinitsownright!. HereisanexamplewherewecanexerciseTheoremNSMS[296]. ExampleRSNS Recastingasubspaceasanullspace Considerthesubsetof C 5 denedas W = 8 > > > > < > > > > : 2 6 6 6 6 4 x 1 x 2 x 3 x 4 x 5 3 7 7 7 7 5 j 3 x 1 + x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 3 +7 x 4 + x 5 =0 ; 4 x 1 +6 x 2 +3 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 5 =0 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 x 1 +4 x 2 +7 x 4 + x 5 =0 9 > > > > = > > > > ; Itispossibletoshowthat W isasubspaceof C 5 bycheckingthethreeconditionsofTheoremTSS[293] directly,butitwillgettediousratherquickly.Instead,give W afreshlookandnoticethatitisasetof solutionstoahomogeneoussystemofequations.Denethematrix A = 2 4 31 )]TJ/F15 10.9091 Tf 8.485 0 Td [(571 463 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(24071 3 5 andthenrecognizethat W = N A .ByTheoremNSMS[296]wecanimmediatelyseethat W isa subspace.Boom! SubsectionTSS TheSpanofaSet ThespanofasetofcolumnvectorsgotaheavyworkoutinChapterV[83]andChapterM[182].The denitionofthespandependedonlyonbeingabletoformulatelinearcombinations.Inanyofourmore generalvectorspaceswealwayshaveadenitionofvectoradditionandofscalarmultiplication.Sowe canbuildlinearcombinationsandmanufacturespans.Thissubsectioncontainstwodenitionsthatare justmildvariantsofdenitionswehaveseenearlierforcolumnvectors.Ifyouhaven'talready,compare themwithDenitionLCCV[90]andDenitionSSCV[112]. DenitionLC LinearCombination Supposethat V isavectorspace.Given n vectors u 1 ; u 2 ; u 3 ;:::; u n and n scalars 1 ; 2 ; 3 ;:::; n their linearcombination isthevector 1 u 1 + 2 u 2 + 3 u 3 + + n u n : 4 ExampleLCM Alinearcombinationofmatrices Inthevectorspace M 23 of2 3matrices,wehavethevectors x = 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 207 y = 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 551 z = 42 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 111 Version2.02

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SubsectionS.TSSTheSpanofaSet299 andwecanformlinearcombinationssuchas 2 x +4 y + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 z =2 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 207 +4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 551 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 42 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 111 = 26 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 4014 + 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(48 20204 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(24 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 = 1008 231917 or, 4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 y +3 z =4 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 207 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 551 +3 42 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 111 = 412 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 8028 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(62 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 + 126 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 333 = 1020 )]TJ/F15 10.9091 Tf 8.485 0 Td [(24 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(729 Whenwerealizethatwecanformlinearcombinationsinanyvectorspace,thenitisnaturaltorevisit ourdenitionofthespanofaset,sinceitisthesetof all possiblelinearcombinationsofasetofvectors. DenitionSS SpanofaSet Supposethat V isavectorspace.Givenasetofvectors S = f u 1 ; u 2 ; u 3 ;:::; u t g ,their span h S i ,isthe setofallpossiblelinearcombinationsof u 1 ; u 2 ; u 3 ;:::; u t .Symbolically, h S i = f 1 u 1 + 2 u 2 + 3 u 3 + + t u t j i 2 C ; 1 i t g = t X i =1 i u i j i 2 C ; 1 i t 4 TheoremSSS SpanofaSetisaSubspace Suppose V isavectorspace.Givenasetofvectors S = f u 1 ; u 2 ; u 3 ;:::; u t g V ,theirspan, h S i ,isa subspace. Proof WewillverifythethreeconditionsofTheoremTSS[293].First, 0 = 0 + 0 + 0 + ::: + 0 PropertyZ[280]for V =0 u 1 +0 u 2 +0 u 3 + +0 u t TheoremZSSM[286] Sowehavewritten 0 asalinearcombinationofthevectorsin S andbyDenitionSS[298] ; 0 2h S i and therefore S 6 = ; Second,suppose x 2h S i and y 2h S i .Canweconcludethat x + y 2h S i ?Whatdoweknowabout x and y byvirtueoftheirmembershipin h S i ?Theremustbescalarsfrom C 1 ; 2 ; 3 ;:::; t and 1 ; 2 ; 3 ;:::; t sothat x = 1 u 1 + 2 u 2 + 3 u 3 + + t u t y = 1 u 1 + 2 u 2 + 3 u 3 + + t u t Version2.02

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SubsectionS.TSSTheSpanofaSet300 Then x + y = 1 u 1 + 2 u 2 + 3 u 3 + + t u t + 1 u 1 + 2 u 2 + 3 u 3 + + t u t = 1 u 1 + 1 u 1 + 2 u 2 + 2 u 2 + 3 u 3 + 3 u 3 + + t u t + t u t PropertyAA[279],PropertyC[279] = 1 + 1 u 1 + 2 + 2 u 2 + 3 + 3 u 3 + + t + t u t PropertyDSA[280] Sinceeach i + i isagainascalarfrom C wehaveexpressedthevectorsum x + y asalinearcombination ofthevectorsfrom S ,andthereforebyDenitionSS[298]wecansaythat x + y 2h S i Third,suppose 2 C and x 2h S i .Canweconcludethat x 2h S i ?Whatdoweknowabout x by virtueofitsmembershipin h S i ?Theremustbescalarsfrom C 1 ; 2 ; 3 ;:::; t sothat x = 1 u 1 + 2 u 2 + 3 u 3 + + t u t Then x = 1 u 1 + 2 u 2 + 3 u 3 + + t u t = 1 u 1 + 2 u 2 + 3 u 3 + + t u t PropertyDVA[280] = 1 u 1 + 2 u 2 + 3 u 3 + + t u t PropertySMA[280] Sinceeach i isagainascalarfrom C wehaveexpressedthescalarmultiple x asalinearcombination ofthevectorsfrom S ,andthereforebyDenitionSS[298]wecansaythat x 2h S i WiththethreeconditionsofTheoremTSS[293]met,wecansaythat h S i isasubspaceandsois alsovectorspace,DenitionVS[279].SeeExerciseSS.T20[125],ExerciseSS.T21[125],ExerciseSS.T22 [125]. ExampleSSP Spanofasetofpolynomials InExampleSP4[294]weprovedthat W = f p x j p 2 P 4 ;p =0 g isasubspaceof P 4 ,thevectorspaceofpolynomialsofdegreeatmost4.Since W isavectorspaceitself, let'sconstructaspanwithin W .Firstlet S = x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 3 +5 x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 ; 2 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 2 +6 x +4 andverifythat S isasubsetof W bycheckingthateachofthesetwopolynomialshas x =2asaroot. Now,ifwedene U = h S i ,thenTheoremSSS[298]tellsusthat U isasubspaceof W .Soquitequickly wehavebuiltachainofsubspaces, U inside W ,and W inside P 4 Ratherthandwellonhowquicklywecanbuildsubspaces,let'strytogainabetterunderstandingof justhowthespanconstructioncreatessubspaces,inthecontextofthisexample.Wecanquicklybuild representativeelementsof U 3 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 3 +5 x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2+5 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 2 +6 x +4=13 x 4 )]TJ/F15 10.9091 Tf 10.91 0 Td [(27 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(15 x 2 +27 x +14 and )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 x 4 )]TJ/F15 10.9091 Tf 10.91 0 Td [(4 x 3 +5 x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.91 0 Td [(2+8 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 2 +6 x +4=14 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(16 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(58 x 2 +50 x +36 Version2.02

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SubsectionS.TSSTheSpanofaSet301 andeachofthesepolynomialsmustbein W sinceitisclosedunderadditionandscalarmultiplication. Butyoumightcheckforyourselfthatbothofthesepolynomialshave x =2asaroot. Icantellyouthat y =3 x 4 )]TJ/F15 10.9091 Tf 11.06 0 Td [(7 x 3 )]TJ/F21 10.9091 Tf 11.06 0 Td [(x 2 +7 x )]TJ/F15 10.9091 Tf 11.059 0 Td [(2isnotin U ,butwouldyoubelieveme?Arstcheck showsthat y doeshave x =2asaroot,butthatonlyshowsthat y 2 W .Whatdoes y havetodotogain membershipin U = h S i ?Itmustbealinearcombinationofthevectorsin S x 4 )]TJ/F15 10.9091 Tf 10.927 0 Td [(4 x 3 +5 x 2 )]TJ/F21 10.9091 Tf 10.927 0 Td [(x )]TJ/F15 10.9091 Tf 10.927 0 Td [(2and 2 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 2 +6 x +4.Solet'ssupposethat y issuchalinearcombination, y =3 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 2 +7 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 = 1 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 3 +5 x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2+ 2 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 2 +6 x +4 = 1 +2 2 x 4 + )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 x 3 + 1 )]TJ/F15 10.9091 Tf 10.91 0 Td [(6 2 x 2 + )]TJ/F21 10.9091 Tf 8.485 0 Td [( 1 +6 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 +4 2 Noticethatoperationsabovearedoneinaccordancewiththedenitionofthevectorspaceofpolynomials ExampleVSP[281].Now,ifweequatecoecients,whichisthedenitionofequalityforpolynomials, thenweobtainthesystemofvelinearequationsintwovariables 1 +2 2 =3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 5 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F21 10.9091 Tf 8.485 0 Td [( 1 +6 2 =7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 +4 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 Buildanaugmentedmatrixfromthesystemandrow-reduce, 2 6 6 6 6 4 123 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(7 5 )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(167 )]TJ/F15 10.9091 Tf 8.485 0 Td [(24 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 3 7 7 7 7 5 RREF )443()223()222()443(! 2 6 6 6 6 6 4 1 00 0 1 0 00 1 000 000 3 7 7 7 7 7 5 Withaleading1inthenalcolumnoftherow-reducedaugmentedmatrix,TheoremRCLS[53]tellsus thesystemofequationsisinconsistent.Therefore,therearenoscalars, 1 and 2 ,toestablish y asa linearcombinationoftheelementsin U .So y 62 U Let'sagainexaminemembershipinaspan. ExampleSM32 Asubspaceof M 32 Thesetofall3 2matricesformsavectorspacewhenweusetheoperationsofmatrixadditionDenition MA[182]andscalarmatrixmultiplicationDenitionMSM[183],aswasshowinExampleVSM[281]. Considerthesubset S = 8 < : 2 4 31 42 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 5 ; 2 4 11 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 14 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 3 5 ; 2 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(19 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 3 5 ; 2 4 42 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 5 ; 2 4 31 )]TJ/F15 10.9091 Tf 8.485 0 Td [(40 )]TJ/F15 10.9091 Tf 8.485 0 Td [(177 3 5 9 = ; anddeneanewsubsetofvectors W in M 32 usingthespanDenitionSS[298], W = h S i .Soby TheoremSSS[298]weknowthat W isasubspaceof M 32 .While W isaninniteset,andthisisaprecise description,itwouldstillbeworthwhiletoinvestigatewhetherornot W containscertainelements. First,is y = 2 4 93 73 10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 3 5 Version2.02

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SubsectionS.TSSTheSpanofaSet302 in W ?Toanswerthis,wewanttodetermineif y canbewrittenasalinearcombinationofthevematrices in S .Canwendscalars, 1 ; 2 ; 3 ; 4 ; 5 sothat 2 4 93 73 10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 3 5 = 1 2 4 31 42 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 5 + 2 2 4 11 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 5 + 3 2 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(19 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 3 5 + 4 2 4 42 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 5 + 5 2 4 31 )]TJ/F15 10.9091 Tf 8.485 0 Td [(40 )]TJ/F15 10.9091 Tf 8.484 0 Td [(177 3 5 = 2 4 3 1 + 2 +3 3 +4 4 +3 5 1 + 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [( 3 +2 4 + 5 4 1 +2 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [( 3 + 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 5 2 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [( 2 +2 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 4 5 1 +14 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(19 3 +14 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(17 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [( 2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(11 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 4 +7 5 3 5 UsingourdenitionofmatrixequalityDenitionME[182]wecantranslatethisstatementintosix equationsintheveunknowns, 3 1 + 2 +3 3 +4 4 +3 5 =9 1 + 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [( 3 +2 4 + 5 =3 4 1 +2 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [( 3 + 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 5 =7 2 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [( 2 +2 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 4 =3 5 1 +14 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(19 3 +14 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(17 5 =10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [( 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(11 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 4 +7 5 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 Thisisalinearsystemofequations,whichwecanrepresentwithanaugmentedmatrixandrow-reducein searchofsolutions.Thematrixthatisrow-equivalenttotheaugmentedmatrixis 2 6 6 6 6 6 6 6 4 1 000 5 8 2 0 1 00 )]TJ/F19 7.9701 Tf 6.587 0 Td [(19 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 00 1 0 )]TJ/F19 7.9701 Tf 6.587 0 Td [(7 8 0 000 1 17 8 1 000000 000000 3 7 7 7 7 7 7 7 5 Sowerecognizethatthesystemisconsistentsincethereisnoleading1inthenalcolumnTheoremRCLS [53],andcompute n )]TJ/F21 10.9091 Tf 10.432 0 Td [(r =5 )]TJ/F15 10.9091 Tf 10.432 0 Td [(4=1freevariablesTheoremFVCS[55].Whilethereareinnitelymany solutions,weareonlyinpursuitofasinglesolution,solet'schoosethefreevariable 5 =0forsimplicity's sake.Thenweeasilyseethat 1 =2, 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1, 3 =0, 4 =1.Sothescalars 1 =2, 2 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1, 3 =0, 4 =1, 5 =0willprovidealinearcombinationoftheelementsof S thatequals y ,aswecanverifyby checking, 2 4 93 73 10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 3 5 =2 2 4 31 42 5 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 3 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 4 11 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 5 + 2 4 42 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 5 Sowithoneparticularlinearcombinationinhand,weareconvincedthat y deservestobeamemberof W = h S i .Second,is x = 2 4 21 31 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 5 in W ?Toanswerthis,wewanttodetermineif x canbewrittenasalinearcombinationofthevematrices in S .Canwendscalars, 1 ; 2 ; 3 ; 4 ; 5 sothat 2 4 21 31 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 5 = 1 2 4 31 42 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 5 + 2 2 4 11 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 5 + 3 2 4 3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(19 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 3 5 + 4 2 4 42 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 5 + 5 2 4 31 )]TJ/F15 10.9091 Tf 8.485 0 Td [(40 )]TJ/F15 10.9091 Tf 8.485 0 Td [(177 3 5 Version2.02

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SubsectionS.SCSubspaceConstructions303 = 2 4 3 1 + 2 +3 3 +4 4 +3 5 1 + 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [( 3 +2 4 + 5 4 1 +2 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [( 3 + 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 5 2 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [( 2 +2 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 4 5 1 +14 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(19 3 +14 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(17 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [( 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(11 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 4 +7 5 3 5 UsingourdenitionofmatrixequalityDenitionME[182]wecantranslatethisstatementintosix equationsintheveunknowns, 3 1 + 2 +3 3 +4 4 +3 5 =2 1 + 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [( 3 +2 4 + 5 =1 4 1 +2 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [( 3 + 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 5 =3 2 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [( 2 +2 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 4 =1 5 1 +14 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(19 3 +14 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(17 5 =4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [( 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(11 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 4 +7 5 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 Thisisalinearsystemofequations,whichwecanrepresentwithanaugmentedmatrixandrow-reducein searchofsolutions.Thematrixthatisrow-equivalenttotheaugmentedmatrixis 2 6 6 6 6 6 6 6 4 1 000 5 8 0 0 1 00 )]TJ/F19 7.9701 Tf 9.68 4.295 Td [(38 8 0 00 1 0 )]TJ/F19 7.9701 Tf 9.68 4.295 Td [(7 8 0 000 1 )]TJ/F19 7.9701 Tf 9.68 4.296 Td [(17 8 0 00000 1 000000 3 7 7 7 7 7 7 7 5 Withaleading1inthelastcolumnTheoremRCLS[53]tellsusthatthesystemisinconsistent.Therefore, therearenovaluesforthescalarsthatwillplace x in W ,andsoweconcludethat x 62 W NoticehowExampleSSP[299]andExampleSM32[300]containedquestionsaboutmembershipina span,butthesequestionsquicklybecamequestionsaboutsolutionstoasystemoflinearequations.This willbeacommonthemegoingforward. SubsectionSC SubspaceConstructions SeveralofthesubsetsofvectorsspacesthatweworkedwithinChapterM[182]arealsosubspaces|they areclosedundervectoradditionandscalarmultiplicationin C m TheoremCSMS ColumnSpaceofaMatrixisaSubspace Supposethat A isan m n matrix.Then C A isasubspaceof C m Proof DenitionCSM[236]showsusthat C A isasubsetof C m ,andthatitisdenedasthespanof asetofvectorsfrom C m thecolumnsofthematrix.Since C A isaspan,TheoremSSS[298]saysitis asubspace. Thatwaseasy!Noticethatwecouldhaveusedthissameapproachtoprovethatthenullspaceisa subspace,sinceTheoremSSNS[118]providedadescriptionofthenullspaceofamatrixasthespanofa setofvectors.However,ImuchpreferthecurrentproofofTheoremNSMS[296].Speakingofeasy,here isaveryeasytheoremthatexposesanotherofourconstructionsascreatingsubspaces. Version2.02

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SubsectionS.READReadingQuestions304 TheoremRSMS RowSpaceofaMatrixisaSubspace Supposethat A isan m n matrix.Then R A isasubspaceof C n Proof DenitionRSM[243]says R A = C )]TJ/F21 10.9091 Tf 5 -8.836 Td [(A t ,sotherowspaceofamatrixisacolumnspace,and everycolumnspaceisasubspacebyTheoremCSMS[302].That'senough. Onemore. TheoremLNSMS LeftNullSpaceofaMatrixisaSubspace Supposethat A isan m n matrix.Then L A isasubspaceof C m Proof DenitionLNS[257]says L A = N )]TJ/F21 10.9091 Tf 5 -8.836 Td [(A t ,sotheleftnullspaceisanullspace,andeverynull spaceisasubspacebyTheoremNSMS[296].Done. Sothespanofasetofvectors,andthenullspace,columnspace,rowspaceandleftnullspaceofa matrixareallsubspaces,andhenceareallvectorspaces,meaningtheyhaveallthepropertiesdetailed inDenitionVS[279]andinthebasictheoremspresentedinSectionVS[279].Wehaveworkedwith theseobjectsasjustsetsinChapterV[83]andChapterM[182],butnowweunderstandthattheyhave muchmorestructure.Inparticular,beingclosedundervectoradditionandscalarmultiplicationmeansa subspaceisalsoclosedunderlinearcombinations. SubsectionREAD ReadingQuestions 1.Summarizethethreeconditionsthatallowustoquicklytestifasetisasubspace. 2.Considerthesetofvectors W = 8 < : 2 4 a b c 3 5 j 3 a )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 b + c =5 9 = ; Istheset W asubspaceof C 3 ?Explainyouranswer. 3.Namevegeneralconstructionsofsetsofcolumnvectorssubsetsof C m thatwenowknowas subspaces. Version2.02

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SubsectionS.EXCExercises305 SubsectionEXC Exercises C20 Workingwithinthevectorspace P 3 ofpolynomialsofdegree3orless,determineif p x = x 3 +6 x +4 isinthesubspace W below. W = x 3 + x 2 + x;x 3 +2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 ;x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 ContributedbyRobertBeezerSolution[305] C21 Considerthesubspace W = 21 3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 40 23 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(31 21 ofthevectorspaceof2 2matrices, M 22 .Is C = )]TJ/F15 10.9091 Tf 8.485 0 Td [(33 6 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 anelementof W ? ContributedbyRobertBeezerSolution[305] C25 Showthattheset W = x 1 x 2 j 3 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 2 =12 fromExampleNSC2Z[295]failsPropertyAC [279]andPropertySC[279]. ContributedbyRobertBeezer C26 Showthattheset Y = x 1 x 2 j x 1 2 Z ;x 2 2 Z fromExampleNSC2S[296]hasPropertyAC[279]. ContributedbyRobertBeezer M20 In C 3 ,thevectorspaceofcolumnvectorsofsize3,provethattheset Z isasubspace. Z = 8 < : 2 4 x 1 x 2 x 3 3 5 j 4 x 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 2 +5 x 3 =0 9 = ; ContributedbyRobertBeezerSolution[306] T20 Asquarematrix A ofsize n isuppertriangularif[ A ] ij =0whenever i>j .Let UT n betheset ofalluppertriangularmatricesofsize n .Provethat UT n isasubspaceofthevectorspaceofallsquare matricesofsize n M nn ContributedbyRobertBeezerSolution[306] Version2.02

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SubsectionS.SOLSolutions306 SubsectionSOL Solutions C20 ContributedbyRobertBeezerStatement[304] Thequestionisif p canbewrittenasalinearcombinationofthevectorsin W .Tocheckthis,weset p equaltoalinearcombinationandmassagewiththedenitionsofvectoradditionandscalarmultiplication thatwegetwith P 3 ExampleVSP[281] p x = a 1 x 3 + x 2 + x + a 2 x 3 +2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6+ a 3 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 3 +6 x +4= a 1 + a 2 x 3 + a 1 + a 3 x 2 + a 1 +2 a 2 x + )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 a 3 Equatingcoecientsofequalpowersof x ,wegetthesystemofequations, a 1 + a 2 =1 a 1 + a 3 =0 a 1 +2 a 2 =6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 a 3 =4 Theaugmentedmatrixofthissystemofequationsrow-reducesto 2 6 6 6 4 1 000 0 1 00 00 1 0 000 1 3 7 7 7 5 Thereisaleading1inthelastcolumn,soTheoremRCLS[53]impliesthatthesystemisinconsistent.So thereisnowayfor p togainmembershipin W ,so p 62 W C21 ContributedbyRobertBeezerStatement[304] Inordertobelongto W ,wemustbeabletoexpress C asalinearcombinationoftheelementsinthe spanningsetof W .Sowebeginwithsuchanexpression,usingtheunknowns a;b;c forthescalarsinthe linearcombination. C = )]TJ/F15 10.9091 Tf 8.484 0 Td [(33 6 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 = a 21 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 + b 40 23 + c )]TJ/F15 10.9091 Tf 8.484 0 Td [(31 21 Massagingtheright-handside,accordingtothedenitionofthevectorspaceoperationsin M 22 Example VSM[281],wendthematrixequality, )]TJ/F15 10.9091 Tf 8.485 0 Td [(33 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 = 2 a +4 b )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 ca + c 3 a +2 b +2 c )]TJ/F21 10.9091 Tf 8.485 0 Td [(a +3 b + c Matrixequalityallowsustoformasystemoffourequationsinthreevariables,whoseaugmentedmatrix row-reducesasfollows, 2 6 6 4 24 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1013 3226 )]TJ/F15 10.9091 Tf 8.485 0 Td [(131 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 7 7 5 RREF )443()223()222()443(! 2 6 6 4 1 002 0 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 00 1 1 0000 3 7 7 5 SincethissystemofequationsisconsistentTheoremRCLS[53],asolutionwillprovidevaluesfor a;b and c thatallowustorecognize C asanelementof W Version2.02

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SubsectionS.SOLSolutions307 M20 ContributedbyRobertBeezerStatement[304] Themembershipcriteriafor Z isasinglelinearequation,whichcomprisesahomogeneoussystemof equations.Assuch,wecanrecognize Z asthesolutionstothissystem,andtherefore Z isanullspace. Specically, Z = N \002 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 .EverynullspaceisasubspacebyTheoremNSMS[296]. AlessdirectsolutionappealstoTheoremTSS[293]. First,wewanttobecertain Z isnon-empty.Thezerovectorof C 3 0 = 2 4 0 0 0 3 5 ,isagoodcandidate, sinceifitfailstobein Z ,wewillknowthat Z is not avectorspace.Checkthat 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(+5=0 sothat 0 2 Z Suppose x = 2 4 x 1 x 2 x 3 3 5 and y = 2 4 y 1 y 2 y 3 3 5 arevectorsfrom Z .Thenweknowthatthesevectorscannotbe totallyarbitrary,theymusthavegainedmembershipin Z byvirtueofmeetingthemembershiptest.For example,weknowthat x mustsatisfy4 x 1 )]TJ/F21 10.9091 Tf 11.156 0 Td [(x 2 +5 x 3 =0while y mustsatisfy4 y 1 )]TJ/F21 10.9091 Tf 11.156 0 Td [(y 2 +5 y 3 =0.Our secondcriteriaasksthequestion,is x + y 2 Z ?Noticerstthat x + y = 2 4 x 1 x 2 x 3 3 5 + 2 4 y 1 y 2 y 3 3 5 = 2 4 x 1 + y 1 x 2 + y 2 x 3 + y 3 3 5 andwecantestthisvectorformembershipin Z asfollows, 4 x 1 + y 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x 2 + y 2 +5 x 3 + y 3 =4 x 1 +4 y 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(y 2 +5 x 3 +5 y 3 = x 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 2 +5 x 3 + y 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(y 2 +5 y 3 =0+0 x 2 Z; y 2 Z =0 andbythiscomputationweseethat x + y 2 Z If isascalarand x 2 Z ,isitalwaystruethat x 2 Z ?Tocheckourthirdcriteria,weexamine x = 2 4 x 1 x 2 x 3 3 5 = 2 4 x 1 x 2 x 3 3 5 andwecantestthisvectorformembershipin Z with 4 x 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [( x 2 +5 x 3 = x 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 2 +5 x 3 = 0 x 2 Z =0 andweseethatindeed x 2 Z .WiththethreeconditionsofTheoremTSS[293]fullled,wecanconclude that Z isasubspaceof C 3 T20 ContributedbyRobertBeezerStatement[304] ApplyTheoremTSS[293]. First,thezerovectorof M nn isthezeromatrix, O ,whoseentriesareallzeroDenitionZM[185]. Thismatrixthenmeetstheconditionthat[ O ] ij =0for i>j andsoisanelementof UT n Version2.02

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SubsectionS.SOLSolutions308 Suppose A;B 2 UT n .Is A + B 2 UT n ?Weexaminetheentriesof A + B below"thediagonal.That is,inthefollowing,assumethat i>j [ A + B ] ij =[ A ] ij +[ B ] ij DenitionMA[182] =0+0 A;B 2 UT n =0 whichqualies A + B formembershipin UT n Suppose 2 C and A 2 UT n .Is A 2 UT n ?Weexaminetheentriesof A below"thediagonal. Thatis,inthefollowing,assumethat i>j [ A ] ij = [ A ] ij DenitionMSM[183] = 0 A 2 UT n =0 whichqualies A formembershipin UT n HavingfullledthethreeconditionsofTheoremTSS[293]weseethat UT n isasubspaceof M nn Version2.02

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SectionLISSLinearIndependenceandSpanningSets309 SectionLISS LinearIndependenceandSpanningSets Avectorspaceisdenedasasetwithtwooperations,meetingtenpropertiesDenitionVS[279].Just asthedenitionofspanofasetofvectorsonlyrequiredknowinghowtoaddvectorsandhowtomultiply vectorsbyscalars,soitiswithlinearindependence.Adenitionofalinearindependentsetofvectorsin anarbitraryvectorspaceonlyrequiresknowinghowtoformlinearcombinationsandequatingthesewith thezerovector.SinceeveryvectorspacemusthaveazerovectorPropertyZ[280],wealwayshavea zerovectoratourdisposal. Inthissectionwewillalsoputatwistonthenotionofthespanofasetofvectors.Ratherthan beginningwithasetofvectorsandcreatingasubspacethatisthespan,wewillinsteadbeginwitha subspaceandlookforasetofvectorswhosespanequalsthesubspace. Thecombinationoflinearindependenceandspanningwillbeveryimportantgoingforward. SubsectionLI LinearIndependence OurpreviousdenitionoflinearindependenceDenitionLI[308]employedarelationoflineardependence thatwasalinearcombinationononesideofanequalityandazerovectorontheotherside.Asa linearcombinationinavectorspaceDenitionLC[297]dependsonlyonvectoradditionandscalar multiplication,andeveryvectorspacemusthaveazerovectorPropertyZ[280],wecanextendour denitionoflinearindependencefromthesettingof C m tothesettingofageneralvectorspace V with almostnochanges.ComparethesenexttwodenitionswithDenitionRLDCV[132]andDenitionLICV [132]. DenitionRLD RelationofLinearDependence Supposethat V isavectorspace.Givenasetofvectors S = f u 1 ; u 2 ; u 3 ;:::; u n g ,anequationofthe form 1 u 1 + 2 u 2 + 3 u 3 + + n u n = 0 isa relationoflineardependence on S .Ifthisequationisformedinatrivialfashion,i.e. i =0, 1 i n ,thenwesayitisa trivialrelationoflineardependence on S 4 DenitionLI LinearIndependence Supposethat V isavectorspace.Thesetofvectors S = f u 1 ; u 2 ; u 3 ;:::; u n g from V is linearly dependent ifthereisarelationoflineardependenceon S thatisnottrivial.Inthecasewherethe only relationoflineardependenceon S isthetrivialone,then S isa linearlyindependent setofvectors. 4 Noticetheemphasisonthewordonly."Thismightremindyouofthedenitionofanonsingular matrix,whereifthematrixisemployedasthecoecientmatrixofahomogeneoussystemthenthe only solutionisthe trivial one. ExampleLIP4 Linearindependencein P 4 Inthevectorspaceofpolynomialswithdegree4orless, P 4 ExampleVSP[281]considertheset S = 2 x 4 +3 x 3 +2 x 2 )]TJ/F21 10.9091 Tf 10.91 0 Td [(x +10 ; )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 3 + x 2 +5 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 ; 2 x 4 + x 3 +10 x 2 +17 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 : Version2.02

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SubsectionLISS.LILinearIndependence310 Isthissetofvectorslinearlyindependentordependent?Considerthat 3 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(2 x 4 +3 x 3 +2 x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x +10 +4 )]TJ/F24 10.9091 Tf 5 -8.836 Td [()]TJ/F21 10.9091 Tf 8.485 0 Td [(x 4 )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 x 3 + x 2 +5 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(8 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(2 x 4 + x 3 +10 x 2 +17 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 =0 x 4 +0 x 3 +0 x 2 +0 x +0= 0 ThisisanontrivialrelationoflineardependenceDenitionRLD[308]ontheset S andsoconvincesus that S islinearlydependentDenitionLI[308]. Now,Ihearyousay,Wheredid those scalarscomefrom?"Donotworryaboutthatrightnow,just besureyouunderstandwhytheaboveexplanationissucienttoprovethat S islinearlydependent.The remainderoftheexamplewilldemonstratehowwemightndthesescalarsiftheyhadnotbeenprovided soreadily.Let'slookatanothersetofvectorspolynomialsfrom P 4 .Let T = 3 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 3 +4 x 2 +6 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x 4 +1 x 3 +0 x 2 +4 x +2 ; 4 x 4 +5 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 +3 x +1 ; 2 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x 3 +4 x 2 +2 x +1 Supposewehavearelationoflineardependenceonthisset, 0 =0 x 4 +0 x 3 +0 x 2 +0 x +0 = 1 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(3 x 4 )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 x 3 +4 x 2 +6 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 + 2 )]TJ/F24 10.9091 Tf 5 -8.836 Td [()]TJ/F15 10.9091 Tf 8.485 0 Td [(3 x 4 +1 x 3 +0 x 2 +4 x +2 + 3 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(4 x 4 +5 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 +3 x +1 + 4 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(2 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x 3 +4 x 2 +2 x +1 Usingourdenitionsofvectoradditionandscalarmultiplicationin P 4 ExampleVSP[281],wearriveat, 0 x 4 +0 x 3 +0 x 2 +0 x +0= 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 +4 3 +2 4 x 4 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 + 2 +5 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 4 x 3 + 1 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 +4 4 x 2 + 1 +4 2 +3 3 +2 4 x + )]TJ/F21 10.9091 Tf 8.485 0 Td [( 1 +2 2 + 3 + 4 : Equatingcoecients,wearriveatthehomogeneoussystemofequations, 3 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 +4 3 +2 4 =0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 + 2 +5 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 4 =0 4 1 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 +4 4 =0 6 1 +4 2 +3 3 +2 4 =0 )]TJ/F21 10.9091 Tf 8.485 0 Td [( 1 +2 2 + 3 + 4 =0 Weformthecoecientmatrixofthishomogeneoussystemofequationsandrow-reducetond 2 6 6 6 6 6 4 1 000 0 1 00 00 1 0 000 1 0000 3 7 7 7 7 7 5 WeexpectedthesystemtobeconsistentTheoremHSC[62]andsocancompute n )]TJ/F21 10.9091 Tf 10.936 0 Td [(r =4 )]TJ/F15 10.9091 Tf 10.936 0 Td [(4=0and TheoremCSRN[54]tellsusthatthesolutionisunique.Sincethisisahomogeneoussystem,thisunique solutionisthetrivialsolutionDenitionTSHSE[62], 1 =0, 2 =0, 3 =0, 4 =0.SobyDenition LI[308]theset T islinearlyindependent. Afewobservations.Ifwehaddiscoveredinnitelymanysolutions,thenwecouldhaveusedoneof thenon-trivialonestoprovidealinearcombinationinthemannerweusedtoshowthat S waslinearly dependent.Itisimportanttorealizethatitisnotinterestingthatwecancreatearelationoflinear dependencewithzeroscalars|wecan always dothat|butthatfor T ,thisisthe only waytocreatea Version2.02

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SubsectionLISS.LILinearIndependence311 relationoflineardependence.Itwasnoaccidentthatwearrivedatahomogeneoussystemofequations inthisexample,itisrelatedtoouruseofthezerovectorindeningarelationoflineardependence.Itis easytopresentaconvincingstatementthatasetislinearlydependentjustexhibitanontrivialrelationof lineardependencebutaconvincingstatementoflinearindependencerequiresdemonstratingthatthereis norelationoflineardependenceotherthanthetrivialone.NoticehowwereliedontheoremsfromChapter SLE[2]toprovidethisdemonstration.Whew!There'salotgoingoninthisexample.Spendsometime withit,we'llbewaitingpatientlyrightherewhenyougetback. ExampleLIM32 Linearindependencein M 32 Considerthetwosetsofvectors R and S fromthevectorspaceofall3 2matrices, M 32 ExampleVSM [281] R = 8 < : 2 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 14 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 3 5 ; 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 3 5 ; 2 4 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 3 5 ; 2 4 79 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 25 3 5 9 = ; S = 8 < : 2 4 20 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 13 3 5 ; 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(40 )]TJ/F15 10.9091 Tf 8.485 0 Td [(22 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 3 5 ; 2 4 11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 24 3 5 ; 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(53 )]TJ/F15 10.9091 Tf 8.485 0 Td [(107 20 3 5 9 = ; Onesetislinearlyindependent,theotherisnot.Whichiswhich?Let'sexamine R rst.Buildageneric relationoflineardependenceDenitionRLD[308], 1 2 4 3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 14 6 )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 3 5 + 2 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 3 5 + 3 2 4 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 3 5 + 4 2 4 79 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 25 3 5 = 0 Massagingtheleft-handsidewithourdenitionsofvectoradditionandscalarmultiplicationin M 32 ExampleVSM[281]weobtain, 2 4 3 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 +6 3 +7 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 +3 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 3 +9 4 1 1 +1 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [( 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 4 4 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 4 6 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 +7 3 +2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 3 +5 4 3 5 = 2 4 00 00 00 3 5 UsingourdenitionofmatrixequalityDenitionME[182]andequatingcorrespondingentriesweget thehomogeneoussystemofsixequationsinfourvariables, 3 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 +6 3 +7 4 =0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 +3 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 3 +9 4 =0 1 1 +1 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [( 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 4 =0 4 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 4 =0 6 1 )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 2 +7 3 +2 4 =0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 1 )]TJ/F15 10.9091 Tf 10.91 0 Td [(6 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 3 +5 4 =0 Formthecoecientmatrixofthishomogeneoussystemandrow-reducetoobtain 2 6 6 6 6 6 6 6 4 1 000 0 1 00 00 1 0 000 1 0000 0000 3 7 7 7 7 7 7 7 5 Version2.02

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SubsectionLISS.LILinearIndependence312 Analyzingthismatrixweareledtoconcludethat 1 =0, 2 =0, 3 =0, 4 =0.Thismeansthereis only atrivialrelationoflineardependenceonthevectorsof R andsowecall R alinearlyindependentset DenitionLI[308]. Soitmustbethat S islinearlydependent.Let'sseeifwecanndanon-trivialrelationoflinear dependenceon S .Wewillbeginaswith R ,byconstructingarelationoflineardependenceDenition RLD[308]withunknownscalars, 1 2 4 20 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 13 3 5 + 2 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(40 )]TJ/F15 10.9091 Tf 8.485 0 Td [(22 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 3 5 + 3 2 4 11 )]TJ/F15 10.9091 Tf 8.484 0 Td [(21 24 3 5 + 4 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(53 )]TJ/F15 10.9091 Tf 8.485 0 Td [(107 20 3 5 = 0 Massagingtheleft-handsidewithourdenitionsofvectoradditionandscalarmultiplicationin M 32 ExampleVSM[281]weobtain, 2 4 2 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 2 + 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 4 3 +3 4 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(10 4 )]TJ/F21 10.9091 Tf 8.485 0 Td [( 1 +2 2 + 3 +7 4 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 +2 3 +2 4 3 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 2 +4 3 3 5 = 2 4 00 00 00 3 5 UsingourdenitionofmatrixequalityDenitionME[182]andequatingcorrespondingentriesweget thehomogeneoussystemofsixequationsinfourvariables, 2 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 2 + 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 4 =0 + 3 +3 4 =0 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(10 4 =0 )]TJ/F21 10.9091 Tf 8.485 0 Td [( 1 +2 2 + 3 +7 4 =0 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 2 +2 3 +2 4 =0 3 1 )]TJ/F15 10.9091 Tf 10.91 0 Td [(6 2 +4 3 =0 Formthecoecientmatrixofthishomogeneoussystemandrow-reducetoobtain 2 6 6 6 6 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 00 1 3 0000 0000 0000 0000 3 7 7 7 7 7 7 5 Analyzingthisweseethatthesystemisconsistentweexpectedthissincethesystemishomogeneous, TheoremHSC[62]andhas n )]TJ/F21 10.9091 Tf 11.287 0 Td [(r =4 )]TJ/F15 10.9091 Tf 11.287 0 Td [(2=2freevariables,namely 2 and 4 .Thismeansthereare innitelymanysolutions,andinparticular,wecanndanon-trivialsolution,solongaswedonotpickall ofourfreevariablestobezero.Themerepresenceofanontrivialsolutionforthesescalarsisenoughto concludethat S isalinearlydependentsetDenitionLI[308].Butlet'sgoaheadandexplicitlyconstruct anon-trivialrelationoflineardependence. Choose 2 =1and 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Thereisnothingspecialaboutthischoice,thereareinnitelymany possibilities,someeasier"thanthisone,justavoidpickingbothvariablestobezero.Thenwendthe correspondingdependentvariablestobe 1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2and 3 =3.Sotherelationoflineardependence, )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 4 20 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 13 3 5 + 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(40 )]TJ/F15 10.9091 Tf 8.485 0 Td [(22 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 3 5 + 2 4 11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 24 3 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(53 )]TJ/F15 10.9091 Tf 8.485 0 Td [(107 20 3 5 = 2 4 00 00 00 3 5 Version2.02

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SubsectionLISS.SSSpanningSets313 isaniron-claddemonstrationthat S islinearlydependent.Canyouconstructanothersuchdemonstration? ExampleLIC Linearlyindependentsetinthecrazyvectorspace Istheset R = f ; 0 ; ; 3 g linearlyindependentinthecrazyvectorspace C ExampleCVS[283]?We beginwithanarbitraryrelationoflinearindependenceon R 0 = a 1 ; 0+ a 2 ; 3DenitionRLD[308] andthenmassageittoapointwherewecanapplythedenitionofequalityin C .Recallthedenitions ofvectoradditionandscalarmultiplicationin C arenotwhatyouwouldexpect. )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1= 0 ExampleCVS[283] = a 1 ; 0+ a 2 ; 3DenitionRLD[308] = a 1 + a 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ; 0 a 1 + a 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+ a 2 + a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ; 3 a 2 + a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1ExampleCVS[283] = a 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ;a 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+ a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ; 4 a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = a 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+7 a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+1 ;a 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+4 a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+1ExampleCVS[283] = a 1 +7 a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ;a 1 +4 a 2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(1 Equalityin C ExampleCVS[283]thenyieldsthetwoequations, 2 a 1 +7 a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1= )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 a 1 +4 a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1= )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 whichbecomesthehomogeneoussystem 2 a 1 +7 a 2 =0 a 1 +4 a 2 =0 Sincethecoecientmatrixofthissystemisnonsingularcheckthis!thesystemhasonlythetrivial solution a 1 = a 2 =0.ByDenitionLI[308]theset R islinearlyindependent.Noticethateventhoughthe zerovectorof C isnotwhatwemightrstsuspected,aquestionaboutlinearindependencestillconcludes withaquestionaboutahomogeneoussystemofequations.Hmmm. SubsectionSS SpanningSets Inavectorspace V ,supposewearegivenasetofvectors S V .Thenwecanimmediatelyconstructa subspace, h S i ,usingDenitionSS[298]andthenbeassuredbyTheoremSSS[298]thattheconstruction doesprovideasubspace.Wenowturnthesituationupside-down.Supposewearerstgivenasubspace W V .Canwendaset S sothat h S i = W ?Typically W isinniteandwearesearchingforanite setofvectors S thatwecancombineinlinearcombinationsandbuild"allof W Iliketothinkof S astherawmaterialsthataresucientfortheconstructionof W .Ifyouhave nails,lumber,wire,copperpipe,drywall,plywood,carpet,shingles,paintandafewotherthings,then youcancombinetheminmanydierentwaystocreateahouseorinnitelymanydierenthousesfor thatmatter.Afast-foodrestaurantmayhavebeef,chicken,beans,cheese,tortillas,tacoshellsandhot sauceandfromthissmalllistofingredientsbuildawidevarietyofitemsforsale.Ormaybeabetter Version2.02

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SubsectionLISS.SSSpanningSets314 analogycomesfromBenCordes|theadditiveprimarycolorsred,greenandbluecanbecombinedto createmanydierentcolorsbyvaryingtheintensityofeach.Theintensityislikeascalarmultiple,and thecombinationofthethreeintensitiesislikevectoraddition.Thethreeindividualcolors,red,greenand blue,aretheelementsofthespanningset. Becausewewillusetermslikespannedby"andspanningset,"thereisthepotentialforconfusion withthespan."Comebackandrereadtherstparagraphofthissubsectionwheneveryouareuncertain aboutthedierence.Here'stheworkingdenition. DenitionTSVS ToSpanaVectorSpace Suppose V isavectorspace.Asubset S of V isa spanningset for V if h S i = V .Inthiscase,wealso say S spans V 4 Thedenitionofaspanningsetrequiresthattwosetssubspacesactuallybeequal.If S isasubsetof V ,then h S i V ,always.Thusitisusuallyonlynecessarytoprovethat V h S i .Nowwouldbeagood timetoreviewDenitionSE[684]. ExampleSSP4 Spanningsetin P 4 InExampleSP4[294]weshowedthat W = f p x j p 2 P 4 ;p =0 g isasubspaceof P 4 ,thevectorspaceofpolynomialswithdegreeatmost4ExampleVSP[281].Inthis example,wewillshowthattheset S = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 ;x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +4 ;x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 2 +12 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 ;x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 x 3 +24 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(32 x +16 isaspanningsetfor W .Todothis,werequirethat W = h S i .Thisisanequalityofsets.Wecancheck thateverypolynomialin S has x =2asarootandtherefore S W .Since W isclosedunderaddition andscalarmultiplication, h S i W also. Soitremainstoshowthat W h S i DenitionSE[684].Todothis,beginbychoosinganarbitrary polynomialin W ,say r x = ax 4 + bx 3 + cx 2 + dx + e 2 W .Thispolynomialisnotasarbitraryasitwould appear,sincewealsoknowitmusthave x =2asaroot.Thistranslatesto 0= a 4 + b 3 + c 2 + d + e =16 a +8 b +4 c +2 d + e asaconditionon r Wewishtoshowthat r isapolynomialin h S i ,thatis,wewanttoshowthat r canbewrittenasa linearcombinationofthevectorspolynomialsin S .Solet'stry. r x = ax 4 + bx 3 + cx 2 + dx + e = 1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2+ 2 )]TJ/F21 10.9091 Tf 5 -8.837 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +4 + 3 )]TJ/F21 10.9091 Tf 5 -8.837 Td [(x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 2 +12 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 + 4 )]TJ/F21 10.9091 Tf 5 -8.836 Td [(x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 x 3 +24 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(32 x +16 = 4 x 4 + 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 4 x 3 + 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 3 +24 2 x 2 + 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 2 +12 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(32 4 x + )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 +4 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 3 +16 4 Equatingcoecientsvectorequalityin P 4 givesthesystemofveequationsinfourvariables, 4 = a 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 4 = b 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 3 +24 2 = c 1 )]TJ/F15 10.9091 Tf 10.91 0 Td [(4 2 +12 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(32 4 = d Version2.02

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SubsectionLISS.SSSpanningSets315 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 1 +4 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 3 +16 4 = e Anysolutiontothissystemofequationswillprovidethelinearcombinationweneedtodetermineif r 2h S i butweneedtobeconvincedthereisasolutionforanyvaluesof a;b;c;d;e thatqualify r tobeamember of W .Sothequestionis:isthissystemofequationsconsistent?Wewillformtheaugmentedmatrix,and row-reduce.Weprobablyneedtodothisbyhand,sincethematrixissymbolic|reversingtheorderof therstfourrowsisthebestwaytostart.Weobtainamatrixinreducedrow-echelonform 2 6 6 6 6 6 4 1 00032 a +12 b +4 c + d 0 1 0024 a +6 b + c 00 1 08 a + b 000 1 a 000016 a +8 b +4 c +2 d + e 3 7 7 7 7 7 5 = 2 6 6 6 6 6 4 1 00032 a +12 b +4 c + d 0 1 0024 a +6 b + c 00 1 08 a + b 000 1 a 00000 3 7 7 7 7 7 5 Foryourresultstomatchourrstmatrix,youmaynditnecessarytomultiplythenalrowofyour row-reducedmatrixbytheappropriatescalar,and/oraddmultiplesofthisrowtosomeoftheotherrows. Toobtainthesecondversionofthematrix,thelastentryofthelastcolumnhasbeensimpliedtozero accordingtotheoneconditionwewereabletoimposeonanarbitrarypolynomialfrom W .Sowith noleading1'sinthelastcolumn,TheoremRCLS[53]tellsusthissystemisconsistent.Therefore, any polynomialfrom W canbewrittenasalinearcombinationofthepolynomialsin S ,so W h S i .Therefore, W = h S i and S isaspanningsetfor W byDenitionTSVS[313]. Noticethatanalternativetorow-reducingtheaugmentedmatrixbyhandwouldbetoappealto TheoremFS[263]byexpressingthecolumnspaceofthecoecientmatrixasanullspace,andthen verifyingthattheconditionon r guaranteesthat r isinthecolumnspace,thusimplyingthatthesystem isalwaysconsistent.Giveitatry,we'llwait.Thishasbeenacomplicatedexample,butworthstudying carefully. Givenasubspaceandasetofvectors,asinExampleSSP4[313]itcantakesomeworktodetermine thatthesetactuallyisaspanningset.Anevenharderproblemistobeconfrontedwithasubspaceand requiredtoconstructaspanningsetwithnoguidance.Wewillnowworkanexampleofthisavor,but someofthestepswillbeunmotivated.Fortunately,wewillhavesomebettertoolsforthistypeofproblem lateron. ExampleSSM22 Spanningsetin M 22 Inthespaceofall2 2matrices, M 22 considerthesubspace Z = ab cd j a +3 b )]TJ/F21 10.9091 Tf 10.909 0 Td [(c )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 d =0 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 a )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 b +3 c +14 d =0 andndaspanningsetfor Z Weneedtoconstructalimitednumberofmatricesin Z sothateverymatrixin Z canbeexpressedas alinearcombinationofthislimitednumberofmatrices.Supposethat B = ab cd isamatrixin Z .Then wecanformacolumnvectorwiththeentriesof B andwrite 2 6 6 4 a b c d 3 7 7 5 2N 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6314 Version2.02

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SubsectionLISS.SSSpanningSets316 Row-reducingthismatrixandapplyingTheoremREMES[28]weobtaintheequivalentstatement, 2 6 6 4 a b c d 3 7 7 5 2N 1 30 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 00 1 4 Wecanthenexpressthesubspace Z inthefollowingequalforms, Z = ab cd j a +3 b )]TJ/F21 10.9091 Tf 10.909 0 Td [(c )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 d =0 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 a )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 b +3 c +14 d =0 = ab cd j a +3 b )]TJ/F21 10.9091 Tf 10.909 0 Td [(d =0 ;c +4 d =0 = ab cd j a = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 b + d;c = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 d = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 b + db )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 dd j b;d 2 C = )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 bb 00 + d 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 dd j b;d 2 C = b )]TJ/F15 10.9091 Tf 8.484 0 Td [(31 00 + d 10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(41 j b;d 2 C = )]TJ/F15 10.9091 Tf 8.485 0 Td [(31 00 ; 10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(41 Sotheset Q = )]TJ/F15 10.9091 Tf 8.485 0 Td [(31 00 ; 10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(41 spans Z byDenitionTSVS[313]. ExampleSSC Spanningsetinthecrazyvectorspace InExampleLIC[312]wedeterminedthattheset R = f ; 0 ; ; 3 g islinearlyindependentinthecrazy vectorspace C ExampleCVS[283].Wenowshowthat R isaspanningsetfor C Givenanarbitraryvector x;y 2 C wedesiretoshowthatitcanbewrittenasalinearcombination oftheelementsof R .Inotherwords,aretherescalars a 1 and a 2 sothat x;y = a 1 ; 0+ a 2 ; 3 Wewillactasifthisequationistrueandtrytodeterminejustwhat a 1 and a 2 wouldbeasfunctionsof x and y x;y = a 1 ; 0+ a 2 ; 3 = a 1 + a 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ; 0 a 1 + a 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+ a 2 + a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ; 3 a 2 + a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1Scalarmultin C = a 1 )]TJ/F15 10.9091 Tf 10.91 0 Td [(1 ;a 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+ a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ; 4 a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = a 1 )]TJ/F15 10.9091 Tf 10.91 0 Td [(1+7 a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+1 ;a 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+4 a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+1Additionin C = a 1 +7 a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ;a 1 +4 a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 Equalityin C thenyieldsthetwoequations, 2 a 1 +7 a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1= x Version2.02

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SubsectionLISS.VRVectorRepresentation317 a 1 +4 a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1= y whichbecomesthelinearsystemwithamatrixrepresentation 27 14 a 1 a 2 = x +1 y +1 Thecoecientmatrixofthissystemisnonsingular,henceinvertibleTheoremNI[228],andwecan employitsinversetondasolutionTheoremTTMI[214],TheoremSNCM[229], a 1 a 2 = 27 14 )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 x +1 y +1 = 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F15 10.9091 Tf 8.484 0 Td [(12 x +1 y +1 = 4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 )]TJ/F21 10.9091 Tf 8.484 0 Td [(x +2 y +1 Wecouldchasethroughtheaboveimplicationsbackwardsandtaketheexistenceofthesesolutionsas sucientevidencefor R beingaspanningsetfor C .Instead,letusviewtheaboveassimplyscratchwork andnowgetseriouswithasimpledirectproofthat R isaspanningset.Ready?Suppose x;y isany vectorfrom C ,thencomputethefollowinglinearcombinationusingthedenitionsoftheoperationsin C x )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 ; 0+ )]TJ/F21 10.9091 Tf 8.485 0 Td [(x +2 y +1 ; 3 = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(3+ x )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ; 0 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(3+ x )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+ )]TJ/F21 10.9091 Tf 8.485 0 Td [(x +2 y +1+ )]TJ/F21 10.9091 Tf 8.485 0 Td [(x +2 y +1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ; 3 )]TJ/F21 10.9091 Tf 8.485 0 Td [(x +2 y +1+ )]TJ/F21 10.9091 Tf 8.485 0 Td [(x +2 y +1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(14 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 ; 4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(4+ )]TJ/F15 10.9091 Tf 8.484 0 Td [(7 x +14 y +6 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 x +8 y +3 = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(14 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(7+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 x +14 y +6+1 ; x )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 y )]TJ/F15 10.9091 Tf 10.91 0 Td [(4+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 x +8 y +3+1 = x;y Thisnalsequenceofcomputationsin C issucienttodemonstratethatanyelementof C can bewritten orexpressedasalinearcombinationofthetwovectorsin R ,so C h R i .Sincethereverseinclusion h R i C istriviallytrue, C = h R i andwesay R spans C DenitionTSVS[313].Noticethatthis demonstrationisnomoreorlessvalidifwehidefromthereaderourscratchworkthatsuggested a 1 = 4 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 y )]TJ/F15 10.9091 Tf 10.909 0 Td [(3and a 2 = )]TJ/F21 10.9091 Tf 8.485 0 Td [(x +2 y +1. SubsectionVR VectorRepresentation InChapterR[530]wewilltakeupthematterofrepresentationsfully,whereTheoremVRRB[317]will becriticalforDenitionVR[530].Wewillnowmotivateandproveacriticaltheoremthattellsushow torepresent"avector.Thistheoremcouldwait,butworkingwithitnowwillprovidesomeextrainsight intothenatureoflinearlyindependentspanningsets.Firstanexample,thenthetheorem. ExampleAVR Avectorrepresentation Considertheset S = 8 < : 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 5 1 3 5 ; 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 5 0 3 5 ; 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 7 4 3 5 9 = ; fromthevectorspace C 3 .Let A bethematrixwhosecolumnsaretheset S ,andverifythat A isnonsingular. ByTheoremNMLIC[138]theelementsof S formalinearlyindependentset.Supposethat b 2 C 3 .Then LS A; b hasauniquesolutionTheoremNMUS[74]andhenceisconsistent.ByTheoremSLSLC[93], b 2h S i .Since b isarbitrary,thisisenoughtoshowthat h S i = C 3 ,andtherefore S isaspanningsetfor Version2.02

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SubsectionLISS.VRVectorRepresentation318 C 3 DenitionTSVS[313].ThissetcomesfromthecolumnsofthecoecientmatrixofArchetypeB [707]. Nowexaminethesituationforaparticularchoiceof b ,say b = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(33 24 5 3 5 .Because S isaspanningset for C 3 ,weknowwecanwrite b asalinearcombinationofthevectorsin S 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(33 24 5 3 5 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 5 1 3 5 + 2 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 5 0 3 5 + 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 7 4 3 5 : Thenonsingularityofthematrix A tellsthatthescalarsinthislinearcombinationareunique.More precisely,itisthelinearindependenceof S thatprovidestheuniqueness.Wewillrefertothescalars a 1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3, a 2 =5, a 3 =2asarepresentationof b relativeto S ."Inotherwords,oncewesettleon S as alinearlyindependentsetthatspans C 3 ,thevector b isrecoverablejustbyknowingthescalars a 1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3, a 2 =5, a 3 =2usethesescalarsinalinearcombinationofthevectorsin S .Thisisallanillustrationof thefollowingimportanttheorem,whichweproveinthesettingofageneralvectorspace. TheoremVRRB VectorRepresentationRelativetoaBasis Supposethat V isavectorspaceand B = f v 1 ; v 2 ; v 3 ;:::; v m g isalinearlyindependentsetthatspans V .Let w beanyvectorin V .Thenthereexist unique scalars a 1 ;a 2 ;a 3 ;:::;a m suchthat w = a 1 v 1 + a 2 v 2 + a 3 v 3 + + a m v m : Proof That w canbewrittenasalinearcombinationofthevectorsin B followsfromthespanning propertyofthesetDenitionTSVS[313].Thisisgood,butnotthemeatofthistheorem.Wenowknow thatforanychoiceofthevector w thereexist some scalarsthatwillcreate w asalinearcombinationof thebasisvectors.Therealquestionis:Isthere more thanonewaytowrite w asalinearcombinationof f v 1 ; v 2 ; v 3 ;:::; v m g ?Arethescalars a 1 ;a 2 ;a 3 ;:::;a m unique?TechniqueU[693] Assumetherearetwowaystoexpress w asalinearcombinationof f v 1 ; v 2 ; v 3 ;:::; v m g .Inother wordsthereexistscalars a 1 ;a 2 ;a 3 ;:::;a m and b 1 ;b 2 ;b 3 ;:::;b m sothat w = a 1 v 1 + a 2 v 2 + a 3 v 3 + + a m v m w = b 1 v 1 + b 2 v 2 + b 3 v 3 + + b m v m : Thennoticethat 0 = w + )]TJ/F36 10.9091 Tf 8.485 0 Td [(w PropertyAI[280] = w + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 w TheoremAISM[287] = a 1 v 1 + a 2 v 2 + a 3 v 3 + + a m v m + )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 b 1 v 1 + b 2 v 2 + b 3 v 3 + + b m v m = a 1 v 1 + a 2 v 2 + a 3 v 3 + + a m v m + )]TJ/F21 10.9091 Tf 8.484 0 Td [(b 1 v 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 2 v 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 3 v 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(::: )]TJ/F21 10.9091 Tf 10.909 0 Td [(b m v m PropertyDVA[280] = a 1 )]TJ/F21 10.9091 Tf 10.91 0 Td [(b 1 v 1 + a 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 2 v 2 + a 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 3 v 3 + + a m )]TJ/F21 10.9091 Tf 10.909 0 Td [(b m v m PropertyC[279],PropertyDSA[280] ButthisisarelationoflineardependenceonalinearlyindependentsetofvectorsDenitionRLD[308]! Nowweareusingtheotherassumptionabout B ,that f v 1 ; v 2 ; v 3 ;:::; v m g isalinearlyindependentset. SobyDenitionLI[308]it must happenthatthescalarsareallzero.Thatis, a 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 1 =0 a 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 2 =0 a 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 3 =0 ::: a m )]TJ/F21 10.9091 Tf 10.909 0 Td [(b m =0 Version2.02

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SubsectionLISS.READReadingQuestions319 a 1 = b 1 a 2 = b 2 a 3 = b 3 :::a m = b m : Andsowendthatthescalarsareunique. Thisisaverytypicaluseofthehypothesisthatasetislinearlyindependent|obtainarelationof lineardependenceandthenconcludethatthescalars must allbezero.Theresultofthistheoremtells usthatwecanwriteanyvectorinavectorspaceasalinearcombinationofthevectorsinalinearly independentspanningset,butonlyjust.Thereisonlyenoughrawmaterialinthespanningsettowrite eachvectoronewayasalinearcombination.Sointhissense,wecouldcallalinearlyindependentspanning setaminimalspanningset."Thesesetsaresoimportantthatwewillgivethemasimplernamebasis" andexploretheirpropertiesfurtherinthenextsection. SubsectionREAD ReadingQuestions 1.Isthesetofmatricesbelowlinearlyindependentorlinearlydependentinthevectorspace M 22 ?Why orwhynot? 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(24 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(23 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 ; 09 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 2.Explainthedierencebetweenthefollowingtwousesofthetermspan": a S isasubsetofthevectorspace V andthespanof S isasubspaceof V b W issubspaceofthevectorspace Y and T spans W 3.Theset S = 8 < : 2 4 6 2 1 3 5 ; 2 4 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 3 5 ; 2 4 5 8 2 3 5 9 = ; islinearlyindependentandspans C 3 .Writethevector x = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 2 2 3 5 alinearcombinationoftheelements of S .Howmanywaysaretheretoanswerthisquestion,andwhichtheoremallowsyoutosayso? Version2.02

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SubsectionLISS.EXCExercises320 SubsectionEXC Exercises C20 Inthevectorspaceof2 2matrices, M 22 ,determineiftheset S belowislinearlyindependent. S = 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 13 ; 04 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 ; 42 13 ContributedbyRobertBeezerSolution[321] C21 Inthecrazyvectorspace C ExampleCVS[283],istheset S = f ; 2 ; ; 8 g linearlyindependent? ContributedbyRobertBeezerSolution[321] C22 Inthevectorspaceofpolynomials P 3 ,determineiftheset S islinearlyindependentorlinearly dependent. S = 2+ x )]TJ/F15 10.9091 Tf 10.91 0 Td [(3 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 x 3 ; 1+ x + x 2 +5 x 3 ; 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x 3 ContributedbyRobertBeezerSolution[322] C23 Determineiftheset S = f ; 1 ; ; 3 g islinearlyindependentinthecrazyvectorspace C Example CVS[283]. ContributedbyRobertBeezerSolution[322] C30 InExampleLIM32[310],ndanothernontrivialrelationoflineardependenceonthelinearlydependentsetof3 2matrices, S ContributedbyRobertBeezer C40 Determineiftheset T = x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x +5 ; 4 x 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 2 +5 x; 3 x +2 spansthevectorspaceofpolynomials withdegree4orless, P 4 ContributedbyRobertBeezerSolution[322] C41 Theset W isasubspaceof M 22 ,thevectorspaceofall2 2matrices.Provethat S isaspanning setfor W W = ab cd j 2 a )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 b +4 c )]TJ/F21 10.9091 Tf 10.909 0 Td [(d =0 S = 10 02 ; 01 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; 00 14 ContributedbyRobertBeezerSolution[322] C42 Determineiftheset S = f ; 1 ; ; 3 g spansthecrazyvectorspace C ExampleCVS[283]. ContributedbyRobertBeezerSolution[323] M10 HalfwaythroughExampleSSP4[313],weneedtoshowthatthesystemofequations LS 0 B B B B @ 2 6 6 6 6 4 0001 001 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(624 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(412 )]TJ/F15 10.9091 Tf 8.485 0 Td [(32 )]TJ/F15 10.9091 Tf 8.485 0 Td [(24 )]TJ/F15 10.9091 Tf 8.485 0 Td [(816 3 7 7 7 7 5 ; 2 6 6 6 6 4 a b c d e 3 7 7 7 7 5 1 C C C C A isconsistentforeverychoiceofthevectorofconstantssatisfying16 a +8 b +4 c +2 d + e =0. Expressthecolumnspaceofthecoecientmatrixofthissystemasanullspace,usingTheoremFS [263].FromthisuseTheoremCSCS[237]toestablishthatthesystemisalwaysconsistent.Noticethat Version2.02

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SubsectionLISS.EXCExercises321 thisapproachremovesfromExampleSSP4[313]theneedtorow-reduceasymbolicmatrix. ContributedbyRobertBeezerSolution[323] T40 ProvethefollowingvariationofTheoremEMMVP[196]:Supposethat B = f u 1 ; u 2 ; u 3 ;:::; u n g isabasisfor C n .Supposealsothat A and B are m n matricessuchthat A u i = B u i forevery1 i n Then A = B .CanyoumodifythehypothesisfurtherandobtainageneralizationofTheoremEMMVP [196]? ContributedbyRobertBeezer T50 Supposethat V isavectorspaceand u ; v 2 V aretwovectorsin V .Usethedenitionoflinear independencetoprovethat S = f u ; v g isalinearlydependentsetifandonlyifoneofthetwovectorsis ascalarmultipleoftheother.Provethisdirectlyinthecontextofanabstractvectorspace V ,without simplygivinganupgradedversionofTheoremDLDS[152]forthespecialcaseofjusttwovectors. ContributedbyRobertBeezerSolution[323] Version2.02

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SubsectionLISS.SOLSolutions322 SubsectionSOL Solutions C20 ContributedbyRobertBeezerStatement[319] Beginwitharelationoflineardependenceonthevectorsin S andmassageitaccordingtothedenitions ofvectoradditionandscalarmultiplicationin M 22 O = a 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 13 + a 2 04 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 + a 3 42 13 00 00 = 2 a 1 +4 a 3 )]TJ/F21 10.9091 Tf 8.485 0 Td [(a 1 +4 a 2 +2 a 3 a 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(a 2 + a 3 3 a 1 +2 a 2 +3 a 3 ByourdenitionofmatrixequalityDenitionME[182]wearriveatahomogeneoussystemoflinear equations, 2 a 1 +4 a 3 =0 )]TJ/F21 10.9091 Tf 8.484 0 Td [(a 1 +4 a 2 +2 a 3 =0 a 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(a 2 + a 3 =0 3 a 1 +2 a 2 +3 a 3 =0 Thecoecientmatrixofthissystemrow-reducestothematrix, 2 6 6 4 1 00 0 1 0 00 1 000 3 7 7 5 andfromthisweconcludethattheonlysolutionis a 1 = a 2 = a 3 =0.Sincetherelationoflinear dependenceDenitionRLD[308]istrivial,theset S islinearlyindependentDenitionLI[308]. C21 ContributedbyRobertBeezerStatement[319] Webeginwitharelationoflineardependenceusingunknownscalars a and b .Wewishtoknowifthese scalars must bothbezero.Recallthatthezerovectorin C is )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(1andthatthedenitionsofvector additionandscalarmultiplicationarenotwhatwemightexpect. 0 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 = a ; 2+ b ; 8DenitionRLD[308] = a + a )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ; 2 a + a )]TJ/F15 10.9091 Tf 10.91 0 Td [(1+ b + b )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ; 8 b + b )]TJ/F15 10.9091 Tf 10.909 0 Td [(1Scalarmult.,ExampleCVS[283] = a )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ; 3 a )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+ b )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ; 9 b )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = a )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+3 b )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+1 ; 3 a )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+9 b )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+1Vectoraddition,ExampleCVS[283] = a +3 b )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ; 3 a +9 b )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 Fromthisweobtaintwoequalities,whichcanbeconvertedtoahomogeneoussystemofequations, )]TJ/F15 10.9091 Tf 8.485 0 Td [(1= a +3 b )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 a +3 b =0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1=3 a +9 b )]TJ/F15 10.9091 Tf 10.909 0 Td [(13 a +9 b =0 ThishomogeneoussystemhasasingularcoecientmatrixTheoremSMZD[389],andsohasmorethan justthetrivialsolutionDenitionNM[71].Anynontrivialsolutionwillgiveusanontrivialrelationof lineardependenceon S .So S islinearlydependentDenitionLI[308]. Version2.02

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SubsectionLISS.SOLSolutions323 C22 ContributedbyRobertBeezerStatement[319] BeginwitharelationoflineardependenceDenitionRLD[308], a 1 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(2+ x )]TJ/F15 10.9091 Tf 10.91 0 Td [(3 x 2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(8 x 3 + a 2 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(1+ x + x 2 +5 x 3 + a 3 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x 3 = 0 Massageaccordingtothedenitionsofscalarmultiplicationandvectoradditioninthedenitionof P 3 ExampleVSP[281]andusethezerovectordrothisvectorspace, a 1 + a 2 +3 a 3 + a 1 + a 2 x + )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 a 1 + a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 a 3 x 2 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 a 1 +5 a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 a 3 x 3 =0+0 x +0 x 2 +0 x 3 Thedenitionoftheequalityofpolynomialsallowsustodeducethefollowingfourequations, 2 a 1 + a 2 +3 a 3 =0 a 1 + a 2 =0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 a 1 + a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 a 3 =0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 a 1 +5 a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 a 3 =0 Row-reducingthecoecientmatrixofthishomogeneoussystemleadstotheuniquesolution a 1 = a 2 = a 3 =0.Sotheonlyrelationoflineardependenceon S isthetrivialone,andthisislinearindependence for S DenitionLI[308]. C23 ContributedbyRobertBeezerStatement[319] Notice,ordiscover,thatthefollowinggivesanontrivialrelationoflineardependenceon S in C ,soby DenitionLI[308],theset S islinearlydependent. 2 ; 1+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; 3=7 ; 3+ )]TJ/F15 10.9091 Tf 8.484 0 Td [(9 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(5= )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1= 0 C40 ContributedbyRobertBeezerStatement[319] Thepolynomial x 4 isanelementof P 4 .Canwewritethiselementasalinearcombinationoftheelements of T ?Towit,aretherescalars a 1 a 2 a 3 suchthat x 4 = a 1 )]TJ/F21 10.9091 Tf 5 -8.837 Td [(x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x +5 + a 2 )]TJ/F15 10.9091 Tf 5 -8.837 Td [(4 x 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 2 +5 x + a 3 x +2 Massagingtherightsideofthisequation,accordingtothedenitionsofExampleVSP[281],andthen equatingcoecients,leadstoaninconsistentsystemofequationscheckthis!.Assuch, T isnotaspanning setfor P 4 C41 ContributedbyRobertBeezerStatement[319] Wewanttoshowthat W = h S i DenitionTSVS[313],whichisanequalityofsetsDenitionSE[684]. First,showthat h S i W .Beginbycheckingthateachofthethreematricesin S isamemberofthe set W .Then,since W isavectorspace,theclosurepropertiesPropertyAC[279],PropertySC[279] guaranteethateverylinearcombinationofelementsof S remainsin W Second,showthat W h S i .Wewanttoconvinceourselvesthatanarbitraryelementof W isalinear combinationofelementsof S .Choose x = ab cd 2 W Thevaluesof a;b;c;d arenottotallyarbitrary,sincemembershipin W requiresthat2 a )]TJ/F15 10.9091 Tf 10.137 0 Td [(3 b +4 c )]TJ/F21 10.9091 Tf 10.137 0 Td [(d =0. Now,rewriteasfollows, x = ab cd = ab c 2 a )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 b +4 c 2 a )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 b +4 c )]TJ/F21 10.9091 Tf 10.909 0 Td [(d =0 Version2.02

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SubsectionLISS.SOLSolutions324 = a 0 02 a + 0 b 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 b + 00 c 4 c DenitionMA[182] = a 10 02 + b 01 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 + c 00 14 DenitionMSM[183] 2h S i DenitionSS[298] C42 ContributedbyRobertBeezerStatement[319] Wewilltrytoshowthat S spans C .Let x;y beanarbitraryelementof C andsearchforscalars a 1 and a 2 suchthat x;y = a 1 ; 1+ a 2 ; 3 = a 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ; 2 a 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+ a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ; 4 a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 = a 1 +8 a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 ; 2 a 1 +4 a 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 Equalityin C leadstothesystem 4 a 1 +8 a 2 = x +1 2 a 1 +4 a 2 = y +1 Thissystemhasasingularcoecientmatrixwhosecolumnspaceissimply 2 1 .Soanychoiceof x and y thatcausesthecolumnvector x +1 y +1 tolieoutsidethecolumnspacewillleadtoaninconsistent system,andhencecreateanelement x;y thatisnotinthespanof S .So S doesnotspan C Forexample,choose x =0and y =5,andthenwecanseethat 1 6 62 2 1 andweknowthat ; 5 cannotbewrittenasalinearcombinationofthevectorsin S .Ashortersolutionmightbeginbyasserting that ; 5isnotin h S i andthenestablishingthisclaimalone. M10 ContributedbyRobertBeezerStatement[319] TheoremFS[263]providesthematrix L = 1 1 2 1 4 1 8 1 16 andsoif A denotesthecoecientmatrixofthesystem,then C A = N L .Thesinglehomogeneous equationin LS L; 0 isequivalenttotheconditiononthevectorofconstantsuse a;b;c;d;e asvariables andthenmultiplyby16. T50 ContributedbyRobertBeezerStatement[320] If S islinearlydependent,thentherearescalars and ,notbothzero,suchthat u + v = 0 Supposethat 6 =0,theproofproceedssimilarlyif 6 =0.Now, u =1 u PropertyO[280] = 1 u PropertyMICN[681] = 1 u PropertySMA[280] = 1 u + 0 PropertyZ[280] = 1 u + v )]TJ/F21 10.9091 Tf 10.909 0 Td [( v PropertyAI[280] Version2.02

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SubsectionLISS.SOLSolutions325 = 1 0 )]TJ/F21 10.9091 Tf 10.909 0 Td [( v DenitionLI[308] = 1 )]TJ/F21 10.9091 Tf 8.485 0 Td [( v PropertyZ[280] = )]TJ/F21 10.9091 Tf 8.485 0 Td [( v PropertySMA[280] whichshowsthat u isascalarmultipleof v Supposenowthat u isascalarmultipleof v .Moreprecisely,supposethereisascalar such that u = v .Then )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 u + v = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 u + u = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 u + u PropertyO[280] = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1+1 u PropertyDSA[280] =0 u PropertyAICN[681] = 0 TheoremZSSM[286] Thisisarelationoflinearoflineardependenceon S DenitionRLD[308],whichisnontrivialsinceone ofthescalarsis )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.Therefore S islinearlydependentbyDenitionLI[308]. Becarefulusingthistheorem.Itisonlyapplicabletosetsoftwovectors.Inparticular,lineardependenceinasetofthreeormorevectorscanbemorecomplicatedthanjustonevectorbeingascalar multipleofanother. Version2.02

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SectionBBases326 SectionB Bases Abasisofavectorspaceisoneofthemostusefulconceptsinlinearalgebra.Itoftenprovidesaconcise, nitedescriptionofaninnitevectorspace. SubsectionB Bases Wenowhaveallthetoolsinplacetodeneabasisofavectorspace. DenitionB Basis Suppose V isavectorspace.Thenasubset S V isa basis of V ifitislinearlyindependentandspans V 4 So,abasisisalinearlyindependentspanningsetforavectorspace.Therequirementthattheset spans V insuresthat S hasenoughrawmaterialtobuild V ,whilethelinearindependencerequirement insuresthatwedonothaveanymorerawmaterialthanweneed.AsweshallseesooninSectionD[341], abasisisaminimalspanningset. Youmayhavenoticedthatweusedthetermbasisforsomeofthetitlesofprevioustheoremse.g. TheoremBNS[139],TheoremBCS[239],TheoremBRS[245]andifyourevieweachofthesetheoremsyou willseethattheirconclusionsprovidelinearlyindependentspanningsetsforsetsthatwenowrecognize assubspacesof C m .ExamplesassociatedwiththesetheoremsincludeExampleNSLIL[140],Example CSOCD[240]andExampleIAS[246].Aswewillsee,thesethreetheoremswillcontinuetobepowerful tools,eveninthesettingofmoregeneralvectorspaces. Furthermore,thearchetypescontainanabundanceofbases.Foreachcoecientmatrixofasystem ofequations,andforeacharchetypedenedsimplyasamatrix,thereisabasisforthenullspace, three basesforthecolumnspace,andabasisfortherowspace.Forthisreason,oursubsequentexampleswill concentrateonbasesforvectorspacesotherthan C m .NoticethatDenitionB[325]doesnotpreclude avectorspacefromhavingmanybases,andthisisthecase,ashintedabovebythestatementthatthe archetypescontainthreebasesforthecolumnspaceofamatrix.Moregenerally,wecangrabanybasisfor avectorspace,multiplyanyonebasisvectorbyanon-zeroscalarandcreateaslightlydierentsetthat isstillabasis.Forimportant"vectorspaces,itwillbeconvenienttohaveacollectionofnice"bases. Whenavectorspacehasasingleparticularlynicebasis,itissometimescalledthe standardbasis though thereisnothingpreciseenoughaboutthistermtoallowustodeneitformally|itisaquestionofstyle. Herearesomenicebasesforimportantvectorspaces. TheoremSUVB StandardUnitVectorsareaBasis Thesetofstandardunitvectorsfor C m DenitionSUV[173], B = f e 1 ; e 2 ; e 3 ;:::; e m g = f e i j 1 i m g isabasisforthevectorspace C m Proof Wemustshowthattheset B isbothlinearlyindependentandaspanningsetfor C m .First,the vectorsin B are,byDenitionSUV[173],thecolumnsoftheidentitymatrix,whichweknowisnonsingular sinceitrow-reducestotheidentitymatrix,TheoremNMRRI[72].Andthecolumnsofanonsingular matrixarelinearlyindependentbyTheoremNMLIC[138]. Version2.02

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SubsectionB.BBases327 Supposewegrabanarbitraryvectorfrom C m ,say v = 2 6 6 6 6 6 4 v 1 v 2 v 3 v m 3 7 7 7 7 7 5 : Canwewrite v asalinearcombinationofthevectorsin B ?Yes,andquitesimply. 2 6 6 6 6 6 4 v 1 v 2 v 3 v m 3 7 7 7 7 7 5 = v 1 2 6 6 6 6 6 4 1 0 0 0 3 7 7 7 7 7 5 + v 2 2 6 6 6 6 6 4 0 1 0 0 3 7 7 7 7 7 5 + v 3 2 6 6 6 6 6 4 0 0 1 0 3 7 7 7 7 7 5 + + v m 2 6 6 6 6 6 4 0 0 0 1 3 7 7 7 7 7 5 v = v 1 e 1 + v 2 e 2 + v 3 e 3 + + v m e m thisshowsthat C m h B i ,whichissucienttoshowthat B isaspanningsetfor C m ExampleBP Basesfor P n Thevectorspaceofpolynomialswithdegreeatmost n P n ,hasthebasis B = 1 ;x;x 2 ;x 3 ;:::;x n : Anothernicebasisfor P n is C = 1 ; 1+ x; 1+ x + x 2 ; 1+ x + x 2 + x 3 ;:::; 1+ x + x 2 + x 3 + + x n : Checkingthateachof B and C isalinearlyindependentspanningsetaregoodexercises. ExampleBM Abasisforthevectorspaceofmatrices Inthevectorspace M mn ofmatricesExampleVSM[281]denethematrices B k` ,1 k m ,1 ` n by [ B k` ] ij = 1if k = i;` = j 0otherwise Sothesematriceshaveentriesthatareallzeros,withtheexceptionofaloneentrythatisone.Thesetof all mn ofthem, B = f B k` j 1 k m; 1 ` n g formsabasisfor M mn Thebasesdescribedabovewilloftenbeconvenientonestoworkwith.Howeverabasisdoesn'thave toobviouslylooklikeabasis. ExampleBSP4 Abasisforasubspaceof P 4 InExampleSSP4[313]weshowedthat S = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 ;x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +4 ;x 3 )]TJ/F15 10.9091 Tf 10.91 0 Td [(6 x 2 +12 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 ;x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 x 3 +24 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(32 x +16 Version2.02

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SubsectionB.BBases328 isaspanningsetfor W = f p x j p 2 P 4 ;p =0 g .Wewillnowshowthat S isalsolinearlyindependent in W .Beginwitharelationoflineardependence, 0+0 x +0 x 2 +0 x 3 +0 x 4 = 1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2+ 2 )]TJ/F21 10.9091 Tf 5 -8.836 Td [(x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +4 + 3 )]TJ/F21 10.9091 Tf 5 -8.837 Td [(x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 2 +12 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 + 4 )]TJ/F21 10.9091 Tf 5 -8.837 Td [(x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 x 3 +24 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(32 x +16 = 4 x 4 + 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 4 x 3 + 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 3 +24 4 x 2 + 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 2 +12 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(32 4 x + )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 +4 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 3 +16 4 Equatingcoecientsvectorequalityin P 4 givesthehomogeneoussystemofveequationsinfourvariables, 4 =0 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 4 =0 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 3 +24 4 =0 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 2 +12 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(32 4 =0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 +4 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 3 +16 4 =0 Weformthecoecientmatrix,androw-reducetoobtainamatrixinreducedrow-echelonform 2 6 6 6 6 6 4 1 000 0 1 00 00 1 0 000 1 0000 3 7 7 7 7 7 5 With only thetrivialsolutiontothishomogeneoussystem,weconcludethatonlyscalarsthatwillforma relationoflineardependencearethetrivialones,andthereforetheset S islinearlyindependentDenition LI[308].Finally, S hasearnedtherighttobecalledabasisfor W DenitionB[325]. ExampleBSM22 Abasisforasubspaceof M 22 InExampleSSM22[314]wediscoveredthat Q = )]TJ/F15 10.9091 Tf 8.485 0 Td [(31 00 ; 10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(41 isaspanningsetforthesubspace Z = ab cd j a +3 b )]TJ/F21 10.9091 Tf 10.909 0 Td [(c )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 d =0 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 a )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 b +3 c +14 d =0 ofthevectorspaceofall2 2matrices, M 22 .Ifwecanalsodeterminethat Q islinearlyindependentin Z orin M 22 ,thenitwillqualifyasabasisfor Z .Let'sbeginwitharelationoflineardependence. 00 00 = 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(31 00 + 2 10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(41 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 + 2 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 2 2 UsingourdenitionofmatrixequalityDenitionME[182]weequatecorrespondingentriesandgeta homogeneoussystemoffourequationsintwovariables, )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 + 2 =0 Version2.02

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SubsectionB.BSCVBasesforSpansofColumnVectors329 1 =0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 2 =0 2 =0 Wecouldrow-reducethecoecientmatrixofthishomogeneoussystem,butitisnotnecessary.Thesecond andfourthequationstellusthat 1 =0, 2 =0isthe only solutiontothishomogeneoussystem.This qualiestheset Q asbeinglinearlyindependent,sincetheonlyrelationoflineardependenceistrivial DenitionLI[308].Therefore Q isabasisfor Z DenitionB[325]. ExampleBC Basisforthecrazyvectorspace InExampleLIC[312]andExampleSSC[315]wedeterminedthattheset R = f ; 0 ; ; 3 g fromthe crazyvectorspace, C ExampleCVS[283],islinearlyindependentandisaspanningsetfor C .By DenitionB[325]weseethat R isabasisfor C Wehaveseenthatseveralofthesetsassociatedwithamatrixaresubspacesofvectorspacesofcolumn vectors.SpecicallythesearethenullspaceTheoremNSMS[296],columnspaceTheoremCSMS[302], rowspaceTheoremRSMS[303]andleftnullspaceTheoremLNSMS[303].Assubspacestheyarevector spacesDenitionS[292]anditisnaturaltoaskaboutbasesforthesevectorspaces.TheoremBNS[139], TheoremBCS[239],TheoremBRS[245]eachhaveconclusionsthatprovidelinearlyindependentspanning setsforrespectivelythenullspace,columnspace,androwspace.Noticethateachofthesetheorems containsthewordbasis"initstitle,eventhoughwedidnotknowtheprecisemeaningofthewordat thetime.Tondabasisforaleftnullspacewecanusethedenitionofthissubspaceasanullspace DenitionLNS[257]andapplyTheoremBNS[139].OrTheoremFS[263]tellsusthattheleftnullspace canbeexpressedasarowspaceandwecanthenuseTheoremBRS[245]. TheoremBS[157]isanotherearlyresultthatprovidesalinearlyindependentspanningseti.e.abasis asitsconclusion.Ifavectorspaceofcolumnvectorscanbeexpressedasaspanofasetofcolumnvectors, thenTheoremBS[157]canbeemployedinastraightforwardmannertoquicklyyieldabasis. SubsectionBSCV BasesforSpansofColumnVectors Wehaveseenseveralexamplesofbasesindierentvectorspaces.Inthissubsection,andthenextSubsectionB.BNM[330],wewillconsiderbuildingbasesfor C m anditssubspaces. Supposewehaveasubspaceof C m thatisexpressedasthespanofasetofvectors, S ,and S is notnecessarilylinearlyindependent,orperhapsnotveryattractive.TheoremREMRS[244]saysthat row-equivalentmatriceshaveidenticalrowspaces,whileTheoremBRS[245]saysthenonzerorowsofa matrixinreducedrow-echelonformareabasisfortherowspace.Thesetheoremstogethergiveusagreat computationaltoolforquicklyndingabasisforasubspacethatisexpressedoriginallyasaspan. ExampleRSB Rowspacebasis Whenwerstdenedthespanofasetofcolumnvectors,inExampleSCAD[120]welookedattheset W = 8 < : 2 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 3 5 ; 2 4 1 4 1 3 5 ; 2 4 7 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 4 3 5 ; 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 5 9 = ; + withaneyetowardsrealizing W asthespanofasmallerset.Bybuildingrelationsoflineardependence thoughwedidnotknowthembythatnamethenwewereabletoremovetwovectorsandwrite W as Version2.02

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SubsectionB.BSCVBasesforSpansofColumnVectors330 thespanoftheothertwovectors.Thesetworemainingvectorsformedalinearlyindependentset,even thoughwedidnotknowthatatthetime. Nowweknowthat W isasubspaceandmusthaveabasis.Considerthematrix, C ,whoserowsare thevectorsinthespanningsetfor W C = 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(31 141 7 )]TJ/F15 10.9091 Tf 8.484 0 Td [(54 )]TJ/F15 10.9091 Tf 8.484 0 Td [(7 )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 7 7 5 Then,byDenitionRSM[243],therowspaceof C willbe W R C = W .TheoremBRS[245]tellsus thatifwerow-reduce C ,thenonzerorowsoftherow-equivalentmatrixinreducedrow-echelonformwill beabasisfor R C ,andhenceabasisfor W .Let'sdoit| C row-reducesto 2 6 6 4 1 0 7 11 0 1 1 11 000 000 3 7 7 5 Ifweconvertthetwononzerorowstocolumnvectorsthenwehaveabasis, B = 8 < : 2 4 1 0 7 11 3 5 ; 2 4 0 1 1 11 3 5 9 = ; and W = 8 < : 2 4 1 0 7 11 3 5 ; 2 4 0 1 1 11 3 5 9 = ; + Foraestheticreasons,wemightwishtomultiplyeachvectorin B by11,whichwillnotchangethespanning orlinearindependencepropertiesof B asabasis.Thenwecanalsowrite W = 8 < : 2 4 11 0 7 3 5 ; 2 4 0 11 1 3 5 9 = ; + ExampleIAS[246]providesanotherexampleofthisavor,thoughnowwecannoticethat X isa subspace,andthattheresultingsetofthreevectorsisabasis.Thisissuchapowerfultechniquethatwe shoulddoonemoreexample. ExampleRS Reducingaspan InExampleRSC5[153]webeganwithasetof n =4vectorsfrom C 5 R = f v 1 ; v 2 ; v 3 ; v 4 g = 8 > > > > < > > > > : 2 6 6 6 6 4 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 2 3 7 7 7 7 5 ; 2 6 6 6 6 4 2 1 3 1 2 3 7 7 7 7 5 ; 2 6 6 6 6 4 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 7 7 5 ; 2 6 6 6 6 4 4 1 2 1 6 3 7 7 7 7 5 9 > > > > = > > > > ; anddened V = h R i .Ourgoalinthatproblemwastondarelationoflineardependenceonthevectors in R ,solvetheresultingequationforoneofthevectors,andre-express V asthespanofasetofthree vectors. Version2.02

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SubsectionB.BNMBasesandNonsingularMatrices331 Hereisanotherwaytoaccomplishsomethingsimilar.Therowspaceofthematrix A = 2 6 6 4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(132 21312 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(76 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 41216 3 7 7 5 isequalto h R i .ByTheoremBRS[245]wecanrow-reducethismatrix,ignoreanyzerorows,anduse thenon-zerorowsascolumnvectorsthatareabasisfortherowspaceof A .Row-reducing A createsthe matrix 2 6 6 4 100 )]TJ/F19 7.9701 Tf 11.798 4.295 Td [(1 17 30 17 010 25 17 )]TJ/F19 7.9701 Tf 11.798 4.295 Td [(2 17 001 )]TJ/F19 7.9701 Tf 11.798 4.295 Td [(2 17 )]TJ/F19 7.9701 Tf 11.798 4.295 Td [(8 17 00000 3 7 7 5 So 8 > > > > < > > > > : 2 6 6 6 6 4 1 0 0 )]TJ/F19 7.9701 Tf 11.798 4.295 Td [(1 17 30 17 3 7 7 7 7 5 ; 2 6 6 6 6 4 0 1 0 25 17 )]TJ/F19 7.9701 Tf 11.797 4.295 Td [(2 17 3 7 7 7 7 5 ; 2 6 6 6 6 4 0 0 1 )]TJ/F19 7.9701 Tf 11.798 4.295 Td [(2 17 )]TJ/F19 7.9701 Tf 11.798 4.295 Td [(8 17 3 7 7 7 7 5 9 > > > > = > > > > ; isabasisfor V .Ourtheoremtellsusthisisabasis,thereisnoneedtoverifythatthesubspacespanned bythreevectorsratherthanfouristheidenticalsubspace,andthereisnoneedtoverifythatwehave reachedthelimitinreducingtheset,sincethesetofthreevectorsisguaranteedtobelinearlyindependent. SubsectionBNM BasesandNonsingularMatrices Aquicksourceofdiversebasesfor C m isthesetofcolumnsofanonsingularmatrix. TheoremCNMB ColumnsofNonsingularMatrixareaBasis Supposethat A isasquarematrixofsize m .Thenthecolumnsof A areabasisof C m ifandonlyif A is nonsingular. Proof Supposethatthecolumnsof A areabasisfor C m .ThenDenitionB[325]saysthesetof columnsislinearlyindependent.TheoremNMLIC[138]thensaysthat A isnonsingular. Supposethat A isnonsingular.ThenbyTheoremNMLIC[138]thissetofcolumnsislinearly independent.TheoremCSNM[242]saysthatforanonsingularmatrix, C A = C m .Thisisequivalent tosayingthatthecolumnsof A areaspanningsetforthevectorspace C m .Asalinearlyindependent spanningset,thecolumnsof A qualifyasabasisfor C m DenitionB[325]. ExampleCABAK ColumnsasBasis,ArchetypeK ArchetypeK[746]isthe5 5matrix K = 2 6 6 6 6 4 10182424 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(60 )]TJ/F15 10.9091 Tf 8.485 0 Td [(18 )]TJ/F15 10.9091 Tf 8.485 0 Td [(30 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3039 27303637 )]TJ/F15 10.9091 Tf 8.485 0 Td [(30 18243030 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 3 7 7 7 7 5 Version2.02

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SubsectionB.OBCOrthonormalBasesandCoordinates332 whichisrow-equivalenttothe5 5identitymatrix I 5 .SobyTheoremNMRRI[72], K isnonsingular. ThenTheoremCNMB[330]saystheset 8 > > > > < > > > > : 2 6 6 6 6 4 10 12 )]TJ/F15 10.9091 Tf 8.484 0 Td [(30 27 18 3 7 7 7 7 5 ; 2 6 6 6 6 4 18 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 30 24 3 7 7 7 7 5 ; 2 6 6 6 6 4 24 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 36 30 3 7 7 7 7 5 ; 2 6 6 6 6 4 24 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(30 37 30 3 7 7 7 7 5 ; 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(12 )]TJ/F15 10.9091 Tf 8.484 0 Td [(18 39 )]TJ/F15 10.9091 Tf 8.484 0 Td [(30 )]TJ/F15 10.9091 Tf 8.484 0 Td [(20 3 7 7 7 7 5 9 > > > > = > > > > ; isanovelbasisof C 5 PerhapsweshouldviewthefactthatthestandardunitvectorsareabasisTheoremSUVB[325]as justasimplecorollaryofTheoremCNMB[330]?SeeTechniqueLC[696]. Withanewequivalenceforanonsingularmatrix,wecanupdateourlistofequivalences. TheoremNME5 NonsingularMatrixEquivalences,Round5 Supposethat A isasquarematrixofsize n .Thefollowingareequivalent. 1. A isnonsingular. 2. A row-reducestotheidentitymatrix. 3.Thenullspaceof A containsonlythezerovector, N A = f 0 g 4.Thelinearsystem LS A; b hasauniquesolutionforeverypossiblechoiceof b 5.Thecolumnsof A arealinearlyindependentset. 6. A isinvertible. 7.Thecolumnspaceof A is C n C A = C n 8.Thecolumnsof A areabasisfor C n Proof WithanewequivalenceforanonsingularmatrixinTheoremCNMB[330]wecanexpandTheorem NME4[242]. SubsectionOBC OrthonormalBasesandCoordinates Welearnedaboutorthogonalsetsofvectorsin C m backinSectionO[167],andwealsolearnedthat orthogonalsetsareautomaticallylinearlyindependentTheoremOSLI[174].Whenanorthogonalset alsospansasubspaceof C m ,thenthesetisabasis.Andwhenthesetisorthonormal,thenthesetis anincrediblynicebasis.Wewillbackupthisclaimwithatheorem,butrstconsiderhowyoumight manufacturesuchaset. Supposethat W isasubspaceof C m withbasis B .Then B spans W andisalinearlyindependent setofnonzerovectors.WecanapplytheGram-SchmidtProcedureTheoremGSP[175]andobtaina linearlyindependentset T suchthat h T i = h B i = W and T isorthogonal.Inotherwords, T isabasisfor W ,andisanorthogonalset.Byscalingeachvectorof T tonorm1,wecanconvert T intoanorthonormal set,withoutdestroyingthepropertiesthatmakeitabasisof W .Inshort,wecanconvertanybasisinto anorthonormalbasis.ExampleGSTV[176],followedbyExampleONTV[177],illustratesthisprocess. Version2.02

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SubsectionB.OBCOrthonormalBasesandCoordinates333 UnitarymatricesDenitionUM[229]areanothergoodsourceoforthonormalbasesandviceversa. Supposethat Q isaunitarymatrixofsize n .Thenthe n columnsof Q formanorthonormalsetTheorem CUMOS[230]thatisthereforelinearlyindependentTheoremOSLI[174].Since Q isinvertibleTheorem UMI[230],weknow Q isnonsingularTheoremNI[228],andthenthecolumnsof Q span C n Theorem CSNM[242].Sothecolumnsofaunitarymatrixofsize n areanorthonormalbasisfor C n Whyallthefussaboutorthonormalbases?TheoremVRRB[317]toldusthatanyvectorinavector spacecouldbewritten,uniquely,asalinearcombinationofbasisvectors.Foranorthonormalbasis, ndingthescalarsforthislinearcombinationisextremelyeasy,andthisisthecontentofthenexttheorem. Furthermore,withvectorswrittenthiswayaslinearcombinationsoftheelementsofanorthonormalset certaincomputationsandanalysisbecomemucheasier.Here'sthepromisedtheorem. TheoremCOB CoordinatesandOrthonormalBases Supposethat B = f v 1 ; v 2 ; v 3 ;:::; v p g isanorthonormalbasisofthesubspace W of C m .Forany w 2 W w = h w ; v 1 i v 1 + h w ; v 2 i v 2 + h w ; v 3 i v 3 + + h w ; v p i v p Proof Because B isabasisof W ,TheoremVRRB[317]tellsusthatwecanwrite w uniquelyasa linearcombinationofthevectorsin B .Soitisnotthisaspectoftheconclusionthatmakesthistheorem interesting.Whatisinterestingisthattheparticularscalarsaresoeasytocompute.Noneedtosolvebig systemsofequations|justdoaninnerproductof w with v i toarriveatthecoecientof v i inthelinear combination. Sobegintheproofbywriting w asalinearcombinationofthevectorsin B ,usingunknownscalars, w = a 1 v 1 + a 2 v 2 + a 3 v 3 + + a p v p andcompute, h w ; v i i = p X k =1 a k v k ; v i + TheoremVRRB[317] = p X k =1 h a k v k ; v i i TheoremIPVA[169] = p X k =1 a k h v k ; v i i TheoremIPSM[170] = a i h v i ; v i i + p X i =1 k 6 = i a k h v k ; v i i PropertyC[279] = a i + p X i =1 k 6 = i a k DenitionONS[177] = a i Sotheuniquescalarsforthelinearcombinationareindeedtheinnerproductsadvertisedintheconclusion ofthetheorem'sstatement. ExampleCROB4 Coordinatizationrelativetoanorthonormalbasis, C 4 Version2.02

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SubsectionB.OBCOrthonormalBasesandCoordinates334 Theset f x 1 ; x 2 ; x 3 ; x 4 g = 8 > > < > > : 2 6 6 4 1+ i 1 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i i 3 7 7 5 ; 2 6 6 4 1+5 i 6+5 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 i 3 7 7 5 ; 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7+34 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 10.909 0 Td [(23 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(10+22 i 30+13 i 3 7 7 5 ; 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 i 6+ i 4+3 i 6 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 3 7 7 5 9 > > = > > ; wasproposed,andpartiallyveried,asanorthogonalsetinExampleAOS[173].Let'sscaleeachvector tonorm1,soastoformanorthonormalsetin C 4 .ThenbyTheoremOSLI[174]thesetwillbelinearly independent,andbyTheoremNME5[331]thesetwillbeabasisfor C 4 .So,oncescalkedtonorm1,the adjustedsetwillbeanorthonormalbasisof C 4 .Thenormsare, k x 1 k = p 6 k x 2 k = p 174 k x 3 k = p 3451 k x 4 k = p 119 Soanorthonormalbasisis B = f v 1 ; v 2 ; v 3 ; v 4 g = 8 > > < > > : 1 p 6 2 6 6 4 1+ i 1 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i i 3 7 7 5 ; 1 p 174 2 6 6 4 1+5 i 6+5 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 i 3 7 7 5 ; 1 p 3451 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7+34 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 10.909 0 Td [(23 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(10+22 i 30+13 i 3 7 7 5 ; 1 p 119 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 i 6+ i 4+3 i 6 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 3 7 7 5 9 > > = > > ; Now,toillustrateTheoremCOB[332],chooseanyvectorfrom C 4 ,say w = 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 4 3 7 7 5 ,andcompute h w ; v 1 i = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 i p 6 ; h w ; v 2 i = )]TJ/F15 10.9091 Tf 8.485 0 Td [(19+30 i p 174 ; h w ; v 3 i = 120 )]TJ/F15 10.9091 Tf 10.909 0 Td [(211 i p 3451 ; h w ; v 4 i = 6+12 i p 119 ThenTheoremCOB[332]guaranteesthat 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 4 3 7 7 5 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 i p 6 0 B B @ 1 p 6 2 6 6 4 1+ i 1 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i i 3 7 7 5 1 C C A + )]TJ/F15 10.9091 Tf 8.485 0 Td [(19+30 i p 174 0 B B @ 1 p 174 2 6 6 4 1+5 i 6+5 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F21 10.9091 Tf 10.91 0 Td [(i 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 i 3 7 7 5 1 C C A + 120 )]TJ/F15 10.9091 Tf 10.909 0 Td [(211 i p 3451 0 B B @ 1 p 3451 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7+34 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 10.909 0 Td [(23 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(10+22 i 30+13 i 3 7 7 5 1 C C A + 6+12 i p 119 0 B B @ 1 p 119 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 i 6+ i 4+3 i 6 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 3 7 7 5 1 C C A asyoumightwanttocheckifyouhaveunlimitedpatience. Aslightlylessintimidatingexamplefollows,inthreedimensionsandwithjustrealnumbers. ExampleCROB3 Coordinatizationrelativetoanorthonormalbasis, C 3 Theset f x 1 ; x 2 ; x 3 g = 8 < : 2 4 1 2 1 3 5 ; 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 1 3 5 ; 2 4 2 1 1 3 5 9 = ; isalinearlyindependentset,whichtheGram-SchmidtProcessTheoremGSP[175]convertstoan orthogonalset,andwhichcanthenbeconvertedtotheorthonormalset, B = f v 1 ; v 2 ; v 3 g = 8 < : 1 p 6 2 4 1 2 1 3 5 ; 1 p 2 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 1 3 5 ; 1 p 3 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 3 5 9 = ; Version2.02

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SubsectionB.OBCOrthonormalBasesandCoordinates335 whichisthereforeanorthonormalbasisof C 3 .Withthreevectorsin C 3 ,allwithrealnumberentries, theinnerproductDenitionIP[168]reducestotheusualdotproduct"orscalarproductandthe orthogonalpairsofvectorscanbeinterpretedasperpendicularpairsofdirections.Sothevectorsin B serveasreplacementsforourusual3-Daxes,ortheusual3-Dunitvectors ~ i; ~ j and ~ k .Wewouldlike todecomposearbitraryvectorsintocomponents"inthedirectionsofeachofthesebasisvectors.Itis TheoremCOB[332]thattellsushowtodothis. Supposethatwechoose w = 2 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 5 3 5 .Compute h w ; v 1 i = 5 p 6 h w ; v 2 i = 3 p 2 h w ; v 3 i = 8 p 3 thenTheoremCOB[332]guaranteesthat 2 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 5 3 5 = 5 p 6 0 @ 1 p 6 2 4 1 2 1 3 5 1 A + 3 p 2 0 @ 1 p 2 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 1 3 5 1 A + 8 p 3 0 @ 1 p 3 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 3 5 1 A whichyoushouldbeabletocheckeasily,evenifyoudonothavemuchpatience. Notonlydothecolumnsofaunitarymatrixformanorthonormalbasis,butthereisadeeperconnection betweenorthonormalbasesandunitarymatrices.Informally,thenexttheoremsaysthatifwetransform eachvectorofanorthonormalbasisbymultiplyingitbyaunitarymatrix,thentheresultingsetwillbe anotherorthonormalbasis.Andmoreremarkably,anymatrixwiththispropertymustbeunitary!Asan equivalenceTechniqueE[690]wecouldtakethisasourdeningpropertyofaunitarymatrix,thoughit mightnothavethesameutilityasDenitionUM[229]. TheoremUMCOB UnitaryMatricesConvertOrthonormalBases Let A bean n n matrixand B = f x 1 ; x 2 ; x 3 ;:::; x n g beanorthonormalbasisof C n .Dene C = f A x 1 ;A x 2 ;A x 3 ;:::;A x n g Then A isaunitarymatrixifandonlyif C isanorthonormalbasisof C n Proof Assume A isaunitarymatrixandestablishseveralfactsabout C .Firstwecheckthat C isanorthonormalsetDenitionONS[177].ByTheoremUMPIP[231],for i 6 = j h A x i ;A x j i = h x i ; x j i =0 Similarly,TheoremUMPIP[231]alsogives,for1 i n k A x i k = k x i k =1 As C isanorthogonalsetDenitionOSV[173],TheoremOSLI[174]yieldsthelinearindependenceof C Havingestablishedthatthecolumnvectorson C formalinearlyindependentset,amatrixwhosecolumns arethevectorsof C isnonsingularTheoremNMLIC[138],andhencethesevectorsformabasisof C n byTheoremCNMB[330]. Nowassumethat C isanorthonormalset.Let y beanarbitraryvectorfrom C n .Since B spans C n ,therearescalars, a 1 ;a 2 ;a 3 ;:::;a n ,suchthat y = a 1 x 1 + a 2 x 2 + a 3 x 3 + + a n x n Version2.02

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SubsectionB.OBCOrthonormalBasesandCoordinates336 Now A A y = n X i =1 h A A y ; x i i x i TheoremCOB[332] = n X i =1 A A n X j =1 a j x j ; x i + x i DenitionTSVS[313] = n X i =1 n X j =1 A Aa j x j ; x i + x i TheoremMMDAA[201] = n X i =1 n X j =1 a j A A x j ; x i + x i TheoremMMSMM[201] = n X i =1 n X j =1 h a j A A x j ; x i i x i TheoremIPVA[169] = n X i =1 n X j =1 a j h A A x j ; x i i x i TheoremIPSM[170] = n X i =1 n X j =1 a j h A x j ; A x i i x i TheoremAIP[204] = n X i =1 n X j =1 a j h A x j ;A x i i x i TheoremAA[190] = n X i =1 n X j =1 j 6 = i a j h A x j ;A x i i x i + n X ` =1 a ` h A x ` ;A x ` i x ` PropertyC[279] = n X i =1 n X j =1 j 6 = i a j x i + n X ` =1 a ` x ` DenitionONS[177] = n X i =1 n X j =1 j 6 = i 0 + n X ` =1 a ` x ` TheoremZSSM[286] = n X ` =1 a ` x ` PropertyZ[280] = y = I n y TheoremMMIM[200] Sincethechoiceof y wasarbitrary,TheoremEMMVP[196]tellsusthat A A = I n ,so A isunitary DenitionUM[229]. Version2.02

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SubsectionB.READReadingQuestions337 SubsectionREAD ReadingQuestions 1.Thematrixbelowisnonsingular.Whatcanyounowsayaboutitscolumns? A = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(301 121 516 3 5 2.Writethevector w = 2 4 6 6 15 3 5 asalinearcombinationofthecolumnsofthematrix A above.Howmany waysaretheretoanswerthisquestion? 3.Whyisanorthonormalbasisdesirable? Version2.02

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SubsectionB.EXCExercises338 SubsectionEXC Exercises C40 FromExampleRSB[328],formanarbitraryandnontriviallinearcombinationofthefourvectors intheoriginalspanningsetfor W .Sotheresultofthiscomputationisofcourseanelementof W .As such,thisvectorshouldbealinearcombinationofthebasisvectorsin B .Findtheuniquescalarsthat providethislinearcombination.Repeatwithanotherlinearcombinationoftheoriginalfourvectors. ContributedbyRobertBeezerSolution[339] C80 Provethat f ; 2 ; ; 3 g isabasisforthecrazyvectorspace C ExampleCVS[283]. ContributedbyRobertBeezer M20 InExampleBM[326]providethevericationslinearindependenceandspanningtoshowthat B isabasisof M mn ContributedbyRobertBeezerSolution[338] T50 TheoremUMCOB[334]saysthatunitarymatricesarecharacterizedasthosematricesthatcarry" orthonormalbasestoorthonormalbases.Thisproblemasksyoutoproveasimilarresult:nonsingular matricesarecharacterizedasthosematricesthatcarry"basestobases. Moreprecisely,supposethat A isasquarematrixofsize n and B = f x 1 ; x 2 ; x 3 ;:::; x n g isabasisof C n .Provethat A isnonsingularifandonlyif C = f A x 1 ;A x 2 ;A x 3 ;:::;A x n g isabasisof C n .Seealso ExercisePD.T33[366],ExerciseMR.T20[564]. ContributedbyRobertBeezerSolution[339] T51 UsetheresultofExerciseB.T50[337]tobuildaveryconciseproofofTheoremCNMB[330].Hint: makeajudiciouschoiceforthebasis B ContributedbyRobertBeezerSolution[340] Version2.02

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SubsectionB.SOLSolutions339 SubsectionSOL Solutions M20 ContributedbyRobertBeezerStatement[337] Weneedtoestablishthelinearindependenceandspanningpropertiesoftheset B = f B k` j 1 k m; 1 ` n g relativetothevectorspace M mn Thisproofismoretransparentifyouwriteoutindividualmatricesinthebasiswithlotsofzerosand dotsandaloneone.Butwedon'thaveroomforthathere,sowewillusesummationnotation.Think carefullyabouteachstep,especiallywhenthedoublesummationsseemtodisappear."Beginwitha relationoflineardependence,usingdoublesubscriptsonthescalarstoalignwiththebasiselements. O = m X k =1 n X ` =1 k` B k` Nowconsidertheentryinrow i andcolumn j fortheseequalmatrices, 0=[ O ] ij DenitionZM[185] = m X k =1 n X ` =1 k` B k` # ij DenitionME[182] = m X k =1 n X ` =1 [ k` B k` ] ij DenitionMA[182] = m X k =1 n X ` =1 k` [ B k` ] ij DenitionMSM[183] = ij [ B ij ] ij [ B k` ] ij =0when k;` 6 = i;j = ij [ B ij ] ij =1 = ij Since i and j werearbitrary,wendthateachscalariszeroandso B islinearlyindependentDenition LI[308]. Toestablishthespanningpropertyof B weneedonlyshowthatanarbitrarymatrix A canbewritten asalinearcombinationoftheelementsof B .Sosupposethat A isanarbitrary m n matrixandconsider thematrix C denedasalinearcombinationoftheelementsof B by C = m X k =1 n X ` =1 [ A ] k` B k` Then, [ C ] ij = m X k =1 n X ` =1 [ A ] k` B k` # ij DenitionME[182] = m X k =1 n X ` =1 [[ A ] k` B k` ] ij DenitionMA[182] = m X k =1 n X ` =1 [ A ] k` [ B k` ] ij DenitionMSM[183] Version2.02

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SubsectionB.SOLSolutions340 =[ A ] ij [ B ij ] ij [ B k` ] ij =0when k;` 6 = i;j =[ A ] ij [ B ij ] ij =1 =[ A ] ij SobyDenitionME[182], A = C ,andtherefore A 2h B i .ByDenitionB[325],theset B isabasisof thevectorspace M mn C40 ContributedbyRobertBeezerStatement[337] Anarbitrarylinearcombinationis y =3 2 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 3 5 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 4 1 4 1 3 5 +1 2 4 7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 4 3 5 + )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 5 = 2 4 25 )]TJ/F15 10.9091 Tf 8.484 0 Td [(10 15 3 5 Youprobablyusedadierentcollectionofscalars.Wewanttowrite y asalinearcombinationof B = 8 < : 2 4 1 0 7 11 3 5 ; 2 4 0 1 1 11 3 5 9 = ; Wecouldsetthisupasvectorequationwithvariablesasscalarsinalinearcombinationofthevectors in B ,butsincethersttwoslotsof B havesuchanicepatternofzerosandones,wecandeterminethe necessaryscalarseasilyandthendouble-checkouranswerwithacomputationinthethirdslot, 25 2 4 1 0 7 11 3 5 + )]TJ/F15 10.9091 Tf 8.484 0 Td [(10 2 4 0 1 1 11 3 5 = 2 4 25 )]TJ/F15 10.9091 Tf 8.484 0 Td [(10 7 11 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 1 11 3 5 = 2 4 25 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 15 3 5 = y Noticehowtheuniquenessofthesescalarsarises.Theyare forced tobe25and )]TJ/F15 10.9091 Tf 8.485 0 Td [(10. T50 ContributedbyRobertBeezerStatement[337] Ourrstproofreliesmostlyondenitionsoflinearindependenceandspanning,whichisagoodexercise. Thesecondproofisshorterandturnsonatechnicalresultfromourworkwithmatrixinverses,Theorem NPNT[226]. Assumethat A isnonsingularandprovethat C isabasisof C n .Firstshowthat C islinearly independent.Workonarelationoflineardependenceon C 0 = a 1 A x 1 + a 2 A x 2 + a 3 A x 3 + + a n A x n DenitionRLD[308] = Aa 1 x 1 + Aa 2 x 2 + Aa 3 x 3 + + Aa n x n TheoremMMSMM[201] = A a 1 x 1 + a 2 x 2 + a 3 x 3 + + a n x n TheoremMMDAA[201] Since A isnonsingular,DenitionNM[71]andTheoremSLEMM[195]allowsustoconcludethat a 1 x 1 + a 2 x 2 + + a n x n = 0 Butthisisarelationoflineardependenceofthelinearlyindependentset B ,sothescalarsaretrivial, a 1 = a 2 = a 3 = = a n =0.ByDenitionLI[308],theset C islinearlyindependent. Nowprovethat C spans C n .Givenanarbitraryvector y 2 C n ,canitbeexpressedasalinear combinationofthevectorsin C ?Since A isanonsingularmatrixwecandenethevector w tobethe uniquesolutionofthesystem LS A; y TheoremNMUS[74].Since w 2 C n wecanwrite w asalinear combinationofthevectorsinthebasis B .Sotherearescalars, b 1 ;b 2 ;b 3 ;:::;b n suchthat w = b 1 x 1 + b 2 x 2 + b 3 x 3 + + b n x n Version2.02

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SubsectionB.SOLSolutions341 Then, y = A w TheoremSLEMM[195] = A b 1 x 1 + b 2 x 2 + b 3 x 3 + + b n x n DenitionTSVS[313] = Ab 1 x 1 + Ab 2 x 2 + Ab 3 x 3 + + Ab n x n TheoremMMDAA[201] = b 1 A x 1 + b 2 A x 2 + b 3 A x 3 + + b n A x n TheoremMMSMM[201] Sowecanwriteanarbitraryvectorof C n asalinearcombinationoftheelementsof C .Inotherwords, C spans C n DenitionTSVS[313].ByDenitionB[325],theset C isabasisfor C n Assumethat C isabasisandprovethat A isnonsingular.Let x beasolutiontothehomogeneous system LS A; 0 .Since B isabasisof C n therearescalars, a 1 ;a 2 ;a 3 ;:::;a n ,suchthat x = a 1 x 1 + a 2 x 2 + a 3 x 3 + + a n x n Then 0 = A x TheoremSLEMM[195] = A a 1 x 1 + a 2 x 2 + a 3 x 3 + + a n x n DenitionTSVS[313] = Aa 1 x 1 + Aa 2 x 2 + Aa 3 x 3 + + Aa n x n TheoremMMDAA[201] = a 1 A x 1 + a 2 A x 2 + a 3 A x 3 + + a n A x n TheoremMMSMM[201] Thisisarelationoflineardependenceonthelinearlyindependentset C ,sothescalarsmustallbezero, a 1 = a 2 = a 3 = = a n =0.Thus, x = a 1 x 1 + a 2 x 2 + a 3 x 3 + + a n x n =0 x 1 +0 x 2 +0 x 3 + +0 x n = 0 : ByDenitionNM[71]weseethat A isnonsingular. Nowforasecondproof.Takethevectorsfor B andusethemasthecolumnsofamatrix, G = [ x 1 j x 2 j x 3 j ::: j x n ].ByTheoremCNMB[330],becausewehavethehypothesisthat B isabasisof C n G is anonsingularmatrix.Noticethatthecolumnsof AG areexactlythevectorsintheset C ,byDenition MM[197]. A nonsingular AG nonsingularTheoremNPNT[226] C basisfor C n TheoremCNMB[330] Thatwaseasy! T51 ContributedbyRobertBeezerStatement[337] Choose B tobethesetofstandardunitvectors,aparticularlynicebasisof C n TheoremSUVB[325]. Foravector e j DenitionSUV[173]fromthisbasis,whatis A e j ? Version2.02

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SectionDDimension342 SectionD Dimension AlmosteveryvectorspacewehaveencounteredhasbeeninniteinsizeanexceptionisExampleVSS [283].Butsomearebiggerandricherthanothers.Dimension,oncesuitablydened,willbeameasureof thesizeofavectorspace,andausefultoolforstudyingitsproperties.Youprobablyalreadyhavearough notionofwhatamathematicaldenitionofdimensionmightbe|trytoforgettheseimpreciseideasand gowiththenewonesgivenhere. SubsectionD Dimension DenitionD Dimension Supposethat V isavectorspaceand f v 1 ; v 2 ; v 3 ;:::; v t g isabasisof V .Thenthe dimension of V is denedbydim V = t .If V hasnonitebases,wesay V hasinnitedimension. ThisdenitioncontainsNotationD. 4 Thisisaverysimpledenition,whichbeliesitspower.Grababasis,anybasis,andcountupthe numberofvectorsitcontains.That'sthedimension.However,thissimplicitycausesaproblem.Givena vectorspace,youandIcouldeachconstructdierentbases|rememberthatavectorspacemighthave manybases.Andwhatifyourbasisandmybasishaddierentsizes?ApplyingDenitionD[341]we wouldarriveatdierentnumbers!Withourcurrentknowledgeaboutvectorspaces,wewouldhavetosay thatdimensionisnotwell-dened."Fortunately,thereisatheoremthatwillcorrectthisproblem. Inastrictlylogicalprogression,thenexttwotheoremswould precede thedenitionofdimension.Many subsequenttheoremswilltracetheirlineagebacktothefollowingfundamentalresult. TheoremSSLD SpanningSetsandLinearDependence Supposethat S = f v 1 ; v 2 ; v 3 ;:::; v t g isanitesetofvectorswhichspansthevectorspace V .Thenany setof t +1ormorevectorsfrom V islinearlydependent. Proof Wewanttoprovethatanysetof t +1ormorevectorsfrom V islinearlydependent.Sowewill beginwithatotallyarbitrarysetofvectorsfrom V R = f u 1 ; u 2 ; u 3 ;:::; u m g ,where m>t .Wewillnow constructanontrivialrelationoflineardependenceon R Eachvector u 1 ; u 2 ; u 3 ;:::; u m canbewrittenasalinearcombinationof v 1 ; v 2 ; v 3 ;:::; v t since S is aspanningsetof V .Thismeansthereexistscalars a ij ,1 i t ,1 j m ,sothat u 1 = a 11 v 1 + a 21 v 2 + a 31 v 3 + + a t 1 v t u 2 = a 12 v 1 + a 22 v 2 + a 32 v 3 + + a t 2 v t u 3 = a 13 v 1 + a 23 v 2 + a 33 v 3 + + a t 3 v t u m = a 1 m v 1 + a 2 m v 2 + a 3 m v 3 + + a tm v t Nowweform,unmotivated,thehomogeneoussystemof t equationsinthe m variables, x 1 ;x 2 ;x 3 ;:::;x m wherethecoecientsarethejust-discoveredscalars a ij a 11 x 1 + a 12 x 2 + a 13 x 3 + + a 1 m x m =0 Version2.02

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SubsectionD.DDimension343 a 21 x 1 + a 22 x 2 + a 23 x 3 + + a 2 m x m =0 a 31 x 1 + a 32 x 2 + a 33 x 3 + + a 3 m x m =0 a t 1 x 1 + a t 2 x 2 + a t 3 x 3 + + a tm x m =0 Thisisahomogeneoussystemwithmorevariablesthanequationsourhypothesisisexpressedas m>t sobyTheoremHMVEI[64]thereareinnitelymanysolutions.Chooseanontrivialsolutionanddenote itby x 1 = c 1 ;x 2 = c 2 ;x 3 = c 3 ;:::;x m = c m .Asasolutiontothehomogeneoussystem,wethenhave a 11 c 1 + a 12 c 2 + a 13 c 3 + + a 1 m c m =0 a 21 c 1 + a 22 c 2 + a 23 c 3 + + a 2 m c m =0 a 31 c 1 + a 32 c 2 + a 33 c 3 + + a 3 m c m =0 a t 1 c 1 + a t 2 c 2 + a t 3 c 3 + + a tm c m =0 Asacollectionofnontrivialscalars, c 1 ;c 2 ;c 3 ;:::;c m willprovidethenontrivialrelationoflineardependencewedesire, c 1 u 1 + c 2 u 2 + c 3 u 3 + + c m u m = c 1 a 11 v 1 + a 21 v 2 + a 31 v 3 + + a t 1 v t DenitionTSVS[313] + c 2 a 12 v 1 + a 22 v 2 + a 32 v 3 + + a t 2 v t + c 3 a 13 v 1 + a 23 v 2 + a 33 v 3 + + a t 3 v t + c m a 1 m v 1 + a 2 m v 2 + a 3 m v 3 + + a tm v t = c 1 a 11 v 1 + c 1 a 21 v 2 + c 1 a 31 v 3 + + c 1 a t 1 v t PropertyDVA[280] + c 2 a 12 v 1 + c 2 a 22 v 2 + c 2 a 32 v 3 + + c 2 a t 2 v t + c 3 a 13 v 1 + c 3 a 23 v 2 + c 3 a 33 v 3 + + c 3 a t 3 v t + c m a 1 m v 1 + c m a 2 m v 2 + c m a 3 m v 3 + + c m a tm v t = c 1 a 11 + c 2 a 12 + c 3 a 13 + + c m a 1 m v 1 PropertyDSA[280] + c 1 a 21 + c 2 a 22 + c 3 a 23 + + c m a 2 m v 2 + c 1 a 31 + c 2 a 32 + c 3 a 33 + + c m a 3 m v 3 + c 1 a t 1 + c 2 a t 2 + c 3 a t 3 + + c m a tm v t = a 11 c 1 + a 12 c 2 + a 13 c 3 + + a 1 m c m v 1 PropertyCMCN[680] + a 21 c 1 + a 22 c 2 + a 23 c 3 + + a 2 m c m v 2 + a 31 c 1 + a 32 c 2 + a 33 c 3 + + a 3 m c m v 3 + a t 1 c 1 + a t 2 c 2 + a t 3 c 3 + + a tm c m v t =0 v 1 +0 v 2 +0 v 3 + +0 v t c j assolution Version2.02

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SubsectionD.DDimension344 = 0 + 0 + 0 + + 0 TheoremZSSM[286] = 0 PropertyZ[280] Thatdoesit. R hasbeenundeniablyshowntobealinearlydependentset. Theproofjustgivenhassomemonstrousexpressionsinit,mostlyowingtothedoublesubscripts present.Nowisagreatopportunitytoshowthevalueofamorecompactnotation.Wewillrewritethekey stepsofthepreviousproofusingsummationnotation,resultinginamoreeconomicalpresentation,and evengreaterinsightintothekeyaspectsoftheproof.Sohereisanalternateproof|studyitcarefully. ProofAlternateProofofTheoremSSLD Wewanttoprovethatanysetof t +1ormore vectorsfrom V islinearlydependent.Sowewillbeginwithatotallyarbitrarysetofvectorsfrom V R = f u j j 1 j m g ,where m>t .Wewillnowconstructanontrivialrelationoflineardependenceon R Eachvector u j ,1 j m canbewrittenasalinearcombinationof v i ,1 i t since S isaspanning setof V .Thismeanstherearescalars a ij ,1 i t ,1 j m ,sothat u j = t X i =1 a ij v i 1 j m Nowweform,unmotivated,thehomogeneoussystemof t equationsinthe m variables, x j ,1 j m wherethecoecientsarethejust-discoveredscalars a ij m X j =1 a ij x j =01 i t Thisisahomogeneoussystemwithmorevariablesthanequationsourhypothesisisexpressedas m>t sobyTheoremHMVEI[64]thereareinnitelymanysolutions.Chooseoneofthesesolutionsthatisnot trivialanddenoteitby x j = c j ,1 j m .Asasolutiontothehomogeneoussystem,wethenhave P m j =1 a ij c j =0for1 i t .Asacollectionofnontrivialscalars, c j ,1 j m ,willprovidethenontrivial relationoflineardependencewedesire, m X j =1 c j u j = m X j =1 c j t X i =1 a ij v i DenitionTSVS[313] = m X j =1 t X i =1 c j a ij v i PropertyDVA[280] = t X i =1 m X j =1 c j a ij v i PropertyCMCN[680] = t X i =1 m X j =1 a ij c j v i Commutativityin C = t X i =1 0 @ m X j =1 a ij c j 1 A v i PropertyDSA[280] = t X i =1 0 v i c j assolution = t X i =1 0 TheoremZSSM[286] Version2.02

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SubsectionD.DDimension345 = 0 PropertyZ[280] Thatdoesit. R hasbeenundeniablyshowntobealinearlydependentset. Noticehowtheswapofthetwosummationsissomucheasierinthethirdstepabove,asopposedto alltherearrangingandregroupingthattakesplaceinthepreviousproof.Inabouthalfthespace.And therearenoellipses ::: TheoremSSLD[341]canbeviewedasageneralizationofTheoremMVSLD[137].Weknowthat C m hasabasiswith m vectorsinitTheoremSUVB[325],soitisasetof m vectorsthatspans C m .By TheoremSSLD[341],anysetofmorethan m vectorsfrom C m willbelinearlydependent.Butthisis exactlytheconclusionwehaveinTheoremMVSLD[137].Maybethisisnotatotalshock,astheproofs ofboththeoremsrelyheavilyonTheoremHMVEI[64].ThebeautyofTheoremSSLD[341]isthatit appliesinanyvectorspace.Weillustratethegeneralityofthistheorem,andhintatitspower,inthenext example. ExampleLDP4 Linearlydependentsetin P 4 InExampleSSP4[313]weshowedthat S = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 ;x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +4 ;x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 2 +12 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 ;x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 x 3 +24 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(32 x +16 isaspanningsetfor W = f p x j p 2 P 4 ;p =0 g .SowecanapplyTheoremSSLD[341]to W with t =4.Hereisasetofvevectorsfrom W ,asyoumaycheckbyverifyingthateachisapolynomialof degree4orlessandhas x =2asaroot, T = f p 1 ;p 2 ;p 3 ;p 4 ;p 5 g W p 1 = x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 3 +2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 x +8 p 2 = )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 3 +6 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 p 3 =2 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x 3 +5 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x +2 p 4 = )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 4 +4 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x 2 +6 x p 5 =4 x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 x 2 +5 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 ByTheoremSSLD[341]weconcludethat T islinearlydependent,withnofurthercomputations. TheoremSSLD[341]isindeedpowerful,butourmainpurposeinprovingitrightnowwastomake surethatourdenitionofdimensionDenitionD[341]iswell-dened.Here'sthetheorem. TheoremBIS BaseshaveIdenticalSizes Supposethat V isavectorspacewithanitebasis B andasecondbasis C .Then B and C havethesame size. Proof Supposethat C hasmorevectorsthan B .Allowingforthepossibilitythat C isinnite,wecan replace C byasubsetthathasmorevectorsthan B .Asabasis, B isaspanningsetfor V DenitionB [325],soTheoremSSLD[341]saysthat C islinearlydependent.However,thiscontradictsthefactthat asabasis C islinearlyindependentDenitionB[325].So C mustalsobeaniteset,withsizelessthan, orequalto,thatof B Supposethat B hasmorevectorsthan C .Asabasis, C isaspanningsetfor V DenitionB[325],so TheoremSSLD[341]saysthat B islinearlydependent.However,thiscontradictsthefactthatasabasis B islinearlyindependentDenitionB[325].So C cannotbestrictlysmallerthan B Theonlypossibilityleftforthesizesof B and C isforthemtobeequal. TheoremBIS[344]tellsusthatifwendonenitebasisinavectorspace,thentheyallhavethesame size.ThisnallymakesDenitionD[341]unambiguous. Version2.02

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SubsectionD.DVSDimensionofVectorSpaces346 SubsectionDVS DimensionofVectorSpaces Wecannowcollectthedimensionofsomecommon,andnotsocommon,vectorspaces. TheoremDCM Dimensionof C m Thedimensionof C m ExampleVSCV[281]is m Proof TheoremSUVB[325]providesabasiswith m vectors. TheoremDP Dimensionof P n Thedimensionof P n ExampleVSP[281]is n +1. Proof ExampleBP[326]provides two baseswith n +1vectors.Takeyourpick. TheoremDM Dimensionof M mn Thedimensionof M mn ExampleVSM[281]is mn Proof ExampleBM[326]providesabasiswith mn vectors. ExampleDSM22 Dimensionofasubspaceof M 22 Itshouldnowbeplausiblethat Z = ab cd j 2 a + b +3 c +4 d =0 ; )]TJ/F21 10.9091 Tf 8.485 0 Td [(a +3 b )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 c )]TJ/F21 10.9091 Tf 10.909 0 Td [(d =0 isasubspaceofthevectorspace M 22 ExampleVSM[281].Itis.Tondthedimensionof Z wemust rstndabasis,thoughanyoldbasiswilldo. Firstconcentrateontheconditionsrelating a;b;c and d .Theyformahomogeneoussystemoftwo equationsinfourvariableswithcoecientmatrix 2134 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 Wecanrow-reducethismatrixtoobtain 1 022 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 Rewritethetwoequationsrepresentedbyeachrowofthismatrix,expressingthedependentvariables a and b intermsofthefreevariables c and d ,andweobtain, a = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 c )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 d b = c Wecannowwriteatypicalentryof Z strictlyintermsof c and d ,andwecandecomposetheresult, ab cd = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 c )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 dc cd = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 cc c 0 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 d 0 0 d = c )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 10 + d )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 01 Version2.02

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SubsectionD.DVSDimensionofVectorSpaces347 thisequationsaysthatanarbitrarymatrixin Z canbewrittenasalinearcombinationofthetwovectors in S = )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 10 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 01 soweknowthat Z = h S i = )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 10 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(20 01 Arethesetwomatricesvectorsalsolinearlyindependent?Beginwitharelationoflineardependenceon S a 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 10 + a 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 01 = O )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 a 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 a 2 a 1 a 1 a 2 = 00 00 Fromtheequalityofthetwoentriesinthelastrow,weconcludethat a 1 =0, a 2 =0.Thustheonly possiblerelationoflineardependenceisthetrivialone,andtherefore S islinearlyindependentDenition LI[308].So S isabasisfor V DenitionB[325].Finally,wecanconcludethatdim Z =2Denition D[341]since S hastwoelements. ExampleDSP4 Dimensionofasubspaceof P 4 InExampleBSP4[326]weshowedthat S = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 ;x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +4 ;x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 2 +12 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 ;x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 x 3 +24 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(32 x +16 isabasisfor W = f p x j p 2 P 4 ;p =0 g .Thus,thedimensionof W isfour,dim W =4. Notethatdim P 4 =5byTheoremDP[345],so W isasubspaceofdimension4withinthevector space P 4 ofdimension5,illustratingtheupcomingTheoremPSSD[358]. ExampleDC Dimensionofthecrazyvectorspace InExampleBC[328]wedeterminedthattheset R = f ; 0 ; ; 3 g fromthecrazyvectorspace, C ExampleCVS[283],isabasisfor C .ByDenitionD[341]weseethat C hasdimension2,dim C =2. Itispossibleforavectorspacetohavenonitebases,inwhichcasewesayithasinnitedimension. Manyofthebestexamplesofthisarevectorspacesoffunctions,whichleadtoconstructionslikeHilbert spaces.Wewillfocusexclusivelyonnite-dimensionalvectorspaces.OK,oneinnite-dimensionalexample, and then wewillfocusexclusivelyonnite-dimensionalvectorspaces. ExampleVSPUD Vectorspaceofpolynomialswithunboundeddegree Denetheset P by P = f p j p x isapolynomialin x g Ouroperationswillbethesameasthosedenedfor P n ExampleVSP[281]. Withnorestrictionsonthepossibledegreesofourpolynomials,anynitesetthatisacandidatefor spanning P willcomeupshort.WewillgiveaproofbycontradictionTechniqueCD[692].Tothisend, supposethatthedimensionof P isnite,saydim P = n Theset T = 1 ;x;x 2 ;:::;x n isalinearlyindependentsetcheckthis!containing n +1polynomials from P .However,abasisof P willbeaspanningsetof P containing n vectors.Thissituationisa contradictionofTheoremSSLD[341],soourassumptionthat P hasnitedimensionisfalse.Thus,we saydim P = 1 Version2.02

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SubsectionD.RNMRankandNullityofaMatrix348 SubsectionRNM RankandNullityofaMatrix Foranymatrix,wehaveseenthatwecanassociateseveralsubspaces|thenullspaceTheoremNSMS [296],thecolumnspaceTheoremCSMS[302],rowspaceTheoremRSMS[303]andtheleftnullspace TheoremLNSMS[303].Asvectorspaces,eachofthesehasadimension,andforthenullspaceand columnspace,theyareimportantenoughtowarrantnames. DenitionNOM NullityOfaMatrix Supposethat A isan m n matrix.Thenthe nullity of A isthedimensionofthenullspaceof A n A =dim N A ThisdenitioncontainsNotationNOM. 4 DenitionROM RankOfaMatrix Supposethat A isan m n matrix.Thenthe rank of A isthedimensionofthecolumnspaceof A r A =dim C A ThisdenitioncontainsNotationROM. 4 ExampleRNM Rankandnullityofamatrix Let'scomputetherankandnullityof A = 2 6 6 6 6 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1321 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(200401 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2410 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21161 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(114 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(123 )]TJ/F15 10.9091 Tf 8.485 0 Td [(163 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 7 7 7 7 7 7 5 Todothis,wewillrstrow-reducethematrixsincethatwillhelpusdeterminebasesforthenullspace andcolumnspace. 2 6 6 6 6 6 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(200401 00 1 030 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 000 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 00000 1 1 0000000 0000000 3 7 7 7 7 7 7 7 5 Fromthisrow-equivalentmatrixinreducedrow-echelonformwerecord D = f 1 ; 3 ; 4 ; 6 g and F = f 2 ; 5 ; 7 g Foreachindexin D ,TheoremBCS[239]createsasinglebasisvector.Intotalthebasiswillhave4 vectors,sothecolumnspaceof A willhavedimension4andwewrite r A =4. Foreachindexin F ,TheoremBNS[139]createsasinglebasisvector.Intotalthebasiswillhave3 vectors,sothenullspaceof A willhavedimension3andwewrite n A =3. Therewerenoaccidentsorcoincidencesinthepreviousexample|withtherow-reducedversionofa matrixinhand,therankandnullityareeasytocompute. TheoremCRN ComputingRankandNullity Supposethat A isan m n matrixand B isarow-equivalentmatrixinreducedrow-echelonformwith r Version2.02

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SubsectionD.RNNMRankandNullityofaNonsingularMatrix349 nonzerorows.Then r A = r and n A = n )]TJ/F21 10.9091 Tf 10.909 0 Td [(r Proof TheoremBCS[239]providesabasisforthecolumnspacebychoosingcolumnsof A thatcorrespond tothedependentvariablesinadescriptionofthesolutionsto LS A; 0 .Intheanalysisof B ,thereis onedependentvariableforeachleading1,onepernonzerorow,oroneperpivotcolumn.Sothereare r columnvectorsinabasisfor C A TheoremBNS[139]provideabasisforthenullspacebycreatingbasisvectorsofthenullspaceof A fromentriesof B ,oneforeachindependentvariable,onepercolumnwithoutaleading1.Sothereare n )]TJ/F21 10.9091 Tf 10.909 0 Td [(r columnvectorsinabasisfor n A EveryarchetypeAppendixA[698]thatinvolvesamatrixlistsitsrankandnullity.Youmayhave noticedasyoustudiedthearchetypesthatthelargerthecolumnspaceisthesmallerthenullspaceis.A simplecorollarystatesthistrade-osuccinctly.SeeTechniqueLC[696]. TheoremRPNC RankPlusNullityisColumns Supposethat A isan m n matrix.Then r A + n A = n Proof Let r bethenumberofnonzerorowsinarow-equivalentmatrixinreducedrow-echelonform.By TheoremCRN[347], r A + n A = r + n )]TJ/F21 10.9091 Tf 10.909 0 Td [(r = n Whenwerstintroduced r asourstandardnotationforthenumberofnonzerorowsinamatrixin reducedrow-echelonformyoumighthavethought r stoodforrows."Notreally|itstandsforrank"! SubsectionRNNM RankandNullityofaNonsingularMatrix Let'stakealookattherankandnullityofasquarematrix. ExampleRNSM Rankandnullityofasquarematrix Thematrix E = 2 6 6 6 6 6 6 6 6 4 04 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12231 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(39 )]TJ/F15 10.9091 Tf 8.485 0 Td [(39 )]TJ/F15 10.9091 Tf 8.485 0 Td [(19 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(494 )]TJ/F15 10.9091 Tf 8.485 0 Td [(16 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(46 )]TJ/F15 10.9091 Tf 8.485 0 Td [(259 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(38 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(424 8229309 3 7 7 7 7 7 7 7 7 5 isrow-equivalenttothematrixinreducedrow-echelonform, 2 6 6 6 6 6 6 6 6 6 4 1 000000 0 1 00000 00 1 0000 000 1 000 0000 1 00 00000 1 0 000000 1 3 7 7 7 7 7 7 7 7 7 5 Version2.02

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SubsectionD.RNNMRankandNullityofaNonsingularMatrix350 With n =7columnsand r =7nonzerorowsTheoremCRN[347]tellsustherankis r E =7andthe nullityis n E =7 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7=0. Thevalueofeitherthenullityortherankareenoughtocharacterizeanonsingularmatrix. TheoremRNNM RankandNullityofaNonsingularMatrix Supposethat A isasquarematrixofsize n .Thefollowingareequivalent. 1.Aisnonsingular. 2.Therankof A is n r A = n 3.Thenullityof A iszero, n A =0. Proof 2TheoremCSNM[242]saysthatif A isnonsingularthen C A = C n .If C A = C n ,then thecolumnspacehasdimension n byTheoremDCM[345],sotherankof A is n 3Suppose r A = n .ThenTheoremRPNC[348]gives n A = n )]TJ/F21 10.9091 Tf 10.909 0 Td [(r A TheoremRPNC[348] = n )]TJ/F21 10.9091 Tf 10.909 0 Td [(n Hypothesis =0 1Suppose n A =0,soabasisforthenullspaceof A istheemptyset.Thisimpliesthat N A = f 0 g andTheoremNMTNS[74]says A isnonsingular. Withanewequivalenceforanonsingularmatrix,wecanupdateourlistofequivalencesTheorem NME5[331]whichnowbecomesalistrequiringdoubledigitstonumber. TheoremNME6 NonsingularMatrixEquivalences,Round6 Supposethat A isasquarematrixofsize n .Thefollowingareequivalent. 1. A isnonsingular. 2. A row-reducestotheidentitymatrix. 3.Thenullspaceof A containsonlythezerovector, N A = f 0 g 4.Thelinearsystem LS A; b hasauniquesolutionforeverypossiblechoiceof b 5.Thecolumnsof A arealinearlyindependentset. 6. A isinvertible. 7.Thecolumnspaceof A is C n C A = C n 8.Thecolumnsof A areabasisfor C n 9.Therankof A is n r A = n 10.Thenullityof A iszero, n A =0. Proof BuildingonTheoremNME5[331]wecanaddtwoofthestatementsfromTheoremRNNM[349]. Version2.02

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SubsectionD.READReadingQuestions351 SubsectionREAD ReadingQuestions 1.Whatisthedimensionofthevectorspace P 6 ,thesetofallpolynomialsofdegree6orless? 2.Howaretherankandnullityofamatrixrelated? 3.Explainwhywemightsaythatanonsingularmatrixhasfullrank." Version2.02

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SubsectionD.EXCExercises352 SubsectionEXC Exercises C20 Thearchetypeslistedbelowarematrices,orsystemsofequationswithcoecientmatrices.For each,computethenullityandrankofthematrix.Thisinformationislistedforeacharchetypealongwith thenumberofcolumnsinthematrix,soastoillustrateTheoremRPNC[348],andnoticehowitcould havebeencomputedimmediatelyafterthedeterminationofthesets D and F associatedwiththereduced row-echelonformofthematrix. ArchetypeA[702] ArchetypeB[707] ArchetypeC[712] ArchetypeD[716]/ArchetypeE[720] ArchetypeF[724] ArchetypeG[729]/ArchetypeH[733] ArchetypeI[737] ArchetypeJ[741] ArchetypeK[746] ArchetypeL[750] ContributedbyRobertBeezer C30 Forthematrix A below,computethedimensionofthenullspaceof A ,dim N A A = 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3119 121 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 31 )]TJ/F15 10.9091 Tf 8.485 0 Td [(368 212 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 7 7 5 ContributedbyRobertBeezerSolution[353] C31 Theset W belowisasubspaceof C 4 .Findthedimensionof W W = 8 > > < > > : 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 4 1 3 7 7 5 ; 2 6 6 4 3 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 5 ; 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 2 5 3 7 7 5 9 > > = > > ; + ContributedbyRobertBeezerSolution[353] C40 InExampleLDP4[344]wedeterminedthatthesetofvepolynomials, T ,islinearlydependentby asimpleinvocationofTheoremSSLD[341].Provethat T islinearlydependentfromscratch,beginning withDenitionLI[308]. ContributedbyRobertBeezer M20 M 22 isthevectorspaceof2 2matrices.Let S 22 denotethesetofall2 2symmetricmatrices. Thatis S 22 = A 2 M 22 j A t = A aShowthat S 22 isasubspaceof M 22 bExhibitabasisfor S 22 andprovethatithastherequiredproperties. cWhatisthedimensionof S 22 ? ContributedbyRobertBeezerSolution[353] Version2.02

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SubsectionD.EXCExercises353 M21 A2 2matrix B isuppertriangularif[ B ] 21 =0.Let UT 2 bethesetofall2 2uppertriangular matrices.Then UT 2 isasubspaceofthevectorspaceofall2 2matrices, M 22 youmayassumethis. Determinethedimensionof UT 2 providing all ofthenecessaryjusticationsforyouranswer. ContributedbyRobertBeezerSolution[354] Version2.02

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SubsectionD.SOLSolutions354 SubsectionSOL Solutions C30 ContributedbyRobertBeezerStatement[351] Rowreduce A A RREF )443()223()222()443(! 2 6 6 4 1 0011 0 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 00 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 00000 3 7 7 5 So r =3forthismatrix.Then dim N A = n A DenitionNOM[347] = n A + r A )]TJ/F21 10.9091 Tf 10.909 0 Td [(r A =5 )]TJ/F21 10.9091 Tf 10.909 0 Td [(r A TheoremRPNC[348] =5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3TheoremCRN[347] =2 WecouldalsouseTheoremBNS[139]andcreateabasisfor N A with n )]TJ/F21 10.9091 Tf 10.111 0 Td [(r =5 )]TJ/F15 10.9091 Tf 10.111 0 Td [(3=2vectorsbecause thesolutionsaredescribedwith2freevariablesandarriveatthedimensionasthesizeofthisbasis. C31 ContributedbyRobertBeezerStatement[351] WewillappealtoTheoremBS[157]oryoucouldconsiderthisanappealtoTheoremBCS[239].Put thethreecolumnvectorsofthisspanningsetintoamatrixascolumnsandrow-reduce. 2 6 6 4 23 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(30 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 412 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(25 3 7 7 5 RREF )443()223()222()443(! 2 6 6 4 1 01 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 000 000 3 7 7 5 Thepivotcolumnsare D = f 1 ; 2 g sowecankeep"thevectorscorrespondingtothepivotcolumnsand set T = 8 > > < > > : 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 4 1 3 7 7 5 ; 2 6 6 4 3 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 5 9 > > = > > ; andconcludethat W = h T i and T islinearlyindependent.Inotherwords, T isabasiswithtwovectors, so W hasdimension2. M20 ContributedbyRobertBeezerStatement[351] aWewillusethethreecriteriaofTheoremTSS[293].Thezerovectorof M 22 isthezeromatrix, O DenitionZM[185],whichisasymmetricmatrix.So S 22 isnotempty,since O2 S 22 Supposethat A and B aretwomatricesin S 22 .Thenweknowthat A t = A and B t = B .Wewantto knowif A + B 2 S 22 ,sotest A + B formembership, A + B t = A t + B t TheoremTMA[186] = A + BA;B 2 S 22 So A + B issymmetricandqualiesformembershipin S 22 Supposethat A 2 S 22 and 2 C .Is A 2 S 22 ?Weknowthat A t = A .Nowcheckthat, A t = A t TheoremTMSM[187] Version2.02

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SubsectionD.SOLSolutions355 = AA 2 S 22 So A isalsosymmetricandqualiesformembershipin S 22 WiththethreecriteriaofTheoremTSS[293]fullled,weseethat S 22 isasubspaceof M 22 bAnarbitrarymatrixfrom S 22 canbewrittenas ab bd .Wecanexpressthismatrixas ab bd = a 0 00 + 0 b b 0 + 00 0 d = a 10 00 + b 01 10 + d 00 01 thisequationsaysthattheset T = 10 00 ; 01 10 ; 00 01 spans S 22 .Isitalsolinearlyindependent? Writearelationoflineardependenceon S O = a 1 10 00 + a 2 01 10 + a 3 00 01 00 00 = a 1 a 2 a 2 a 3 TheequalityofthesetwomatricesDenitionME[182]tellsusthat a 1 = a 2 = a 3 =0,andtheonly relationoflineardependenceon T istrivial.So T islinearlyindependent,andhenceisabasisof S 22 cThebasis T foundinpartbhassize3.SobyDenitionD[341],dim S 22 =3. M21 ContributedbyRobertBeezerStatement[352] Atypicalmatrixfrom UT 2 lookslike ab 0 c where a;b;c 2 C arearbitraryscalars.Observingthiswecanthenwrite ab 0 c = a 10 00 + b 01 00 + c 00 01 whichsaysthat R = 10 00 ; 01 00 ; 00 01 isaspanningsetfor UT 2 DenitionTSVS[313].Is R islinearlyindependent?Ifso,itisabasisfor UT 2 Soconsiderarelationoflineardependenceon R 1 10 00 + 2 01 00 + 3 00 01 = O = 00 00 Fromthisequation,onerapidlyarrivesattheconclusionthat 1 = 2 = 3 =0.So R isalinearly independentsetDenitionLI[308],andhenceisabasisDenitionB[325]for UT 2 .Now,wesimply countupthesizeoftheset R toseethatthedimensionof UT 2 isdim UT 2 =3. Version2.02

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SectionPDPropertiesofDimension356 SectionPD PropertiesofDimension Oncethedimensionofavectorspaceisknown,thenthedeterminationofwhetherornotasetofvectors islinearlyindependent,orifitspansthevectorspace,canoftenbemucheasier.Inthissectionwewill stateaworkhorsetheoremandthenapplyittothecolumnspaceandrowspaceofamatrix.Itwillalso helpusdescribeasuper-basisfor C m SubsectionGT Goldilocks'Theorem Webeginwithausefultheoremthatwewillneedlater,andintheproofofthemaintheoreminthis subsection.Thistheoremsaysthatwecanextendlinearlyindependentsets,onevectoratatime,by addingvectorsfromoutsidethespanofthelinearlyindependentset,allthewhilepreservingthelinear independenceoftheset. TheoremELIS ExtendingLinearlyIndependentSets Suppose V isvectorspaceand S isalinearlyindependentsetofvectorsfrom V .Suppose w isavector suchthat w 62h S i .Thentheset S 0 = S [f w g islinearlyindependent. Proof Suppose S = f v 1 ; v 2 ; v 3 ;:::; v m g andbeginwitharelationoflineardependenceon S 0 a 1 v 1 + a 2 v 2 + a 3 v 3 + + a m v m + a m +1 w = 0 : Therearetwocasestoconsider.Firstsupposethat a m +1 =0.Thentherelationoflineardependenceon S 0 becomes a 1 v 1 + a 2 v 2 + a 3 v 3 + + a m v m = 0 : andbythelinearindependenceoftheset S ,weconcludethat a 1 = a 2 = a 3 = = a m =0.Soallofthe scalarsintherelationoflineardependenceon S 0 arezero. Inthesecondcase,supposethat a m +1 6 =0.Thentherelationoflineardependenceon S 0 becomes a m +1 w = )]TJ/F21 10.9091 Tf 8.484 0 Td [(a 1 v 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(a 2 v 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(a 3 v 3 )-222()]TJ/F21 10.9091 Tf 36.969 0 Td [(a m v m w = )]TJ/F21 10.9091 Tf 16.718 7.38 Td [(a 1 a m +1 v 1 )]TJ/F21 10.9091 Tf 19.143 7.38 Td [(a 2 a m +1 v 2 )]TJ/F21 10.9091 Tf 19.143 7.38 Td [(a 3 a m +1 v 3 )-222()]TJ/F21 10.9091 Tf 43.576 7.38 Td [(a m a m +1 v m Thisequationexpresses w asalinearcombinationofthevectorsin S ,contrarytotheassumptionthat w 62h S i ,sothiscaseleadstoacontradiction. Therstcaseyieldedonlyatrivialrelationoflineardependenceon S 0 andthesecondcaseledtoa contradiction.So S 0 isalinearlyindependentsetsinceanyrelationoflineardependenceistrivial. Inthestory GoldilocksandtheThreeBears ,theyounggirlGoldilocksvisitstheemptyhouseofthe threebearswhileoutwalkinginthewoods.Onebowlofporridgeistoohot,theothertoocold,thethird isjustright.Onechairistoohard,onetoosoft,thethirdisjustright.Soitiswithsetsofvectors|some aretoobiglinearlydependent,somearetoosmalltheydon'tspan,andsomearejustrightbases. Here'sGoldilocks'Theorem. TheoremG Goldilocks Supposethat V isavectorspaceofdimension t .Let S = f v 1 ; v 2 ; v 3 ;:::; v m g beasetofvectorsfrom V .Then Version2.02

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SubsectionPD.GTGoldilocks'Theorem357 1.If m>t ,then S islinearlydependent. 2.If m
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SubsectionPD.GTGoldilocks'Theorem358 werebothbasesfor P n ExampleVSP[281].Supposewehadrstveriedthat B wasabasis,sowe wouldthenknowthatdim P n = n +1.Thesizeof C is n +1,therightsizetobeabasis.Wecould thenverifythat C islinearlyindependent.Wewouldnothavetomakeanyspecialeortstoprovethat C spans P n ,sinceTheoremG[355]wouldallowustoconcludethispropertyof C directly.Thenwewould beabletosaythat C isabasisof P n also. ExampleBDM22 Basisbydimensionin M 22 InExampleDSM22[345]weshowedthat B = )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 10 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 01 isabasisforthesubspace Z of M 22 ExampleVSM[281]givenby Z = ab cd j 2 a + b +3 c +4 d =0 ; )]TJ/F21 10.9091 Tf 8.485 0 Td [(a +3 b )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 c )]TJ/F21 10.9091 Tf 10.909 0 Td [(d =0 Thistellsusthatdim Z =2.Inthisexamplewewillndanotherbasis.Wecanconstructtwonew matricesin Z byforminglinearcombinationsofthematricesin B 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 10 + )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 01 = 22 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 10 +1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 01 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(83 31 Thentheset C = 22 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(83 31 hastherightsizetobeabasisof Z .Let'sseeifitisalinearlyindependentset.Therelationoflinear dependence a 1 22 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 + a 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(83 31 = O 2 a 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 a 2 2 a 1 +3 a 2 2 a 1 +3 a 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 a 1 + a 2 = 00 00 leadstothehomogeneoussystemofequationswhosecoecientmatrix 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 23 23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(31 3 7 7 5 row-reducesto 2 6 6 4 1 0 0 1 00 00 3 7 7 5 Sowith a 1 = a 2 =0astheonlysolution,thesetislinearlyindependent.NowwecanapplyTheoremG [355]toseethat C alsospans Z andthereforeisasecondbasisfor Z ExampleSVP4 Setsofvectorsin P 4 InExampleBSP4[326]weshowedthat B = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 ;x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +4 ;x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 x 2 +12 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 ;x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 x 3 +24 x 2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(32 x +16 Version2.02

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SubsectionPD.RTRanksandTransposes359 isabasisfor W = f p x j p 2 P 4 ;p =0 g .Sodim W =4. Theset 3 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 ; 2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x +6 ;x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 + x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 isasubsetof W checkthisandithappenstobelinearlyindependentcheckthis,too.However,by TheoremG[355]itcannotspan W Theset 3 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 ; 2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x +6 ;x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 + x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 ; )]TJ/F21 10.9091 Tf 8.485 0 Td [(x 4 +2 x 3 +5 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(10 x;x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(16 isanothersubsetof W checkthisandTheoremG[355]tellsusthatitmustbelinearlydependent. Theset x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 ;x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x;x 3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 2 ;x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x 3 isathirdsubsetof W checkthisandislinearlyindependentcheckthis.Sinceithastherightsizeto beabasis,andislinearlyindependent,TheoremG[355]tellsusthatitalsospans W ,andthereforeisa basisof W AsimpleconsequenceofTheoremG[355]istheobservationthatpropersubspaceshavestrictlysmaller dimensions.Hopefullythismayseemintuitivelyobvious,butitstillrequiresproof,andwewillcitethis resultlater. TheoremPSSD ProperSubspaceshaveSmallerDimension Supposethat U and V aresubspacesofthevectorspace W ,suchthat U V .Thendim U < dim V Proof Supposethatdim U = m anddim V = t .Then U hasabasis B ofsize m .If m>t ,thenby TheoremG[355], B islinearlydependent,whichisacontradiction.If m = t ,thenbyTheoremG[355], B spans V .Then U = h B i = V ,alsoacontradiction.Allthatremainsisthat m
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SubsectionPD.RTRanksandTransposes360 TheoremRMRT RankofaMatrixistheRankoftheTranspose Suppose A isan m n matrix.Then r A = r )]TJ/F21 10.9091 Tf 5 -8.836 Td [(A t Proof Supposewerow-reduce A tothematrix B inreducedrow-echelonform,and B has r non-zero rows.Thequantity r tellsusthreethingsabout B :thenumberofleading1's,thenumberofnon-zero rowsandthenumberofpivotcolumns.Forthisproofwewillbeinterestedinthelattertwo. TheoremBRS[245]andTheoremBCS[239]eachhasaconclusionthatprovidesabasis,fortherow spaceandthecolumnspace,respectively.Ineachcase,thesebasescontain r vectors.Thisobservation makesthefollowinggo. r A =dim C A DenitionROM[347] = r TheoremBCS[239] =dim R A TheoremBRS[245] =dim )]TJ/F24 10.9091 Tf 5 -8.837 Td [(C )]TJ/F21 10.9091 Tf 5 -8.837 Td [(A t TheoremCSRST[247] = r )]TJ/F21 10.9091 Tf 5 -8.837 Td [(A t DenitionROM[347] JacobLinenthalhelpedwiththisproof. Thissaysthattherowspaceandthecolumnspaceofamatrixhavethesamedimension,whichshould beverysurprising.Itdoes not saythatcolumnspaceandtherowspaceareidentical.Indeed,ifthematrix isnotsquare,thenthesizesnumberofslotsofthevectorsineachspacearedierent,sothesetsarenot evencomparable. ItisnothardtoconstructbyyourselfexamplesofmatricesthatillustrateTheoremRMRT[359],since itappliesequallywellto any matrix.Grabamatrix,row-reduceit,countthenonzerorowsortheleading 1's.That'stherank.Transposethematrix,row-reducethat,countthenonzerorowsortheleading1's. That'stherankofthetranspose.Thetheoremsaysthetwowillbeequal.Here'sanexampleanyway. ExampleRRTI Rank,rankoftranspose,ArchetypeI ArchetypeI[737]hasa4 7coecientmatrixwhichrow-reducesto 2 6 6 4 1 40021 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 00 1 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(35 000 1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(66 0000000 3 7 7 5 sotherankis3.Row-reducingthetransposeyields 2 6 6 6 6 6 6 6 6 6 4 1 00 )]TJ/F19 7.9701 Tf 9.681 4.295 Td [(31 7 0 1 0 12 7 00 1 13 7 0000 0000 0000 0000 3 7 7 7 7 7 7 7 7 7 5 : demonstratingthattherankofthetransposeisalso3. Version2.02

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SubsectionPD.DFSDimensionofFourSubspaces361 SubsectionDFS DimensionofFourSubspaces Thattherankofamatrixequalstherankofitstransposeisafundamentalandsurprisingresult.However, applyingTheoremFS[263]wecaneasilydeterminethedimensionofallfourfundamentalsubspaces associatedwithamatrix. TheoremDFS DimensionsofFourSubspaces Supposethat A isan m n matrix,and B isarow-equivalentmatrixinreducedrow-echelonformwith r nonzerorows.Then 1.dim N A = n )]TJ/F21 10.9091 Tf 10.909 0 Td [(r 2.dim C A = r 3.dim R A = r 4.dim L A = m )]TJ/F21 10.9091 Tf 10.909 0 Td [(r Proof If A row-reducestoamatrixinreducedrow-echelonformwith r nonzerorows,thenthematrix C ofextendedechelonformDenitionEEF[261]willbean r n matrixinreducedrow-echelonformwith nozerorowsand r pivotcolumnsTheoremPEEF[262].Similarly,thematrix L ofextendedechelon formDenitionEEF[261]willbean m )]TJ/F21 10.9091 Tf 11.026 0 Td [(r m matrixinreducedrow-echelonformwithnozerorows and m )]TJ/F21 10.9091 Tf 10.909 0 Td [(r pivotcolumnsTheoremPEEF[262]. dim N A =dim N C TheoremFS[263] = n )]TJ/F21 10.9091 Tf 10.909 0 Td [(r TheoremBNS[139] dim C A =dim N L TheoremFS[263] = m )]TJ/F15 10.9091 Tf 10.909 0 Td [( m )]TJ/F21 10.9091 Tf 10.909 0 Td [(r TheoremBNS[139] = r dim R A =dim R C TheoremFS[263] = r TheoremBRS[245] dim L A =dim R L TheoremFS[263] = m )]TJ/F21 10.9091 Tf 10.909 0 Td [(r TheoremBRS[245] Therearemanydierentwaystostateandprovethisresult,andindeed,theequalityofthedimensions ofthecolumnspaceandrowspaceisjustaslightexpansionofTheoremRMRT[359].However,we haverestrictedourtechniquestoapplyingTheoremFS[263]andthendeterminingdimensionswithbases providedbyTheoremBNS[139]andTheoremBRS[245].Thisprovidesanappealingsymmetrytothe resultsandtheproof. Version2.02

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SubsectionPD.DSDirectSums362 SubsectionDS DirectSums SomeofthemoreadvancedideasinlinearalgebraarecloselyrelatedtodecomposingTechniqueDC[694] vectorspacesintodirectsumsofsubspaces.Withourpreviousresultsaboutbasesanddimension,now istherighttimetostateandcollectafewresultsaboutdirectsums,thoughwewillonlymentionthese resultsinpassinguntilwegettoSectionNLT[610],wheretheywillgetaheavyworkout. Adirectsumisashort-handwaytodescribetherelationshipbetweenavectorspaceandtwo,ormore, ofitssubspaces.Aswewilluseit,itisnotawaytoconstructnewvectorspacesfromothers. DenitionDS DirectSum Supposethat V isavectorspacewithtwosubspaces U and W suchthatforevery v 2 V 1.Thereexistsvectors u 2 U w 2 W suchthat v = u + w 2.If v = u 1 + w 1 and v = u 2 + w 2 where u 1 ; u 2 2 U w 1 ; w 2 2 W then u 1 = u 2 and w 1 = w 2 Then V isthe directsum of U and W andwewrite V = U W ThisdenitioncontainsNotationDS. 4 Informally,whenwesay V isthedirectsumofthesubspaces U and W ,wearesayingthateachvector of V canalwaysbeexpressedasthesumofavectorfrom U andavectorfrom W ,andthisexpression canonlybeaccomplishedinonewayi.e.uniquely.Thisstatementshouldbegintofeelsomethinglike ourdenitionsofnonsingularmatricesDenitionNM[71]andlinearindependenceDenitionLI[308]. Itshouldnotbehardtoimaginethenaturalextensionofthisdenitiontothecaseofmorethantwo subspaces.Couldyouprovideacarefuldenitionof V = U 1 U 2 U 3 ::: U m ExercisePD.M50[366]? ExampleSDS Simpledirectsum In C 3 ,dene v 1 = 2 4 3 2 5 3 5 v 2 = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 1 3 5 v 3 = 2 4 2 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 3 5 Then C 3 = hf v 1 ; v 2 gihf v 3 gi .Thisstatementderivesfromthefactthat B = f v 1 ; v 2 ; v 3 g isbasisfor C 3 .Thespanningpropertyof B yieldsthedecompositionofanyvectorintoasumofvectorsfromthetwo subspaces,andthelinearindependenceof B yieldstheuniquenessofthedecomposition.Wewillillustrate theseclaimswithanumericalexample. Choose v = 2 4 10 1 6 3 5 .Then v =2 v 1 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 v 2 +1 v 3 = v 1 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 v 2 + v 3 wherewehaveaddedparenthesesforemphasis.Obviously1 v 3 2hf v 3 gi ,while2 v 1 + )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 v 2 2hf v 1 ; v 2 gi TheoremVRRB[317]providestheuniquenessofthescalarsintheselinearcombinations. ExampleSDS[361]iseasytogeneralizeintoatheorem. TheoremDSFB DirectSumFromaBasis Supposethat V isavectorspacewithabasis B = f v 1 ; v 2 ; v 3 ;:::; v n g .Dene U = hf v 1 ; v 2 ; v 3 ;:::; v m gi W = hf v m +1 ; v m +2 ; v m +3 ;:::; v n gi Version2.02

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SubsectionPD.DSDirectSums363 Then V = U W Proof Chooseanyvector v 2 V .ThenbyTheoremVRRB[317]thereareuniquescalars, a 1 ;a 2 ;a 3 ;:::;a n suchthat v = a 1 v 1 + a 2 v 2 + a 3 v 3 + + a n v n = a 1 v 1 + a 2 v 2 + a 3 v 3 + + a m v m + a m +1 v m +1 + a m +2 v m +2 + a m +3 v m +3 + + a n v n = u + w wherewehaveimplicitlydened u and w inthelastline.Itshouldbeclearthat u 2 U ,andsimilarly, w 2 W andnotsimplybythechoiceoftheirnames. Supposewehadanotherdecompositionof v ,say v = u + w .Thenwecouldwrite u asalinear combinationof v 1 through v m ,sayusingscalars b 1 ;b 2 ;b 3 ;:::;b m .Andwecouldwrite w asalinear combinationof v m +1 through v n ,sayusingscalars c 1 ;c 2 ;c 3 ;:::;c n )]TJ/F22 7.9701 Tf 6.586 0 Td [(m .Thesetwocollectionsofscalars wouldthentogethergivealinearcombinationof v 1 through v n thatequals v .Bytheuniquenessof a 1 ;a 2 ;a 3 ;:::;a n a i = b i for1 i m and a m + i = c i for1 i n )]TJ/F21 10.9091 Tf 11.299 0 Td [(m .Fromtheequalityofthese scalarsweconcludethat u = u and w = w .SowithbothconditionsofDenitionDS[361]fullledwe seethat V = U W Givenonesubspaceofavectorspace,wecanalwaysndanothersubspacethatwillpairwiththerst toformadirectsum.Themainideaofthistheorem,anditsproof,istheideaofextendingalinearly independentsubsetintoabasiswithrepeatedapplicationsofTheoremELIS[355]. TheoremDSFOS DirectSumFromOneSubspace Supposethat U isasubspaceofthevectorspace V .Thenthereexistsasubspace W of V suchthat V = U W Proof If U = V ,thenchoose W = f 0 g .Otherwise,chooseabasis B = f v 1 ; v 2 ; v 3 ;:::; v m g for U Thensince B isalinearlyindependentset,TheoremELIS[355]tellsusthereisavector v m +1 in V ,but notin U ,suchthat B [f v m +1 g islinearlyindependent.Denethesubspace U 1 = h B [f v m +1 gi Wecanrepeatthisprocedure,inthecasewere U 1 6 = V ,creatinganewvector v m +2 in V ,but notin U 1 ,andanewsubspace U 2 = h B [f v m +1 ; v m +2 gi .Ifwecontinuerepeatingthisprocedure, eventually, U k = V forsome k ,andwecannolongerapplyTheoremELIS[355].Nomatter,inthiscase B [f v m +1 ; v m +2 ;:::; v m + k g isalinearlyindependentsetthatspans V ,i.e.abasisfor V Dene W = hf v m +1 ; v m +2 ;:::; v m + k gi .WenowareexactlyinpositiontoapplyTheoremDSFB[361] andseethat V = U W Thereareseveraldierentwaystodeneadirectsum.Ournexttwotheoremsgiveequivalences TechniqueE[690]fordirectsums,andthereforecouldhavebeenemployedasdenitions.Therst shouldfurthercementthenotionthatadirectsumhassomeconnectionwithlinearindependence. TheoremDSZV DirectSumsandZeroVectors Suppose U and W aresubspacesofthevectorspace V .Then V = U W ifandonlyif 1.Forevery v 2 V ,thereexistsvectors u 2 U w 2 W suchthat v = u + w 2.Whenever 0 = u + w with u 2 U w 2 W then u = w = 0 Proof Therstconditionisidenticalinthedenitionandthetheorem,soweonlyneedtoestablishthe equivalenceofthesecondconditions. Version2.02

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SubsectionPD.DSDirectSums364 Assumethat V = U W ,accordingtoDenitionDS[361].ByPropertyZ[280], 0 2 V and 0 = 0 + 0 .Ifwealsoassumethat 0 = u + w ,thentheuniquenessofthedecompositiongives u = 0 and w = 0 Supposethat v 2 V v = u 1 + w 1 and v = u 2 + w 2 where u 1 ; u 2 2 U w 1 ; w 2 2 W .Then 0 = v )]TJ/F36 10.9091 Tf 10.909 0 Td [(v PropertyAI[280] = u 1 + w 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [( u 2 + w 2 = u 1 )]TJ/F36 10.9091 Tf 10.909 0 Td [(u 2 + w 1 )]TJ/F36 10.9091 Tf 10.909 0 Td [(w 2 PropertyAA[279] ByPropertyAC[279], u 1 )]TJ/F36 10.9091 Tf 11.343 0 Td [(u 2 2 U and w 1 )]TJ/F36 10.9091 Tf 11.344 0 Td [(w 2 2 W .Wecannowapplyourhypothesis,thesecond statementofthetheorem,toconcludethat u 1 )]TJ/F36 10.9091 Tf 10.909 0 Td [(u 2 = 0w 1 )]TJ/F36 10.9091 Tf 10.909 0 Td [(w 2 = 0 u 1 = u 2 w 1 = w 2 whichestablishestheuniquenessneededforthesecondconditionofthedenition. Oursecondequivalencelendsfurthercredencetocallingadirectsumadecomposition.Thetwo subspacesofadirectsumhavenonontrivialelementsincommon. TheoremDSZI DirectSumsandZeroIntersection Suppose U and W aresubspacesofthevectorspace V .Then V = U W ifandonlyif 1.Forevery v 2 V ,thereexistsvectors u 2 U w 2 W suchthat v = u + w 2. U W = f 0 g Proof Therstconditionisidenticalinthedenitionandthetheorem,soweonlyneedtoestablishthe equivalenceofthesecondconditions. Assumethat V = U W ,accordingtoDenitionDS[361].ByPropertyZ[280]andDenition SI[685], f 0 g U W .Toestablishtheoppositeinclusion,supposethat x 2 U W .Then,since x isan elementofboth U and W ,wecanwritetwodecompositionsof x asavectorfrom U plusavectorfrom W x = x + 0x = 0 + x Bytheuniquenessofthedecomposition,weseetwicethat x = 0 and U W f 0 g .ApplyingDenition SE[684],wehave U W = f 0 g Assumethat U W = f 0 g .Andassumefurtherthat v 2 V issuchthat v = u 1 + w 1 and v = u 2 + w 2 where u 1 ; u 2 2 U w 1 ; w 2 2 W .Dene x = u 1 )]TJ/F36 10.9091 Tf 11.235 0 Td [(u 2 .thenbyPropertyAC[279], x 2 U Also x = u 1 )]TJ/F36 10.9091 Tf 10.909 0 Td [(u 2 = v )]TJ/F36 10.9091 Tf 10.909 0 Td [(w 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [( v )]TJ/F36 10.9091 Tf 10.909 0 Td [(w 2 = v )]TJ/F36 10.9091 Tf 10.909 0 Td [(v )]TJ/F15 10.9091 Tf 10.909 0 Td [( w 1 )]TJ/F36 10.9091 Tf 10.909 0 Td [(w 2 = w 2 )]TJ/F36 10.9091 Tf 10.909 0 Td [(w 1 So x 2 W byPropertyAC[279].Thus, x 2 U W = f 0 g DenitionSI[685].So x = 0 and u 1 )]TJ/F36 10.9091 Tf 10.909 0 Td [(u 2 = 0w 2 )]TJ/F36 10.9091 Tf 10.909 0 Td [(w 1 = 0 u 1 = u 2 w 2 = w 1 Version2.02

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SubsectionPD.DSDirectSums365 yieldingthedesireduniquenessofthesecondconditionofthedenition. IfthestatementofTheoremDSZV[362]didnotremindyouoflinearindependence,thenexttheorem shouldestablishtheconnection. TheoremDSLI DirectSumsandLinearIndependence Suppose U and W aresubspacesofthevectorspace V with V = U W .Supposethat R isalinearly independentsubsetof U and S isalinearlyindependentsubsetof W .Then R [ S isalinearlyindependent subsetof V Proof Let R = f u 1 ; u 2 ; u 3 ;:::; u k g and S = f w 1 ; w 2 ; w 3 ;:::; w ` g .Beginwitharelationoflinear dependenceDenitionRLD[308]ontheset R [ S usingscalars a 1 ;a 2 ;a 3 ;:::;a k and b 1 ;b 2 ;b 3 ;:::;b ` Then, 0 = a 1 u 1 + a 2 u 2 + a 3 u 3 + + a k u k + b 1 w 1 + b 2 w 2 + b 3 w 3 + + b ` w ` = a 1 u 1 + a 2 u 2 + a 3 u 3 + + a k u k + b 1 w 1 + b 2 w 2 + b 3 w 3 + + b ` w ` = u + w wherewehavemadeanimplicitdenitionofthevectors u 2 U w 2 W .ApplyingTheoremDSZV[362] weconcludethat u = a 1 u 1 + a 2 u 2 + a 3 u 3 + + a k u k = 0 w = b 1 w 1 + b 2 w 2 + b 3 w 3 + + b ` w ` = 0 Nowthelinearindependenceof R and S individuallyyields a 1 = a 2 = a 3 = = a k =0 b 1 = b 2 = b 3 = = b ` =0 Forcedtoacknowledgethatonlyatriviallinearcombinationyieldsthezerovector,DenitionLI[308]says theset R [ S islinearlyindependentin V Ourlasttheoreminthiscollectionwillgosomewaystowardsexplainingthewordsum"inthemoniker directsum,"whilealsopartiallyexplainingwhytheseresultsappearinasectiondevotedtoadiscussion ofdimension. TheoremDSD DirectSumsandDimension Suppose U and W aresubspacesofthevectorspace V with V = U W .Thendim V =dim U +dim W Proof Wewillestablishthisequalityofpositiveintegerswithtwoinequalities.Wewillneedabasisof U callit B andabasisof W callit C First,notethat B and C havesizesequaltothedimensionsoftherespectivesubspaces.Theunion ofthesetwolinearlyindependentsets, B [ C willbelinearlyindependentin V byTheoremDSLI[364]. Further,thetwobaseshavenovectorsincommonbyTheoremDSZI[363],since B C f 0 g andthezero vectorisneveranelementofalinearlyindependentsetExerciseLI.T10[144].Sothesizeoftheunionis exactlythesumofthedimensionsof U and W .ByTheoremG[355]thesizeof B [ C cannotexceedthe dimensionof V withoutbeinglinearlydependent.Theseobservationsgiveusdim U +dim W dim V Grabanyvector v 2 V .ThenbyTheoremDSZI[363]wecanwrite v = u + w with u 2 U and w 2 W Individually,wecanwrite u asalinearcombinationofthebasiselementsin B ,andsimilarly,wecanwrite w asalinearcombinationofthebasiselementsin C ,sincethebasesarespanningsetsfortheirrespective subspaces.Thesetwosetsofscalarswillprovidealinearcombinationofallofthevectorsin B [ C which Version2.02

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SubsectionPD.READReadingQuestions366 willequal v .Theupshotofthisisthat B [ C isaspanningsetfor V .ByTheoremG[355],thesizeof B [ C cannotbesmallerthanthedimensionof V withoutfailingtospan V .Theseobservationsgiveus dim U +dim W dim V Thereisacertainappeallingsymmetryinthepreviousproof,wherebothlinearindependenceand spanningpropertiesofthebasesareused,bothofthersttwoconclusionsofTheoremG[355]areemployed, andwehavequotedbothofthetwoconditionsofTheoremDSZI[363]. Onenaltheoremtellsusthatwecansuccessivelydecomposedirectsumsintosumsofsmallerand smallersubspaces. TheoremRDS RepeatedDirectSums Suppose V isavectorspacewithsubspaces U and W with V = U W .Supposethat X and Y are subspacesof W with W = X Y .Then V = U X Y Proof Supposethat v 2 V .Thendueto V = U W ,thereexistvectors u 2 U and w 2 W suchthat v = u + w .Dueto W = X Y ,thereexistvectors x 2 X and y 2 Y suchthat w = x + y .Alltogether, v = u + w = u + x + y whichwouldbetherstconditionofadenitionofa3-waydirectproduct.Nowconsidertheuniqueness. Supposethat v = u 1 + x 1 + y 1 v = u 2 + x 2 + y 2 Because x 1 + y 1 2 W x 2 + y 2 2 W ,and V = U W ,weconcludethat u 1 = u 2 x 1 + y 1 = x 2 + y 2 Fromthesecondequality,anapplicationof W = X Y yieldstheconclusions x 1 = x 2 and y 1 = y 2 .This establishestheuniquenessofthedecompositionof v intoasumofvectorsfrom U X and Y Rememberthatwhenwewrite V = U W therealwaysneedstobeasuperspace,"inthiscase V .The statement U W ismeaningless.Writing V = U W issimplyashorthandforasomewhatcomplicated relationshipbetween V U and W ,asdescribedinthetwoconditionsofDenitionDS[361],orTheorem DSZV[362],orTheoremDSZI[363].TheoremDSFB[361]andTheoremDSFOS[362]givesussure-re waystobuilddirectsums,whileTheoremDSLI[364],TheoremDSD[364]andTheoremRDS[365]tellus interestingpropertiesofdirectsums.Thissubsectionhasbeenlongontheoremsandshortonexamples. Ifweweretousethetermlemma"wemighthavechosentolabelsomeoftheseresultsassuch,since theywillbeimportanttoolsinotherproofs,butmaynothavemuchinterestontheirownseeTechnique LC[696].Wewillbereferencingtheseresultsheavilyinlatersections,andwillremindyouthentocome backforasecondlook. SubsectionREAD ReadingQuestions 1.WhydoesTheoremG[355]havethetitleitdoes? 2.WhatissosurprisingaboutTheoremRMRT[359]? 3.Row-reducethematrix A toreducedrow-echelonform.Withoutanyfurthercomputations,compute thedimensionsofthefoursubspaces, N A C A R A and L A A = 2 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1285 1114 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 02 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 20184 3 7 7 5 Version2.02

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SubsectionPD.EXCExercises367 SubsectionEXC Exercises C10 ExampleSVP4[357]leavesseveraldetailsforthereadertocheck.Verifytheseveclaims. ContributedbyRobertBeezer C40 Determineiftheset T = x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x +5 ; 4 x 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 2 +5 x; 3 x +2 spansthevectorspaceofpolynomials withdegree4orless, P 4 .ComparethesolutiontothisexercisewithSolutionLISS.C40[322]. ContributedbyRobertBeezerSolution[367] M50 MimicDenitionDS[361]andconstructareasonabledenitionof V = U 1 U 2 U 3 ::: U m ContributedbyRobertBeezer T05 Trivially,if U and V aretwosubspacesof W ,thendim U =dim V .Combinethisfact,Theorem PSSD[358],andTheoremEDYES[358]allintoonegrandcombinedtheorem.YoumightlooktoTheorem PIP[172]stylisticinspiration.Noticethisproblemdoesnotaskyoutoproveanything.Itjustasksyou torollupthreetheoremsintoonecompact,logicallyequivalentstatement. ContributedbyRobertBeezer T10 Provethefollowingtheorem,whichcouldbeviewedasareformulationofpartsandof TheoremG[355],ormoreappropriatelyasacorollaryofTheoremG[355]TechniqueLC[696]. Suppose V isavectorspaceand S isasubsetof V suchthatthenumberofvectorsin S equalsthe dimensionof V .Then S islinearlyindependentifandonlyif S spans V ContributedbyRobertBeezer T15 Supposethat A isan m n matrixandletmin m;n denotetheminimumof m and n .Provethat r A min m;n ContributedbyRobertBeezer T20 Supposethat A isan m n matrixand b 2 C m .Provethatthelinearsystem LS A; b isconsistent ifandonlyif r A = r [ A j b ]. ContributedbyRobertBeezerSolution[367] T25 Supposethat V isavectorspacewithnitedimension.Let W beanysubspaceof V .Provethat W hasnitedimension. ContributedbyRobertBeezer T33 PartofExerciseB.T50[337]isthehalfoftheproofwhereweassumethematrix A isnonsingular andprovethatasetisbasis.InSolutionB.T50[339]weproveddirectlythatthesetwasbothlinearly independentandaspanningset.ShortenthispartoftheproofbyapplyingTheoremG[355].Becareful, thereisonesubtlety. ContributedbyRobertBeezerSolution[367] T60 Supposethat W isavectorspacewithdimension5,and U and V aresubspacesof W ,eachof dimension3.Provethat U V containsanon-zerovector.Stateamoregeneralresult. ContributedbyJoeRiegseckerSolution[367] Version2.02

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SubsectionPD.SOLSolutions368 SubsectionSOL Solutions C40 ContributedbyRobertBeezerStatement[366] Thevectorspace P 4 hasdimension5byTheoremDP[345].Since T containsonly3vectors,and3 < 5, TheoremG[355]tellsusthat T doesnotspan P 5 T20 ContributedbyRobertBeezerStatement[366] Supposerstthat LS A; b isconsistent.ThenbyTheoremCSCS[237], b 2C A .Thismeansthat C A = C [ A j b ]andsoitfollowsthat r A = r [ A j b ]. Addingacolumntoamatrixwillonlyincreasethesizeofitscolumnspace,soinallcases, C A C [ A j b ].However,ifweassumethat r A = r [ A j b ],thenbyTheoremEDYES[358]we concludethat C A = C [ A j b ].Then b 2C [ A j b ]= C A sobyTheoremCSCS[237], LS A; b is consistent. T33 ContributedbyRobertBeezerStatement[366] ByTheoremDCM[345]weknowthat C n hasdimension n .SobyTheoremG[355]weneedonlyestablish thattheset C islinearlyindependentoraspanningset.However,thehypothesesalsorequirethat C beofsize n .Weassumedthat B = f x 1 ; x 2 ; x 3 ;:::; x n g hadsize n ,butthereisnoguaranteethat C = f A x 1 ;A x 2 ;A x 3 ;:::;A x n g willhavesize n .Therecouldbesomecollapsing"orcollisions." Supposeweestablishthat C islinearlyindependent.Then C musthave n distinctelementsorelsewe couldfashionanontrivialrelationoflineardependenceinvolvingduplicateelements. Ifweinsteadtochoosetoprovethat C isaspanningset,thenwecouldestablishtheuniquenessofthe elementsof C quiteeasily.Supposethat A x i = A x j .Then A x i )]TJ/F36 10.9091 Tf 10.909 0 Td [(x j = A x i )]TJ/F21 10.9091 Tf 10.909 0 Td [(A x j = 0 Since A isnonsingular,weconcludethat x i )]TJ/F36 10.9091 Tf 10.909 0 Td [(x j = 0 ,or x i = x j ,contrarytoourdescriptionof B T60 ContributedbyRobertBeezerStatement[366] Let f u 1 ; u 2 ; u 3 g and f v 1 ; v 2 ; v 3 g bebasesfor U and V respectively.Then,theset f u 1 ; u 2 ; u 3 ; v 1 ; v 2 ; v 3 g islinearlydependent,sinceTheoremG[355]sayswecannothave6linearlyindependentvectorsinavector spaceofdimension5.Sowecanassertthatthereisanon-trivialrelationoflineardependence, a 1 u 1 + a 2 u 2 + a 3 u 3 + b 1 v 1 + b 2 v 2 + b 3 v 3 = 0 where a 1 ;a 2 ;a 3 and b 1 ;b 2 ;b 3 arenotallzero. Wecanrearrangethisequationas a 1 u 1 + a 2 u 2 + a 3 u 3 = )]TJ/F21 10.9091 Tf 8.485 0 Td [(b 1 v 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 2 v 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 3 v 3 Thisisanequalityoftwovectors,sowecangivethiscommonvectoraname,say w w = a 1 u 1 + a 2 u 2 + a 3 u 3 = )]TJ/F21 10.9091 Tf 8.485 0 Td [(b 1 v 1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 2 v 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 3 v 3 Thisisthedesirednon-zerovector,aswewillnowshow. First,since w = a 1 u 1 + a 2 u 2 + a 3 u 3 ,wecanseethat w 2 U .Similarly, w = )]TJ/F21 10.9091 Tf 8.485 0 Td [(b 1 v 1 )]TJ/F21 10.9091 Tf 11.012 0 Td [(b 2 v 2 )]TJ/F21 10.9091 Tf 11.012 0 Td [(b 3 v 3 ,so w 2 V .Thisestablishesthat w 2 U V DenitionSI[685]. Is w 6 = 0 ?Supposenot,inotherwords,suppose w = 0 .Then 0 = w = a 1 u 1 + a 2 u 2 + a 3 u 3 Because f u 1 ; u 2 ; u 3 g isabasisfor U ,itisalinearlyindependentsetandtherelationoflineardependence abovemeanswemustconcludethat a 1 = a 2 = a 3 =0.Byasimilarprocess,wewouldconcludethat Version2.02

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SubsectionPD.SOLSolutions369 b 1 = b 2 = b 3 =0.Butthisisacontradictionsince a 1 ;a 2 ;a 3 ;b 1 ;b 2 ;b 3 werechosensothatsomewere nonzero.So w 6 = 0 Howdoesthisgeneralize?Allwereallyneededwastheoriginalrelationoflineardependencethat resultedbecausewehadtoomany"vectorsin W .Amoregeneralstatementwouldbe:Supposethat W isavectorspacewithdimension n U isasubspaceofdimension p and V isasubspaceofdimension q .If p + q>n ,then U V containsanon-zerovector. Version2.02

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AnnotatedAcronymsPD.VSVectorSpaces370 AnnotatedAcronymsVS VectorSpaces DenitionVS[279] Themostfundamentalobjectinlinearalgebraisavectorspace.Orelsethemostfundamentalobjectis avector,andavectorspaceisimportantbecauseitisacollectionofvectors.Eitherway,DenitionVS [279]iscritical.Allofourremainingtheoremsthatassumeweareworkingwithavectorspacecantrace theirlineagebacktothisdenition. TheoremTSS[293] CheckalltenpropertiesofavectorspaceDenitionVS[279]cangettedious.Butifyouhaveasubset ofa known vectorspace,thenTheoremTSS[293]considerablyshortenstheverication.Also,proofsof closurethelasttrwoconditionsinTheoremTSS[293]areagoodwaytppracticeacommonstyleof proof. TheoremVRRB[317] Theproofofuniquenessinthistheoremisaverytypicalemploymentofthehypothesisoflinearindependence.Butthat'snotwhywementionithere.Thistheoremiscriticaltoourrstsectionabout representations,SectionVR[530],viaDenitionVR[530]. TheoremCNMB[330] HavingjustdenedabasisDenitionB[325]wediscoverthatthecolumnsofanonsingularmatrixform abasisof C m .Muchofwhatweknowaboutnonsingularmatricesiseithercontainedinthisstatement,or muchmoreevidentbecauseofit. TheoremSSLD[341] Thistheoremisakeyjunctureinourdevelopmentoflinearalgebra.Youhaveprobablyalreadyrealized howusefulTheoremG[355]is.AllfourpartsofTheoremG[355]haveproofsthatnishwithanapplication ofTheoremSSLD[341]. TheoremRPNC[348] Thissimplerelationshipbetweentherank,nullityandnumberofcolumnsofamatrixmightbesurprising. Butinsimplicitycomespower,asthistheoremcanbeveryuseful.Itwillbegeneralizedintheverylast theoremofChapterLT[452],TheoremRPNDD[517]. TheoremG[355] Awhimsicaltitle,buttheintentistomakesureyoudon'tmissthisone.Muchoftheinteractionbetween bases,dimension,linearindependenceandspanningiscapturedinthistheorem. TheoremRMRT[359] Thisoneisarealsurprise.Whyshouldamatrix,anditstranspose,bothrow-reducetothesamenumber ofnon-zerorows? Version2.02

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ChapterD Determinants Thedeterminantisafunctionthattakesasquarematrixasaninputandproducesascalarasanoutput. Sounlikeavectorspace,itisnotanalgebraicstructure.However,ithasmanybenecialpropertiesfor studyingvectorspaces,matricesandsystemsofequations,soitishardtoignorethoughsomehavetried. Whilethepropertiesofadeterminantcanbeveryuseful,theyarealsocomplicatedtoprove. SectionDM DeterminantofaMatrix First,aslightdetour,asweintroduceelementarymatrices,whichwillbringusbacktothebeginningof thecourseandouroldfriend,rowoperations. SubsectionEM ElementaryMatrices Elementarymatricesareverysimple,asyoumighthavesuspectedfromtheirname.Theirpurposeis toeectrowoperationsDenitionRO[28]onamatrixthroughmatrixmultiplicationDenitionMM [197].Theirdenitionslookmorecomplicatedthantheyreallyare,sobesuretoreadaheadafteryou readthedenitionforsomeexplanationsandanexample. DenitionELEM ElementaryMatrices 1.For i 6 = j E i;j isthesquarematrixofsize n with [ E i;j ] k` = 8 > > > > > > > > > < > > > > > > > > > : 0 k 6 = i;k 6 = j;` 6 = k 1 k 6 = i;k 6 = j;` = k 0 k = i;` 6 = j 1 k = i;` = j 0 k = j;` 6 = i 1 k = j;` = i 371

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SubsectionDM.EMElementaryMatrices372 2.For 6 =0, E i isthesquarematrixofsize n with [ E i ] k` = 8 > < > : 0 k 6 = i;` 6 = k 1 k 6 = i;` = k k = i;` = i 3.For i 6 = j E i;j isthesquarematrixofsize n with [ E i;j ] k` = 8 > > > > > > < > > > > > > : 0 k 6 = j;` 6 = k 1 k 6 = j;` = k 0 k = j;` 6 = i;` 6 = j 1 k = j;` = j k = j;` = i ThisdenitioncontainsNotationELEM. 4 Again,thesematricesarenotascomplicatedastheyappear,sincetheyaremostlyperturbationsof the n n identitymatrixDenitionIM[72]. E i;j istheidentitymatrixwithrowsorcolumns i and j tradingplaces, E i istheidentitymatrixwherethediagonalentryinrow i andcolumn i hasbeen replacedby ,and E i;j istheidentitymatrixwheretheentryinrow j andcolumn i hasbeenreplaced by .Yes,thosesubscriptslookbackwardsinthedescriptionof E i;j .Noticethatournotationmakes noreferencetothesizeoftheelementarymatrix,sincethiswillalwaysbeapparentfromthecontext,or unimportant. The raisond'^etre forelementarymatricesistodo"rowoperationsonmatriceswithmatrixmultiplication.Sohereisanexamplewherewewillbothseesomeelementarymatricesandseehowtheycan accomplishrowoperations. ExampleEMRO Elementarymatricesandrowoperations WewillperformasequenceofrowoperationsDenitionRO[28]onthe3 4matrix A ,whilealso multiplyingthematrixontheleftbytheappropriate3 3elementarymatrix. A = 2 4 2131 1324 5031 3 5 R 1 $ R 3 : 2 4 5031 1324 2131 3 5 E 1 ; 3 : 2 4 001 010 100 3 5 2 4 2131 1324 5031 3 5 = 2 4 5031 1324 2131 3 5 2 R 2 : 2 4 5031 2648 2131 3 5 E 2 : 2 4 100 020 001 3 5 2 4 5031 1324 2131 3 5 = 2 4 5031 2648 2131 3 5 2 R 3 + R 1 : 2 4 9293 2648 2131 3 5 E 3 ; 1 : 2 4 102 010 001 3 5 2 4 5031 2648 2131 3 5 = 2 4 9293 2648 2131 3 5 Thenextthreetheoremsestablishthateachelementarymatrixeectsarowoperationviamatrix multiplication. Version2.02

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SubsectionDM.EMElementaryMatrices373 TheoremEMDRO ElementaryMatricesDoRowOperations Supposethat A isan m n matrix,and B isamatrixofthesamesizethatisobtainedfrom A byasingle rowoperationDenitionRO[28].Thenthereisanelementarymatrixofsize m thatwillconvert A to B viamatrixmultiplicationontheleft.Moreprecisely, 1.Iftherowoperationswapsrows i and j ,then B = E i;j A 2.Iftherowoperationmultipliesrow i by ,then B = E i A 3.Iftherowoperationmultipliesrow i by andaddstheresulttorow j ,then B = E i;j A Proof Ineachofthethreeconclusions,performingtherowoperationon A willcreatethematrix B whereonlyoneortworowswillhavechanged.Sowewillestablishtheequalityofthematrixentriesrow byrow,rstfortheunchangedrows,thenforthechangedrows,showingineachcasethattheresultof thematrixproductisthesameastheresultoftherowoperation.Herewego. Row k oftheproduct E i;j A ,where k 6 = i k 6 = j ,isunchangedfrom A [ E i;j A ] k` = n X p =1 [ E i;j ] kp [ A ] p` TheoremEMP[198] =[ E i;j ] kk [ A ] k` + n X p =1 p 6 = k [ E i;j ] kp [ A ] p` =1[ A ] k` + n X p =1 p 6 = k 0[ A ] p` DenitionELEM[370] =[ A ] k` Row i oftheproduct E i;j A isrow j of A [ E i;j A ] i` = n X p =1 [ E i;j ] ip [ A ] p` TheoremEMP[198] =[ E i;j ] ij [ A ] j` + n X p =1 p 6 = j [ E i;j ] ip [ A ] p` =1[ A ] j` + n X p =1 p 6 = j 0[ A ] p` DenitionELEM[370] =[ A ] j` Row j oftheproduct E i;j A isrow i of A [ E i;j A ] j` = n X p =1 [ E i;j ] jp [ A ] p` TheoremEMP[198] =[ E i;j ] ji [ A ] i` + n X p =1 p 6 = i [ E i;j ] jp [ A ] p` Version2.02

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SubsectionDM.EMElementaryMatrices374 =1[ A ] i` + n X p =1 p 6 = i 0[ A ] p` DenitionELEM[370] =[ A ] i` Sothematrixproduct E i;j A isthesameastherowoperationthatswapsrows i and j Row k oftheproduct E i A ,where k 6 = i ,isunchangedfrom A [ E i A ] k` = n X p =1 [ E i ] kp [ A ] p` TheoremEMP[198] =[ E i ] kk [ A ] k` + n X p =1 p 6 = k [ E i ] kp [ A ] p` =1[ A ] k` + n X p =1 p 6 = k 0[ A ] p` DenitionELEM[370] =[ A ] k` Row i oftheproduct E i A is timesrow i of A [ E i A ] i` = n X p =1 [ E i ] ip [ A ] p` TheoremEMP[198] =[ E i ] ii [ A ] i` + n X p =1 p 6 = i [ E i ] ip [ A ] p` = [ A ] i` + n X p =1 p 6 = i 0[ A ] p` DenitionELEM[370] = [ A ] i` Sothematrixproduct E i A isthesameastherowoperationthatswapsmultipliesrow i by Row k oftheproduct E i;j A ,where k 6 = j ,isunchangedfrom A [ E i;j A ] k` = n X p =1 [ E i;j ] kp [ A ] p` TheoremEMP[198] =[ E i;j ] kk [ A ] k` + n X p =1 p 6 = k [ E i;j ] kp [ A ] p` =1[ A ] k` + n X p =1 p 6 = k 0[ A ] p` DenitionELEM[370] =[ A ] k` Row j oftheproduct E i;j A ,is timesrow i of A andthenaddedtorow j of A [ E i;j A ] j` = n X p =1 [ E i;j ] jp [ A ] p` TheoremEMP[198] Version2.02

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SubsectionDM.DDDenitionoftheDeterminant375 =[ E i;j ] jj [ A ] j` + [ E i;j ] ji [ A ] i` + n X p =1 p 6 = j;i [ E i;j ] jp [ A ] p` =1[ A ] j` + [ A ] i` + n X p =1 p 6 = j;i 0[ A ] p` DenitionELEM[370] =[ A ] j` + [ A ] i` Sothematrixproduct E i;j A isthesameastherowoperationthatmultipliesrow i by andaddsthe resulttorow j Laterinthissectionwewillneedtwofactsaboutelementarymatrices. TheoremEMN ElementaryMatricesareNonsingular If E isanelementarymatrix,then E isnonsingular. Proof Weshowthatwecanrow-reduceeachelementarymatrixtotheidentitymatrix.Givenan elementarymatrixoftheform E i;j ,performtherowoperationthatswapsrow j withrow i .Givenan elementarymatrixoftheform E i ,with 6 =0,performtherowoperationthatmultipliesrow i by1 = Givenanelementarymatrixoftheform E i;j ,with 6 =0,performtherowoperationthatmultiplies row i by )]TJ/F21 10.9091 Tf 8.485 0 Td [( andaddsittorow j .Ineachcase,theresultofthesinglerowoperationistheidentity matrix.Soeachelementarymatrixisrow-equivalenttotheidentitymatrix,andbyTheoremNMRRI[72] isnonsingular. Noticethatwehavenowmadeuseofthenonzerorestrictionon inthedenitionof E i .Onemore keypropertyofelementarymatrices. TheoremNMPEM NonsingularMatricesareProductsofElementaryMatrices Supposethat A isanonsingularmatrix.Thenthereexistselementarymatrices E 1 ;E 2 ;E 3 ;:::;E t sothat A = E 1 E 2 E 3 :::E t Proof Since A isnonsingular,itisrow-equivalenttotheidentitymatrixbyTheoremNMRRI[72],so thereisasequenceof t rowoperationsthatconverts I to A .Foreachoftheserowoperations,formtheassociatedelementarymatrixfromTheoremEMDRO[372]anddenotethesematricesby E 1 ;E 2 ;E 3 ;:::;E t Applyingtherstrowoperationto I yieldsthematrix E 1 I .Thesecondrowoperationyields E 2 E 1 I andthethirdrowoperationcreates E 3 E 2 E 1 I .Theresultofthefullsequenceof t rowoperationswillyield A ,so A = E t :::E 3 E 2 E 1 I = E t :::E 3 E 2 E 1 Otherthanthecosmeticmatterofre-indexingtheseelementarymatricesintheoppositeorder,thisisthe desiredresult. SubsectionDD DenitionoftheDeterminant We'llnowturntothedenitionofadeterminantanddosomesamplecomputations.Thedenitionofthe determinantfunctionis recursive ,thatis,thedeterminantofalargematrixisdenedintermsofthe determinantofsmallermatrices.Tothisend,wewillmakeafewdenitions. Version2.02

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SubsectionDM.DDDenitionoftheDeterminant376 DenitionSM SubMatrix Supposethat A isan m n matrix.Thenthe submatrix A i j j isthe m )]TJ/F15 10.9091 Tf 10.394 0 Td [(1 n )]TJ/F15 10.9091 Tf 10.394 0 Td [(1matrixobtained from A byremovingrow i andcolumn j ThisdenitioncontainsNotationSM. 4 ExampleSS Somesubmatrices Forthematrix A = 2 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(239 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(201 3521 3 5 wehavethesubmatrices A j 3= 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(29 351 A j 1= )]TJ/F15 10.9091 Tf 8.485 0 Td [(239 )]TJ/F15 10.9091 Tf 8.485 0 Td [(201 DenitionDM DeterminantofaMatrix Suppose A isasquarematrix.Thenits determinant ,det A = j A j ,isanelementof C denedrecursively by: If A isa1 1matrix,thendet A =[ A ] 11 If A isamatrixofsize n with n 2,then det A =[ A ] 11 det A j 1 )]TJ/F15 10.9091 Tf 10.909 0 Td [([ A ] 12 det A j 2+[ A ] 13 det A j 3 )]TJ/F15 10.9091 Tf -279.021 -17.691 Td [([ A ] 14 det A j 4+ + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 n +1 [ A ] 1 n det A j n ThisdenitioncontainsNotationDM. 4 Sotocomputethedeterminantofa5 5matrixwemustbuild5submatrices,eachofsize4.To computethedeterminantsofeachthe4 4matricesweneedtocreate4submatriceseach,thesenowof size3andsoon.Tocomputethedeterminantofa10 10matrixwouldrequirecomputingthedeterminant of10!=10 9 8 7 6 5 4 3 2=3 ; 628 ; 8001 1matrices.Fortunatelytherearebetterways. Howeverthisdoessuggestanexcellentcomputerprogrammingexercisetowritearecursiveprocedureto computeadeterminant. Let'scomputethedeterminantofareasonablesizedmatrixbyhand. ExampleD33M Determinantofa 3 3 matrix Supposethatwehavethe3 3matrix A = 2 4 32 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 416 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 3 5 Then det A = j A j = 32 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 416 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 Version2.02

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SubsectionDM.CDComputingDeterminants377 =3 16 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 46 )]TJ/F15 10.9091 Tf 8.484 0 Td [(32 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 41 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 =3 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(1 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 )]TJ/F15 10.9091 Tf 5 -8.836 Td [(4 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F27 10.9091 Tf 10.909 8.836 Td [()]TJ/F15 10.9091 Tf 5 -8.836 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 =3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 =24 )]TJ/F15 10.9091 Tf 10.909 0 Td [(52+1 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(27 Inpracticeitisabitsillytodecomposea2 2matrixdownintoacoupleof1 1matricesandthen computetheexceedinglyeasydeterminantofthesepunymatrices.Sohereisasimpletheorem. TheoremDMST DeterminantofMatricesofSizeTwo Supposethat A = ab cd .Thendet A = ad )]TJ/F21 10.9091 Tf 10.909 0 Td [(bc Proof ApplyingDenitionDM[375], ab cd = a d )]TJ/F21 10.9091 Tf 10.909 0 Td [(b c = ad )]TJ/F21 10.9091 Tf 10.909 0 Td [(bc Doyourecallseeingtheexpression ad )]TJ/F21 10.9091 Tf 10.909 0 Td [(bc before?Hint:TheoremTTMI[214] SubsectionCD ComputingDeterminants Thereareavarietyofwaystocomputethedeterminant.Wewillestablishrstthatwecanchooseto mimicourdenitionofthedeterminant,butbyusingmatrixentriesandsubmatricesbasedonarowother thantherstone. TheoremDER DeterminantExpansionaboutRows Supposethat A isasquarematrixofsize n .Then det A = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 i +1 [ A ] i 1 det A i j 1+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 i +2 [ A ] i 2 det A i j 2 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 i +3 [ A ] i 3 det A i j 3+ + )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 i + n [ A ] in det A i j n 1 i n whichisknownas expansion aboutrow i Proof First,thestatementofthetheoremcoincideswithDenitionDM[375]when i =1,sothroughout, weneedonlyconsider i> 1. Giventherecursivedenitionofthedeterminant,itshouldbenosurprisethatwewilluseinduction forthisproofTechniqueI[694].When n =1,thereisnothingtoprovesincethereisbutonerow.When n =2,wejustexamineexpansionaboutthesecondrow, )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2+1 [ A ] 21 det A j 1+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2+2 [ A ] 22 det A j 2 = )]TJ/F15 10.9091 Tf 10.303 0 Td [([ A ] 21 [ A ] 12 +[ A ] 22 [ A ] 11 DenitionDM[375] =[ A ] 11 [ A ] 22 )]TJ/F15 10.9091 Tf 10.909 0 Td [([ A ] 12 [ A ] 21 =det A TheoremDMST[376] Version2.02

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SubsectionDM.CDComputingDeterminants378 Sothetheoremistrueformatricesofsize n =1and n =2.Nowassumetheresultistrueforallmatrices ofsize n )]TJ/F15 10.9091 Tf 11.204 0 Td [(1aswederiveanexpressionforexpansionaboutrow i foramatrixofsize n .Wewillabuse ournotationforasubmatrixslightly,so A i 1 ;i 2 j j 1 ;j 2 willdenotethematrixformedbyremovingrows i 1 and i 2 ,alongwithremovingcolumns j 1 and j 2 .Also,aswetakeadeterminantofasubmatrix,wewill needtojumpup"theindexofsummationpartwaythroughasweskipover"amissingcolumn.Todo thissmoothlywewillset `j = 0 `j Now, det A = n X j =1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1+ j [ A ] 1 j det A j j DenitionDM[375] = n X j =1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1+ j [ A ] 1 j X 1 ` n ` 6 = j )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 i )]TJ/F19 7.9701 Tf 6.586 0 Td [(1+ ` )]TJ/F22 7.9701 Tf 6.587 0 Td [( `j [ A ] i` det A ;i j j;` InductionHypothesis = n X j =1 X 1 ` n ` 6 = j )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 j + i + ` )]TJ/F22 7.9701 Tf 6.587 0 Td [( `j [ A ] 1 j [ A ] i` det A ;i j j;` PropertyDCN[681] = n X ` =1 X 1 j n j 6 = ` )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 j + i + ` )]TJ/F22 7.9701 Tf 6.586 0 Td [( `j [ A ] 1 j [ A ] i` det A ;i j j;` PropertyCACN[680] = n X ` =1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 i + ` [ A ] i` X 1 j n j 6 = ` )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 j )]TJ/F22 7.9701 Tf 6.587 0 Td [( `j [ A ] 1 j det A ;i j j;` PropertyDCN[681] = n X ` =1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 i + ` [ A ] i` X 1 j n j 6 = ` )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 `j + j [ A ] 1 j det A i; 1 j `;j 2 `j iseven = n X ` =1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 i + ` [ A ] i` det A i j ` DenitionDM[375] Wecanalsoobtainaformulathatcomputesadeterminantbyexpansionaboutacolumn,butthiswill besimplerifwerstprovearesultabouttheinterplayofdeterminantsandtransposes.Noticehowthe followingproofmakesuseoftheabilitytocomputeadeterminantbyexpandingabout any row. TheoremDT DeterminantoftheTranspose Supposethat A isasquarematrix.Thendet )]TJ/F21 10.9091 Tf 5 -8.836 Td [(A t =det A Proof WithourdenitionofthedeterminantDenitionDM[375]andtheoremslikeTheoremDER [376],usinginductionTechniqueI[694]isanaturalapproachtoprovingpropertiesofdeterminants.And soitishere.Let n bethesizeofthematrix A ,andwewilluseinductionon n For n =1,thetransposeofamatrixisidenticaltotheoriginalmatrix,sovacuously,thedeterminants areequal. Nowassumetheresultistrueformatricesofsize n )]TJ/F15 10.9091 Tf 10.909 0 Td [(1.Then, det )]TJ/F21 10.9091 Tf 5 -8.836 Td [(A t = 1 n n X i =1 det )]TJ/F21 10.9091 Tf 5 -8.836 Td [(A t Version2.02

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SubsectionDM.CDComputingDeterminants379 = 1 n n X i =1 n X j =1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 i + j A t ij det )]TJ/F21 10.9091 Tf 5 -8.837 Td [(A t i j j TheoremDER[376] = 1 n n X i =1 n X j =1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 i + j [ A ] ji det )]TJ/F21 10.9091 Tf 5 -8.837 Td [(A t i j j DenitionTM[185] = 1 n n X i =1 n X j =1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 i + j [ A ] ji det )]TJ/F15 10.9091 Tf 5 -8.836 Td [( A j j i t DenitionTM[185] = 1 n n X i =1 n X j =1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 i + j [ A ] ji det A j j i InductionHypothesis = 1 n n X j =1 n X i =1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 j + i [ A ] ji det A j j i PropertyCACN[680] = 1 n n X j =1 det A TheoremDER[376] =det A Nowwecaneasilygettheresultthatadeterminantcanbecomputedbyexpansionaboutanycolumn aswell. TheoremDEC DeterminantExpansionaboutColumns Supposethat A isasquarematrixofsize n .Then det A = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 1+ j [ A ] 1 j det A j j + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2+ j [ A ] 2 j det A j j + )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 3+ j [ A ] 3 j det A j j + + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 n + j [ A ] nj det A n j j 1 j n whichisknownas expansion aboutcolumn j Proof det A =det )]TJ/F21 10.9091 Tf 5 -8.836 Td [(A t TheoremDT[377] = n X i =1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 j + i A t ji det )]TJ/F21 10.9091 Tf 5 -8.836 Td [(A t j j i TheoremDER[376] = n X i =1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 j + i A t ji det )]TJ/F15 10.9091 Tf 5 -8.836 Td [( A i j j t DenitionTM[185] = n X i =1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 j + i A t ji det A i j j TheoremDT[377] = n X i =1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 i + j [ A ] ij det A i j j DenitionTM[185] Thatthedeterminantofan n n matrixcanbecomputedin2 n dierentalbeitsimilarwaysis nothingshortofremarkable.Forthedoubtersamongus,wewilldoanexample,computinga4 4matrix intwodierentways. Version2.02

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SubsectionDM.CDComputingDeterminants380 ExampleTCSD Twocomputations,samedeterminant Let A = 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2301 9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(201 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4126 3 7 7 5 ThenexpandingaboutthefourthrowTheoremDER[376]with i =4yields, j A j = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4+1 301 )]TJ/F15 10.9091 Tf 8.485 0 Td [(201 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4+2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(201 901 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4+3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(231 9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 13 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4+4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(230 9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(4+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(22+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2+6=92 whileexpandingaboutcolumn3TheoremDEC[378]with j =3gives j A j = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1+3 9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 416 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2+3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(231 13 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 416 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3+3 )]TJ/F15 10.9091 Tf 8.484 0 Td [(231 9 )]TJ/F15 10.9091 Tf 8.484 0 Td [(21 416 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4+3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(231 9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 =0+0+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(107+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2=92 Noticehowmucheasierthesecondcomputationwas.Bychoosingtoexpandaboutthethirdcolumn,we havetwoentriesthatarezero,sotwo3 3determinantsneednotbecomputedatall! Whenamatrixhasallzerosaboveorbelowthediagonal,exploitingthezerosbyexpandingabout theproperroworcolumnmakescomputingadeterminantinsanelyeasy. ExampleDUTM Determinantofanuppertriangularmatrix Supposethat T = 2 6 6 6 6 4 23 )]TJ/F15 10.9091 Tf 8.484 0 Td [(133 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(152 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 00392 000 )]TJ/F15 10.9091 Tf 8.484 0 Td [(13 00005 3 7 7 7 7 5 Wewillcomputethedeterminantofthis5 5matrixbyconsistentlyexpandingabouttherstcolumnfor eachsubmatrixthatarisesanddoesnothaveazeroentrymultiplyingit. det T = 23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(133 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(152 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 00392 000 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 00005 =2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1+1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(152 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0392 00 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 0005 Version2.02

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SubsectionDM.READReadingQuestions381 =2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 1+1 392 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 005 =2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1+1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 05 =2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1+1 5 =2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1=30 Ifyouconsultothertextsinyourstudyofdeterminants,youmayrunintothetermsminor"and cofactor,"especiallyinadiscussioncenteredonexpansionaboutrowsandcolumns.We'vechosennotto makethesedenitionsformallysincewe'vebeenabletogetalongwithoutthem.However,informally,a minor isadeterminantofasubmatrix,specicallydet A i j j andisusuallyreferencedastheminorof [ A ] ij .A cofactor isasignedminor,specicallythecofactorof[ A ] ij is )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 i + j det A i j j SubsectionREAD ReadingQuestions 1.Constructtheelementarymatrixthatwilleecttherowoperation )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 R 2 + R 3 ona4 7matrix. 2.Computethedeterminantofthematrix 2 4 23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 382 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 5 3.Computethedeterminantofthematrix 2 6 6 6 6 4 39 )]TJ/F15 10.9091 Tf 8.485 0 Td [(242 014 )]TJ/F15 10.9091 Tf 8.485 0 Td [(27 00 )]TJ/F15 10.9091 Tf 8.485 0 Td [(252 000 )]TJ/F15 10.9091 Tf 8.485 0 Td [(16 00004 3 7 7 7 7 5 Version2.02

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SubsectionDM.EXCExercises382 SubsectionEXC Exercises C24 Doingthecomputationsbyhand,ndthedeterminantofthematrixbelow. 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 242 3 5 ContributedbyRobertBeezerSolution[382] C25 Doingthecomputationsbyhand,ndthedeterminantofthematrixbelow. 2 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(14 251 206 3 5 ContributedbyRobertBeezerSolution[382] C26 Doingthecomputationsbyhand,ndthedeterminantofthematrix A A = 2 6 6 4 2032 5124 3012 5321 3 7 7 5 ContributedbyRobertBeezerSolution[382] Version2.02

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SubsectionDM.SOLSolutions383 SubsectionSOL Solutions C24 ContributedbyRobertBeezerStatement[381] We'llexpandabouttherstrowsincetherearenozerostoexploit, )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 242 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 42 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(41 22 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 24 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(1+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 10.91 0 Td [(1+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10+ )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12=70 C25 ContributedbyRobertBeezerStatement[381] Wecanexpandaboutanyroworcolumn,sothezeroentryinthemiddleofthelastrowisattractive.Let's expandaboutcolumn2.ByTheoremDER[376]andTheoremDEC[378]youwillgetthesameresultby expandingaboutadierentroworcolumn.WewilluseTheoremDMST[376]twice. 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(14 251 206 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1+2 21 26 + )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 2+2 34 26 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3+2 34 21 =++0=60 C26 ContributedbyRobertBeezerStatement[381] Withtwozerosincolumn2,wechoosetoexpandaboutthatcolumnTheoremDEC[378], det A = 2032 5124 3012 5321 =0 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 524 312 521 +1 232 312 521 +0 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 232 524 521 +3 232 524 312 = )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 )]TJ/F15 10.9091 Tf 10.91 0 Td [(3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5+2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5+ )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4+2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(6+21+2++6 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2=29 Version2.02

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SectionPDMPropertiesofDeterminantsofMatrices384 SectionPDM PropertiesofDeterminantsofMatrices Wehaveseenhowtocomputethedeterminantofamatrix,andtheincrediblefactthatwecanperform expansionabout any row or columntomakethiscomputation.Inthislargelytheoreticalsection,wewill stateandproveseveralmoreintriguingpropertiesaboutdeterminants.Ourmaingoalwillbethetwo resultsinTheoremSMZD[389]andTheoremDRMM[391],butmorespecically,wewillseehowthe valueofadeterminantwillallowustogaininsightintothevariouspropertiesofasquarematrix. SubsectionDRO DeterminantsandRowOperations Westarteasywithastraightforwardtheoremwhoseproofpresagesthestyleofsubsequentproofsinthis subsection. TheoremDZRC DeterminantwithZeroRoworColumn Supposethat A isasquarematrixwitharowwhereeveryentryiszero,oracolumnwhereeveryentryis zero.Thendet A =0. Proof Supposethat A isasquarematrixofsize n androw i haseveryentryequaltozero.Wecompute det A viaexpansionaboutrow i det A = n X j =1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 i + j [ A ] ij det A i j j TheoremDER[376] = n X j =1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 i + j 0det A i j j Row i iszeros = n X j =1 0=0 Theproofforthecaseofazerocolumnisentirelysimilar,orcouldbederivedfromanapplicationof TheoremDT[377]employingthetransposeofthematrix. TheoremDRCS DeterminantforRoworColumnSwap Supposethat A isasquarematrix.Let B bethesquarematrixobtainedfrom A byinterchangingthe locationoftworows,orinterchangingthelocationoftwocolumns.Thendet B = )]TJ/F15 10.9091 Tf 10.303 0 Td [(det A Proof Beginwiththespecialcasewhere A isasquarematrixofsize n andweform B byswapping adjacent rows i and i +1forsome1 i n )]TJ/F15 10.9091 Tf 11.232 0 Td [(1.Noticethattheassumptionaboutswappingadjacent rowsmeansthat B i +1 j j = A i j j forall1 j n ,and[ B ] i +1 ;j =[ A ] ij forall1 j n .Wecompute det B viaexpansionaboutrow i +1. det B = n X j =1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 i +1+ j [ B ] i +1 ;j det B i +1 j j TheoremDER[376] = n X j =1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 i +1+ j [ A ] ij det A i j j Hypothesis Version2.02

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SubsectionPDM.DRODeterminantsandRowOperations385 = n X j =1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 i + j [ A ] ij det A i j j = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 n X j =1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 i + j [ A ] ij det A i j j = )]TJ/F15 10.9091 Tf 10.303 0 Td [(det A TheoremDER[376] Sotheresultholdsforthespecialcasewhereweswapadjacentrowsofthematrix.Asanycomputer scientistknows,wecanaccomplish any rearrangementofanorderedlistbyswappingadjacentelements. Thisprinciplecanbedemonstratedbynavesortingalgorithmssuchasbubblesort."Inanyevent,we don'tneedtodiscusseverypossiblereordering,wejustneedtoconsideraswapoftworows,sayrows s and t with1 s
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SubsectionPDM.DRODeterminantsandRowOperations386 TheoremDERC DeterminantwithEqualRowsorColumns Supposethat A isasquarematrixwithtwoequalrows,ortwoequalcolumns.Thendet A =0. Proof Supposethat A isasquarematrixofsize n wherethetworows s and t areequal.Formthematrix B byswappingrows r and s .Noticethatasaconsequenceofourhypothesis, A = B .Then det A = 1 2 det A +det A = 1 2 det A )]TJ/F15 10.9091 Tf 10.909 0 Td [(det B TheoremDRCS[383] = 1 2 det A )]TJ/F15 10.9091 Tf 10.909 0 Td [(det A Hypothesis, A = B = 1 2 =0 Theproofforthecaseoftwoequalcolumnsisentirelysimilar,orcouldbederivedfromanapplicationof TheoremDT[377]employingthetransposeofthematrix. Nowexplainthethirdrowoperation.Herewego. TheoremDRCMA DeterminantforRoworColumnMultiplesandAddition Supposethat A isasquarematrix.Let B bethesquarematrixobtainedfrom A bymultiplyingarow bythescalar andthenaddingittoanotherrow,orbymultiplyingacolumnbythescalar andthen addingittoanothercolumn.Thendet B =det A Proof Supposethat A isasquarematrixofsize n .Formthematrix B bymultiplyingrow s by and addingittorow t .Let C betheauxiliarymatrixwherewereplacerow t of A byrow s of A .Noticethat A t j j = B t j j = C t j j forall1 j n .Wecomputethedeterminantof B byexpansionaboutrow t det B = n X j =1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 t + j [ B ] tj det B t j j TheoremDER[376] = n X j =1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 t + j [ A ] sj +[ A ] tj det B t j j Hypothesis = n X j =1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 t + j [ A ] sj det B t j j + n X j =1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 t + j [ A ] tj det B t j j = n X j =1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 t + j [ A ] sj det B t j j + n X j =1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 t + j [ A ] tj det B t j j = n X j =1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 t + j [ C ] tj det C t j j + n X j =1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 t + j [ A ] tj det A t j j = det C +det A TheoremDER[376] Version2.02

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SubsectionPDM.DRODeterminantsandRowOperations387 = 0+det A =det A TheoremDERC[385] Theproofforthecaseofaddingamultipleofacolumnisentirelysimilar,orcouldbederivedfroman applicationofTheoremDT[377]employingthetransposeofthematrix. Isthiswhatyouexpected?Wecouldarguethatthethirdrowoperationisthemostpopular,andyetit hasnoeectwhatsoeveronthedeterminantofamatrix!Wecanexploitthis,alongwithourunderstanding oftheothertworowoperations,toprovideanotherapproachtocomputingadeterminant.We'llexplain thisinthecontextofanexample. ExampleDRO Determinantbyrowoperations Supposewedesirethedeterminantofthe4 4matrix A = 2 6 6 4 2023 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 3540 3 7 7 5 Wewillperformasequenceofrowoperationsonthismatrix,shootingforanuppertriangularmatrix, whosedeterminantwillbesimplytheproductofitsdiagonalentries.Foreachrowoperation,wewilltrack theeectonthedeterminantviaTheoremDRCS[383],TheoremDRCM[384],TheoremDRCMA[385]. R 1 $ R 2 )456()222()222()223()456(! A 1 = 2 6 6 4 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 2023 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 3540 3 7 7 5 det A = )]TJ/F15 10.9091 Tf 10.303 0 Td [(det A 1 TheoremDRCS[383] )]TJ/F19 7.9701 Tf 6.587 0 Td [(2 R 1 + R 2 )324()222()223()222()222()324(! A 2 = 2 6 6 4 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(641 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 3540 3 7 7 5 = )]TJ/F15 10.9091 Tf 10.303 0 Td [(det A 2 TheoremDRCMA[385] 1 R 1 + R 3 )348()222()223()222()348(! A 3 = 2 6 6 4 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(641 04 )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 3540 3 7 7 5 = )]TJ/F15 10.9091 Tf 10.303 0 Td [(det A 3 TheoremDRCMA[385] )]TJ/F19 7.9701 Tf 6.587 0 Td [(3 R 1 + R 4 )324()222()223()222()222()324(! A 4 = 2 6 6 4 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(641 04 )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(47 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 7 7 5 = )]TJ/F15 10.9091 Tf 10.303 0 Td [(det A 4 TheoremDRCMA[385] 1 R 3 + R 2 )348()222()223()222()348(! A 5 = 2 6 6 4 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(224 04 )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(47 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 7 7 5 = )]TJ/F15 10.9091 Tf 10.303 0 Td [(det A 5 TheoremDRCMA[385] )]TJ/F20 5.9776 Tf 7.782 3.259 Td [(1 2 R 2 )470()222()222()470(! A 6 = 2 6 6 4 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 04 )]TJ/F15 10.9091 Tf 8.485 0 Td [(23 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(47 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 7 7 5 =2det A 6 TheoremDRCM[384] )]TJ/F19 7.9701 Tf 6.587 0 Td [(4 R 2 + R 3 )324()222()223()222()222()324(! A 7 = 2 6 6 4 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 00211 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(47 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 7 7 5 =2det A 7 TheoremDRCMA[385] Version2.02

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SubsectionPDM.DROEMDeterminants,RowOperations,ElementaryMatrices388 4 R 2 + R 4 )348()222()223()222()348(! A 8 = 2 6 6 4 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 00211 003 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 3 7 7 5 =2det A 8 TheoremDRCMA[385] )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 R 3 + R 4 )324()222()223()222()222()324(! A 9 = 2 6 6 4 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 00211 001 )]TJ/F15 10.9091 Tf 8.485 0 Td [(22 3 7 7 5 =2det A 9 TheoremDRCMA[385] )]TJ/F19 7.9701 Tf 6.586 0 Td [(2 R 4 + R 3 )324()222()222()223()222()324(! A 10 = 2 6 6 4 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 00055 001 )]TJ/F15 10.9091 Tf 8.485 0 Td [(22 3 7 7 5 =2det A 10 TheoremDRCMA[385] R 3 $ R 4 )456()222()222()223()456(! A 11 = 2 6 6 4 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 001 )]TJ/F15 10.9091 Tf 8.485 0 Td [(22 00055 3 7 7 5 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2det A 11 TheoremDRCS[383] 1 55 R 4 )327()222()327(! A 12 = 2 6 6 4 13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 001 )]TJ/F15 10.9091 Tf 8.485 0 Td [(22 0001 3 7 7 5 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(110det A 12 TheoremDRCM[384] Thematrix A 12 isuppertriangular,soexpansionabouttherstcolumnrepeatedlywillresultin det A 12 ==1seeExampleDUTM[379]andthus,det A = )]TJ/F15 10.9091 Tf 8.485 0 Td [(110= )]TJ/F15 10.9091 Tf 8.485 0 Td [(110. Noticethatoursequenceofrowoperationswassomewhat adhoc ,suchasthetransformationto A 5 Wecouldhavebeenevenmoremethodical,andstrictlyfollowedtheprocessthatconvertsamatrixto reducedrow-echelonformTheoremREMEF[30],eventuallyachievingthesamenumericalresultwith analmatrixthatequaledthe4 4identitymatrix.Noticetoothatwecouldhavestoppedwith A 8 sinceatthispointwecouldcomputedet A 8 bytwoexpansionsaboutrstcolumns,followedbyasimple determinantofa2 2matrixTheoremDMST[376]. Thebeautyofthisapproachisthatcomputationallyweshouldalreadyhavewrittenaprocedureto convertmatricestoreduced-rowechelonform,soallweneedtodoistrackthemultiplicativechangesto thedeterminantasthealgorithmproceeds.Further,forasquarematrixofsize n thisapproachrequireson theorderof n 3 multiplications,whilearecursiveapplicationofexpansionaboutaroworcolumnTheorem DER[376],TheoremDEC[378]willrequireinthevicinityof n )]TJ/F15 10.9091 Tf 10.11 0 Td [(1 n !multiplications.Soevenforvery smallmatrices,acomputationalapproachutilizingrowoperationswillhavesuperiorrun-time.Tracking, andcontrolling,theeectsofround-oerrorsisanotherstory,bestsavedforanumericallinearalgebra course. SubsectionDROEM Determinants,RowOperations,ElementaryMatrices Asanalpreparationforourtwomostimportanttheoremsaboutdeterminants,weproveahandfulof factsabouttheinterplayofrowoperationsandmatrixmultiplicationwithelementarymatriceswithregard tothedeterminant.Butrst,asimple,butcrucial,factabouttheidentitymatrix. TheoremDIM DeterminantoftheIdentityMatrix Version2.02

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SubsectionPDM.DROEMDeterminants,RowOperations,ElementaryMatrices389 Forevery n 1,det I n =1. Proof Itmaybeoverkill,butthisisagoodsituationtorunthroughaproofbyinductionon n Technique I[694].Istheresulttruewhen n =1?Yes, det I 1 =[ I 1 ] 11 DenitionDM[375] =1DenitionIM[72] Nowassumethetheoremistruefortheidentitymatrixofsize n )]TJ/F15 10.9091 Tf 11.208 0 Td [(1andinvestigatethedeterminantof theidentitymatrixofsize n withexpansionaboutrow1, det I n = n X j =1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1+ j [ I n ] 1 j det I n j j DenitionDM[375] = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1+1 [ I n ] 11 det I n j 1 + n X j =2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1+ j [ I n ] 1 j det I n j j =1det I n )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 + n X j =2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1+ j 0det I n j j DenitionIM[72] =1+ n X j =2 0=1InductionHypothesis TheoremDEM DeterminantsofElementaryMatrices ForthethreepossibleversionsofanelementarymatrixDenitionELEM[370]wehavethedeterminants, 1.det E i;j = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2.det E i = 3.det E i;j =1 Proof Swappingrows i and j oftheidentitymatrixwillcreate E i;j DenitionELEM[370],so det E i;j = )]TJ/F15 10.9091 Tf 10.303 0 Td [(det I n TheoremDRCS[383] = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1TheoremDIM[387] Multiplyingrow i oftheidentitymatrixby willcreate E i DenitionELEM[370],so det E i = det I n TheoremDRCM[384] = = TheoremDIM[387] Version2.02

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SubsectionPDM.DNMMMDeterminants,NonsingularMatrices,MatrixMultiplication390 Multiplyingrow i oftheidentitymatrixby andaddingtorow j willcreate E i j DenitionELEM [370],so det E i j =det I n TheoremDRCMA[385] =1TheoremDIM[387] TheoremDEMMM Determinants,ElementaryMatrices,MatrixMultiplication Supposethat A isasquarematrixofsize n and E isanyelementarymatrixofsize n .Then det EA =det E det A Proof Theproofprocedesinthreeparts,oneforeachtypeofelementarymatrix,witheachpartvery similartotheothertwo.First,let B bethematrixobtainedfrom A byswappingrows i and j det E i;j A =det B TheoremEMDRO[372] = )]TJ/F15 10.9091 Tf 10.303 0 Td [(det A TheoremDRCS[383] =det E i;j det A TheoremDEM[388] Second,let B bethematrixobtainedfrom A bymultiplyingrow i by det E i A =det B TheoremEMDRO[372] = det A TheoremDRCM[384] =det E i det A TheoremDEM[388] Third,let B bethematrixobtainedfrom A bymultiplyingrow i by andaddingtorow j det E i;j A =det B TheoremEMDRO[372] =det A TheoremDRCMA[385] =det E i;j det A TheoremDEM[388] Sincethedesiredresultholdsforeachvarietyofelementarymatrixindividually,wearedone. SubsectionDNMMM Determinants,NonsingularMatrices,MatrixMultiplication Ifyouaskedsomeonewithsubstantialexperienceworkingwithmatricesaboutthevalueofthedeterminant, they'dbelikelytoquotethefollowingtheoremastherstthingtocometomind. TheoremSMZD SingularMatriceshaveZeroDeterminants Let A beasquarematrix.Then A issingularifandonlyifdet A =0. Proof Ratherthanjumpingintothetwohalvesoftheequivalence,werstestablishafewitems.Let B betheuniquesquarematrixthatisrow-equivalentto A andinreducedrow-echelonformTheorem REMEF[30],TheoremRREFU[32].Foreachoftherowoperationsthatconverts B into A ,thereisan Version2.02

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SubsectionPDM.DNMMMDeterminants,NonsingularMatrices,MatrixMultiplication391 elementarymatrix E i whicheectstherowoperationbymatrixmultiplicationTheoremEMDRO[372]. RepeatedapplicationsofTheoremEMDRO[372]allowustowrite A = E s E s )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 :::E 2 E 1 B Then det A =det E s E s )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 :::E 2 E 1 B =det E s det E s )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 ::: det E 2 det E 1 det B TheoremDEMMM[389] FromTheoremDEM[388]wecaninferthatthedeterminantofanelementarymatrixisneverzeronote thebanon =0for E i inDenitionELEM[370].Sotheproductontherightiscomposedofnonzero scalars,withthepossibleexceptionofdet B .Moreprecisely,wecanarguethatdet A =0ifandonly ifdet B =0.Withthisestablished,wecantakeupthetwohalvesoftheequivalence. If A issingular,thenbyTheoremNMRRI[72], B cannotbetheidentitymatrix.Becausethe numberofpivotcolumnsisequaltothenumberofnonzerorows,noteverycolumnisapivotcolumn, and B issquare,weseethat B musthaveazerorow.ByTheoremDZRC[383]thedeterminantof B iszero,andbytheabove,weconcludethatthedeterminantof A iszero. WewillprovethecontrapositiveTechniqueCP[691].Soassume A isnonsingular,thenby TheoremNMRRI[72], B istheidentitymatrixandTheoremDIM[387]tellsusthatdet B =1 6 =0. Withtheargumentabove,weconcludethatthedeterminantof A isnonzeroaswell. Forthecaseof2 2matricesyoumightcomparetheapplicationofTheoremSMZD[389]withthe combinationoftheresultsstatedinTheoremDMST[376]andTheoremTTMI[214]. ExampleZNDAB Zeroandnonzerodeterminant,ArchetypesAandB ThecoecientmatrixinArchetypeA[702]hasazerodeterminantcheckthis!whilethecoecientmatrix ArchetypeB[707]hasanonzerodeterminantcheckthis,too.Thesematricesaresingularandnonsingular, respectively.ThisisexactlywhatTheoremSMZD[389]says,andcontinuesourlistofcontrastsbetween thesetwoarchetypes. SinceTheoremSMZD[389]isanequivalenceTechniqueE[690]wecanexpandonourgrowinglist ofequivalencesaboutnonsingularmatrices.Theadditionoftheconditiondet A 6 =0isoneofthebest motivationsforlearningaboutdeterminants. TheoremNME7 NonsingularMatrixEquivalences,Round7 Supposethat A isasquarematrixofsize n .Thefollowingareequivalent. 1. A isnonsingular. 2. A row-reducestotheidentitymatrix. 3.Thenullspaceof A containsonlythezerovector, N A = f 0 g 4.Thelinearsystem LS A; b hasauniquesolutionforeverypossiblechoiceof b 5.Thecolumnsof A arealinearlyindependentset. 6. A isinvertible. 7.Thecolumnspaceof A is C n C A = C n 8.Thecolumnsof A areabasisfor C n 9.Therankof A is n r A = n Version2.02

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SubsectionPDM.DNMMMDeterminants,NonsingularMatrices,MatrixMultiplication392 10.Thenullityof A iszero, n A =0. 11.Thedeterminantof A isnonzero,det A 6 =0. Proof TheoremSMZD[389]says A issingularifandonlyifdet A =0.Ifwenegateeachofthese statements,wearriveattwocontrapositivesthatwecancombineastheequivalence, A isnonsingularif andonlyifdet A 6 =0.ThisallowsustoaddanewstatementtothelistfoundinTheoremNME6[349]. Computationally,row-reducingamatrixisthemostecientwaytodetermineifamatrixisnonsingular, thoughtheeectofusingdivisioninacomputercanleadtoround-oerrorsthatconfusesmallquantities withcriticalzeroquantities.Conceptually,thedeterminantmayseemthemostecientwaytodetermineif amatrixisnonsingular.Thedenitionofadeterminantusesjustaddition,subtractionandmultiplication, sodivisionisneveraproblem.Andthenaltestiseasy:isthedeterminantzeroornot?However, thenumberofoperationsinvolvedincomputingadeterminantbythedenitionveryquicklybecomesso excessiveastobeimpractical. Nowforthe coupdegr^ace .WewillgeneralizeTheoremDEMMM[389]tothecaseof any twosquare matrices.Youmayrecallthinkingthatmatrixmultiplicationwasdenedinaneedlesslycomplicated manner.Forsure,thedenitionofadeterminantseemsevenstranger.ThoughTheoremSMZD[389] mightbeforcingyoutoreconsider.Readthestatementofthenexttheoremandcontemplatehownicely matrixmultiplicationanddeterminantsplaywitheachother. TheoremDRMM DeterminantRespectsMatrixMultiplication Supposethat A and B aresquarematricesofthesamesize.Thendet AB =det A det B Proof Thisproofisconstructedintwocases.First,supposethat A issingular.Thendet A =0by TheoremSMZD[389].BythecontrapositiveofTheoremNPNT[226], AB issingularaswell.Sobya secondapplicationofTheoremSMZD[389],det AB =0.Puttingitalltogether det AB =0=0det B =det A det B asdesired. Forthesecondcase,supposethat A isnonsingular.ByTheoremNMPEM[374]thereareelementary matrices E 1 ;E 2 ;E 3 ;:::;E s suchthat A = E 1 E 2 E 3 :::E s .Then det AB =det E 1 E 2 E 3 :::E s B =det E 1 det E 2 det E 3 ::: det E s det B TheoremDEMMM[389] =det E 1 E 2 E 3 :::E s det B TheoremDEMMM[389] =det A det B Itisamazingthatmatrixmultiplicationandthedeterminantinteractthisway.Mightitalsobetrue thatdet A + B =det A +det B ?SeeExercisePDM.M30[393]. Version2.02

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SubsectionPDM.READReadingQuestions393 SubsectionREAD ReadingQuestions 1.Considerthetwomatricesbelow,andsupposeyoualreadyhavecomputeddet A = )]TJ/F15 10.9091 Tf 8.485 0 Td [(120.Whatis det B ?Why? A = 2 6 6 4 083 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(25 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2843 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(42 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 7 7 5 B = 2 6 6 4 083 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(42 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2843 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(25 3 7 7 5 2.StatethetheoremthatallowsustomakeyetanotherextensiontoourNMExseriesoftheorems. 3.Whatisamazingabouttheinteractionbetweenmatrixmultiplicationandthedeterminant? Version2.02

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SubsectionPDM.EXCExercises394 SubsectionEXC Exercises C30 Eachofthearchetypesbelowisasystemofequationswithasquarecoecientmatrix,orisasquare matrixitself.Computethedeterminantofeachmatrix,notinghowTheoremSMZD[389]indicateswhen thematrixissingularornonsingular. ArchetypeA[702] ArchetypeB[707] ArchetypeF[724] ArchetypeK[746] ArchetypeL[750] ContributedbyRobertBeezer M20 Constructa3 3nonsingularmatrixandcallit A .Then,foreachentryofthematrix,compute thecorrespondingcofactor,andcreateanew3 3matrixfullofthesecofactorsbyplacingthecofactorof anentryinthesamelocationastheentryitwasbasedon.Oncecomplete,callthismatrix C .Compute AC t .Anyobservations?Repeatwithanewmatrix,orperhapswitha4 4matrix. ContributedbyRobertBeezerSolution[394] M30 Constructanexampletoshowthatthefollowingstatementisnottrueforallsquarematrices A and B ofthesamesize:det A + B =det A +det B ContributedbyRobertBeezer T10 TheoremNPNT[226]saysthatiftheproductofsquarematrices AB isnonsingular,thenthe individualmatrices A and B arenonsingularalso.Constructanewproofofthisresultmakinguseof theoremsaboutdeterminantsofmatrices. ContributedbyRobertBeezer T15 UseTheoremDRCM[384]toproveTheoremDZRC[383]asacorollary.SeeTechniqueLC[696]. ContributedbyRobertBeezer T20 Supposethat A isasquarematrixofsize n and 2 C isascalar.Provethatdet A = n det A ContributedbyRobertBeezer T25 EmployTheoremDT[377]toconstructthesecondhalfoftheproofofTheoremDRCM[384]the portionaboutamultipleofacolumn. ContributedbyRobertBeezer Version2.02

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SubsectionPDM.SOLSolutions395 SubsectionSOL Solutions M20 ContributedbyRobertBeezerStatement[393] Theresultofthesecomputationsshouldbeamatrixwiththevalueofdet A inthediagonalentriesand zeroselsewhere.Thesuggestionofusinganonsingularmatrixwaspartiallysothatitwasobviousthat thevalueofthedeterminantappearsonthediagonal. Thisresultwhichistrueingeneralprovidesamethodforcomputingtheinverseofanonsingular matrix.Since AC t =det A I n ,wecanmultiplybythereciprocalofthedeterminantwhichisnonzero! andtheinverseof A itexists!toarriveatanexpressionforthematrixinverse: A )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 = 1 det A C t Version2.02

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AnnotatedAcronymsPDM.DDeterminants396 AnnotatedAcronymsD Determinants TheoremEMDRO[372] Themainpurposeofelementarymatricesistoprovideamoreformalfoundationforrowoperations. Withthistheoremwecanconvertthenotionofdoingarowoperation"intotheslightlymoreprecise, andtractable,operationofmatrixmultiplicationbyanelementarymatrix.Theotherbigresultsinthis chapteraremadepossiblebythisconnectionandourpreviousunderstandingofthebehaviorofmatrix multiplicationsuchasresultsinSectionMM[194]. TheoremDER[376] Wedenethedeterminantbyexpansionabouttherstrowandthenproveyoucanexpandaboutanyrow andwithTheoremDEC[378],aboutanycolumn.Amazing.Ifthedeterminantseemscontrived,these resultsmightbegintoconvinceyouthatmaybesomethinginterestingisgoingon. TheoremDRMM[391] TheoremEMDRO[372]connectselementarymatriceswithmatrixmultiplication.Nowweconnectdeterminantswithmatrixmultiplication.Ifyouthoughtthedenitionofmatrixmultiplicationasexemplied byTheoremEMP[198]wasasoutlandishasthedenitionofthedeterminant,thennomore.Theyseem toplaytogetherquitenicely. TheoremSMZD[389] Thistheoremprovidesasimpletestfornonsingularity,eventhoughitisstatedandtitledasatheoremabout singularity.It'llbehelpful,especiallyinconcertwithTheoremDRMM[391],inestablishingupcoming resultsaboutnonsingularmatricesorcreatingalternativeproofsofearlierresults.Youmightevenuse thistheoremasanindicatorofhowoftenamatrixissingular.Createasquarematrixatrandom|what aretheoddsitissingular?Thistheoremsaysthedeterminanthastobezero,whichwemightsuspectis arareoccurrence.Ofcourse,wehavetobealotmorecarefulaboutwordslikerandom,"odds,"and rare"ifwewantpreciseanswerstothisquestion. Version2.02

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ChapterE Eigenvalues Whenwehaveasquarematrixofsize n A ,andwemultiplyitbyavector x from C n toformthematrixvectorproductDenitionMVP[194],theresultisanothervectorin C n .Sowecanadoptafunctional viewofthiscomputation|theactofmultiplyingbyasquarematrixisafunctionthatconvertsonevector x intoanotherone A x ofthesamesize.Forsomevectors,thisseeminglycomplicatedcomputationis reallynomorecomplicatedthanscalarmultiplication.Thevectorsvaryaccordingtothechoiceof A ,so thequestionistodetermine,foranindividualchoiceof A ,ifthereareanysuchvectors,andifso,which ones.Ithappensinavarietyofsituationsthatthesevectorsandthescalarsthatgoalongwiththemare ofspecialinterest. Wewillbesolvingpolynomialequationsinthischapter,whichraisesthespecterofrootsthatare complexnumbers.Thisdistinctpossibilityisourmainreasonforentertainingthecomplexnumbers throughoutthecourse.YoumightbemovedtorevisitSectionCNO[679]andSectionO[167]. SectionEE EigenvaluesandEigenvectors Westartwiththeprincipaldenitionforthischapter. SubsectionEEM EigenvaluesandEigenvectorsofaMatrix DenitionEEM EigenvaluesandEigenvectorsofaMatrix Supposethat A isasquarematrixofsize n x 6 = 0 isavectorin C n ,and isascalarin C .Thenwesay x isan eigenvector of A with eigenvalue if A x = x 4 Beforegoinganyfurther,perhapsweshouldconvinceyouthatsuchthingseverhappenatall.Understandthenextexample,butdonotconcernyourselfwithwherethepiecescomefrom.Wewillhave methodssoonenoughtobeabletodiscovertheseeigenvectorsourselves. ExampleSEE Someeigenvaluesandeigenvectors 397

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SubsectionEE.EEMEigenvaluesandEigenvectorsofaMatrix398 Considerthematrix A = 2 6 6 4 20498 )]TJ/F15 10.9091 Tf 8.485 0 Td [(26 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(280 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1343614 716348 )]TJ/F15 10.9091 Tf 8.485 0 Td [(90 )]TJ/F15 10.9091 Tf 8.485 0 Td [(36 )]TJ/F15 10.9091 Tf 8.485 0 Td [(472 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2326028 3 7 7 5 andthevectors x = 2 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 5 3 7 7 5 y = 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 4 3 7 7 5 z = 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 7 0 8 3 7 7 5 w = 2 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4 0 3 7 7 5 Then A x = 2 6 6 4 20498 )]TJ/F15 10.9091 Tf 8.485 0 Td [(26 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(280 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1343614 716348 )]TJ/F15 10.9091 Tf 8.485 0 Td [(90 )]TJ/F15 10.9091 Tf 8.485 0 Td [(36 )]TJ/F15 10.9091 Tf 8.485 0 Td [(472 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2326028 3 7 7 5 2 6 6 4 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 2 5 3 7 7 5 = 2 6 6 4 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 8 20 3 7 7 5 =4 2 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 5 3 7 7 5 =4 x so x isaneigenvectorof A witheigenvalue =4.Also, A y = 2 6 6 4 20498 )]TJ/F15 10.9091 Tf 8.485 0 Td [(26 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(280 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1343614 716348 )]TJ/F15 10.9091 Tf 8.485 0 Td [(90 )]TJ/F15 10.9091 Tf 8.485 0 Td [(36 )]TJ/F15 10.9091 Tf 8.485 0 Td [(472 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2326028 3 7 7 5 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(10 4 3 7 7 5 = 2 6 6 4 0 0 0 0 3 7 7 5 =0 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 4 3 7 7 5 =0 y so y isaneigenvectorof A witheigenvalue =0.Also, A z = 2 6 6 4 20498 )]TJ/F15 10.9091 Tf 8.485 0 Td [(26 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(280 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1343614 716348 )]TJ/F15 10.9091 Tf 8.485 0 Td [(90 )]TJ/F15 10.9091 Tf 8.485 0 Td [(36 )]TJ/F15 10.9091 Tf 8.485 0 Td [(472 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2326028 3 7 7 5 2 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 7 0 8 3 7 7 5 = 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 14 0 16 3 7 7 5 =2 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 7 0 8 3 7 7 5 =2 z so z isaneigenvectorof A witheigenvalue =2.Also, A w = 2 6 6 4 20498 )]TJ/F15 10.9091 Tf 8.485 0 Td [(26 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(280 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1343614 716348 )]TJ/F15 10.9091 Tf 8.485 0 Td [(90 )]TJ/F15 10.9091 Tf 8.485 0 Td [(36 )]TJ/F15 10.9091 Tf 8.485 0 Td [(472 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2326028 3 7 7 5 2 6 6 4 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 4 0 3 7 7 5 = 2 6 6 4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 8 0 3 7 7 5 =2 2 6 6 4 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 4 0 3 7 7 5 =2 w so w isaneigenvectorof A witheigenvalue =2. Sowehavedemonstratedfoureigenvectorsof A .Aretheremore?Yes,anynonzeroscalarmultipleof aneigenvectorisagainaneigenvector.Inthisexample,set u =30 x .Then A u = A x =30 A x TheoremMMSMM[201] =30 x x aneigenvectorof A =4 x PropertySMAM[184] =4 u sothat u isalsoaneigenvectorof A forthesameeigenvalue, =4. Thevectors z and w arebotheigenvectorsof A forthesameeigenvalue =2,yetthisisnotassimple asthetwovectorsjustbeingscalarmultiplesofeachothertheyaren't.Lookwhathappenswhenwe addthemtogether,toform v = z + w ,andmultiplyby A A v = A z + w Version2.02

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SubsectionEE.PMPolynomialsandMatrices399 = A z + A w TheoremMMDAA[201] =2 z +2 wz w eigenvectorsof A =2 z + w PropertyDVAC[87] =2 v sothat v isalsoaneigenvectorof A fortheeigenvalue =2.Soitwouldappearthatthesetofeigenvectors thatareassociatedwithaxedeigenvalueisclosedunderthevectorspaceoperationsof C n .Hmmm. Thevector y isaneigenvectorof A fortheeigenvalue =0,sowecanuseTheoremZSSM[286]to write A y =0 y = 0 .Butthisalsomeansthat y 2N A .Therewouldappeartobeaconnectionhere also. ExampleSEE[396]hintsatanumberofintriguingproperties,andtherearemanymore.Wewill explorethegeneralpropertiesofeigenvaluesandeigenvectorsinSectionPEE[419],butinthissectionwe willconcernourselveswiththequestionofactuallycomputingeigenvaluesandeigenvectors.Firstweneed abitofbackgroundmaterialonpolynomialsandmatrices. SubsectionPM PolynomialsandMatrices Apolynomialisacombinationofpowers,multiplicationbyscalarcoecients,andadditionwithsubtractionjustbeingtheinverseofaddition.Weneverhaveoccasiontodividewhencomputingthevalueof apolynomial.Soitiswithmatrices.Wecanaddandsubtractmatrices,wecanmultiplymatricesby scalars,andwecanformpowersofsquarematricesbyrepeatedapplicationsofmatrixmultiplication.We donotnormallydividematricesthoughsometimeswecanmultiplybyaninverse.Ifamatrixissquare, alltheoperationsconstitutingapolynomialwillpreservethesizeofthematrix.Soitisnaturaltoconsider evaluatingapolynomialwithamatrix,eectivelyreplacingthevariableofthepolynomialbyamatrix. We'lldemonstratewithanexample, ExamplePM Polynomialofamatrix Let p x =14+19 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x 3 + x 4 D = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(132 10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(311 3 5 andwewillcompute p D .First,thenecessarypowersof D .Noticethat D 0 isdenedtobethemultiplicativeidentity, I 3 ,aswillbethecaseingeneral. D 0 = I 3 = 2 4 100 010 001 3 5 D 1 = D = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(132 10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(311 3 5 D 2 = DD 1 = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(132 10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(311 3 5 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(132 10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(311 3 5 = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 510 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 3 5 D 3 = DD 2 = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(132 10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(311 3 5 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 510 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 3 5 = 2 4 19 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4158 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(411 3 5 Version2.02

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SubsectionEE.EEEExistenceofEigenvaluesandEigenvectors400 D 4 = DD 3 = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(132 10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(311 3 5 2 4 19 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4158 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(411 3 5 = 2 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(74954 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(30 )]TJ/F15 10.9091 Tf 8.485 0 Td [(494743 3 5 Then p D =14+19 D )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 D 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 D 3 + D 4 =14 2 4 100 010 001 3 5 +19 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(132 10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(311 3 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 510 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 3 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 2 4 19 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4158 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(411 3 5 + 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(74954 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(30 )]TJ/F15 10.9091 Tf 8.484 0 Td [(494743 3 5 = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(139193166 27 )]TJ/F15 10.9091 Tf 8.485 0 Td [(98 )]TJ/F15 10.9091 Tf 8.485 0 Td [(124 )]TJ/F15 10.9091 Tf 8.485 0 Td [(19311820 3 5 Noticethat p x factorsas p x =14+19 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x 3 + x 4 = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(7 x +1 2 Because D commuteswithitself DD = DD ,wecanusedistributivityofmatrixmultiplicationacross matrixadditionTheoremMMDAA[201]withoutbeingcarefulwithanyofthematrixproducts,andjust aseasilyevaluate p D usingthefactoredformof p x p D =14+19 D )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 D 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 D 3 + D 4 = D )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 I 3 D )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 I 3 D + I 3 2 = 2 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(332 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(31 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 5 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(832 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(31 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 3 5 2 4 032 11 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(312 3 5 2 = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(139193166 27 )]TJ/F15 10.9091 Tf 8.485 0 Td [(98 )]TJ/F15 10.9091 Tf 8.485 0 Td [(124 )]TJ/F15 10.9091 Tf 8.485 0 Td [(19311820 3 5 Thisexampleisnotmeanttobetooprofound.It is meanttoshowyouthatitisnaturaltoevaluatea polynomialwithamatrix,andthatthefactoredformofthepolynomialisasgoodasormaybebetter thantheexpandedform.Anddonotforgetthatconstanttermsinpolynomialsarereallymultiplesof theidentitymatrixwhenweareevaluatingthepolynomialwithamatrix. SubsectionEEE ExistenceofEigenvaluesandEigenvectors Beforeweembarkoncomputingeigenvaluesandeigenvectors,wewillprovethateverymatrixhasatleast oneeigenvalueandaneigenvectortogowithit.Later,inTheoremMNEM[427],wewilldeterminethe maximumnumberofeigenvaluesamatrixmayhave. ThedeterminantDenitionD[341]willbeapowerfultoolinSubsectionEE.CEE[403]whenitcomes timetocomputeeigenvalues.However,itispossible,withsomemoreadvancedmachinery,tocompute eigenvalueswithoutevermakinguseofthedeterminant.SheldonAxlerdoesjustthatinhisbook, Linear Version2.02

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SubsectionEE.EEEExistenceofEigenvaluesandEigenvectors401 AlgebraDoneRight .Hereandnow,wegiveAxler'sdeterminant-free"proofthateverymatrixhasan eigenvalue.Theresultisnottoostartling,buttheproofismostenjoyable. TheoremEMHE EveryMatrixHasanEigenvalue Suppose A isasquarematrix.Then A hasatleastoneeigenvalue. Proof Supposethat A hassize n ,andchoose x as any nonzerovectorfrom C n .Noticehowmuch latitudewehaveinourchoiceof x .Onlythezerovectoriso-limits.Considertheset S = x ;A x ;A 2 x ;A 3 x ;:::;A n x Thisisasetof n +1vectorsfrom C n ,sobyTheoremMVSLD[137], S islinearlydependent.Let a 0 ;a 1 ;a 2 ;:::;a n beacollectionof n +1scalarsfrom C ,notallzero,thatprovidearelationoflinear dependenceon S .Inotherwords, a 0 x + a 1 A x + a 2 A 2 x + a 3 A 3 x + + a n A n x = 0 Someofthe a i arenonzero.Supposethatjust a 0 6 =0,and a 1 = a 2 = a 3 = = a n =0.Then a 0 x = 0 andbyTheoremSMEZV[287],either a 0 =0or x = 0 ,whicharebothcontradictions.So a i 6 =0forsome i 1.Let m bethelargestintegersuchthat a m 6 =0.Fromthisdiscussionweknowthat m 1.Wecan alsoassumethat a m =1,forifnot,replaceeach a i by a i =a m toobtainscalarsthatserveequallywellin providingarelationoflineardependenceon S Denethepolynomial p x = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + + a m x m Becausewehaveconsistentlyused C asoursetofscalarsratherthan R ,weknowthatwecanfactor p x intolinearfactorsoftheform x )]TJ/F21 10.9091 Tf 10.926 0 Td [(b i ,where b i 2 C .Sotherearescalars, b 1 ;b 2 ;b 3 ;:::;b m ,from C sothat, p x = x )]TJ/F21 10.9091 Tf 10.909 0 Td [(b m x )]TJ/F21 10.9091 Tf 10.909 0 Td [(b m )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 x )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 3 x )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 2 x )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 1 Putitalltogetherand 0 = a 0 x + a 1 A x + a 2 A 2 x + a 3 A 3 x + + a n A n x = a 0 x + a 1 A x + a 2 A 2 x + a 3 A 3 x + + a m A m x a i =0for i>m = )]TJ/F21 10.9091 Tf 5 -8.836 Td [(a 0 I n + a 1 A + a 2 A 2 + a 3 A 3 + + a m A m x TheoremMMDAA[201] = p A x Denitionof p x = A )]TJ/F21 10.9091 Tf 10.909 0 Td [(b m I n A )]TJ/F21 10.9091 Tf 10.909 0 Td [(b m )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 I n A )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 3 I n A )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 2 I n A )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 1 I n x Let k bethesmallestintegersuchthat A )]TJ/F21 10.9091 Tf 10.909 0 Td [(b k I n A )]TJ/F21 10.9091 Tf 10.909 0 Td [(b k )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 I n A )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 3 I n A )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 2 I n A )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 1 I n x = 0 : Fromtheprecedingequation,weknowthat k m .Denethevector z by z = A )]TJ/F21 10.9091 Tf 10.909 0 Td [(b k )]TJ/F19 7.9701 Tf 6.586 0 Td [(1 I n A )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 3 I n A )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 2 I n A )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 1 I n x Noticethatbythedenitionof k ,thevector z mustbenonzero.Inthecasewhere k =1,weunderstand that z isdenedby z = x ,and z isstillnonzero.Now A )]TJ/F21 10.9091 Tf 10.909 0 Td [(b k I n z = A )]TJ/F21 10.9091 Tf 10.909 0 Td [(b k I n A )]TJ/F21 10.9091 Tf 10.909 0 Td [(b k )]TJ/F19 7.9701 Tf 6.587 0 Td [(1 I n A )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 3 I n A )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 2 I n A )]TJ/F21 10.9091 Tf 10.909 0 Td [(b 1 I n x = 0 whichallowsustowrite A z = A + O z PropertyZM[184] Version2.02

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SubsectionEE.EEEExistenceofEigenvaluesandEigenvectors402 = A )]TJ/F21 10.9091 Tf 10.909 0 Td [(b k I n + b k I n z PropertyAIM[184] = A )]TJ/F21 10.9091 Tf 10.909 0 Td [(b k I n z + b k I n z TheoremMMDAA[201] = 0 + b k I n z Deningpropertyof z = b k I n z PropertyZM[184] = b k z TheoremMMIM[200] Since z 6 = 0 ,thisequationsaysthat z isaneigenvectorof A fortheeigenvalue = b k DenitionEEM [396],sowehaveshownthatanysquarematrix A doeshaveatleastoneeigenvalue. TheproofofTheoremEMHE[400]isconstructiveitcontainsanunambiguousprocedurethatleads toaneigenvalue,butitisnotmeanttobepractical.Wewillillustratethetheoremwithanexample,the purposebeingtoprovideacompanionforstudyingtheproofandnottosuggestthisisthebestprocedure forcomputinganeigenvalue. ExampleCAEHW Computinganeigenvaluethehardway ThisexampleillustratestheproofofTheoremEMHE[400],sowillemploythesamenotationastheproof |lookthereforfullexplanations.Itis not meanttobeanexampleofareasonablecomputationalapproach tondingeigenvaluesandeigenvectors.OK,warningsinplace,herewego. Let A = 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(7 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1110 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 41020 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1140 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 82 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 )]TJ/F15 10.9091 Tf 8.484 0 Td [(15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1160 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 3 7 7 7 7 5 andchoose x = 2 6 6 6 6 4 3 0 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 4 3 7 7 7 7 5 Itisimportanttonoticethatthechoiceof x couldbe anything ,solongasitis not thezerovector.We havenotchosen x totallyatrandom,butsoastomakeourillustrationofthetheoremasgeneralas possible.Youcouldreplicatethisexamplewithyourownchoiceandthecomputationsareguaranteedto bereasonable,providedyouhaveacomputationaltoolthatwillfactorafthdegreepolynomialforyou. Theset S = x ;A x ;A 2 x ;A 3 x ;A 4 x ;A 5 x = 8 > > > > < > > > > : 2 6 6 6 6 4 3 0 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 4 3 7 7 7 7 5 ; 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 3 7 7 7 7 5 ; 2 6 6 6 6 4 6 )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 6 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 10 3 7 7 7 7 5 ; 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 14 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(18 3 7 7 7 7 5 ; 2 6 6 6 6 4 18 )]TJ/F15 10.9091 Tf 8.485 0 Td [(30 18 10 34 3 7 7 7 7 5 ; 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(34 62 )]TJ/F15 10.9091 Tf 8.485 0 Td [(34 )]TJ/F15 10.9091 Tf 8.485 0 Td [(26 )]TJ/F15 10.9091 Tf 8.485 0 Td [(66 3 7 7 7 7 5 9 > > > > = > > > > ; isguaranteedtobelinearlydependent,asithassixvectorsfrom C 5 TheoremMVSLD[137].Wewill searchforanon-trivialrelationoflineardependencebysolvingahomogeneoussystemofequationswhose coecientmatrixhasthevectorsof S ascolumnsthroughrowoperations, 2 6 6 6 6 4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(46 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1018 )]TJ/F15 10.9091 Tf 8.484 0 Td [(34 02 )]TJ/F15 10.9091 Tf 8.485 0 Td [(614 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3062 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(46 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1018 )]TJ/F15 10.9091 Tf 8.485 0 Td [(34 )]TJ/F15 10.9091 Tf 8.485 0 Td [(54 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(210 )]TJ/F15 10.9091 Tf 8.485 0 Td [(26 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(610 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1834 )]TJ/F15 10.9091 Tf 8.485 0 Td [(66 3 7 7 7 7 5 RREF )443()223()222()443(! 2 6 6 6 6 4 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(26 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1430 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(37 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1531 000000 000000 000000 3 7 7 7 7 5 Version2.02

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SubsectionEE.EEEExistenceofEigenvaluesandEigenvectors403 Therearefourfreevariablesfordescribingsolutionstothishomogeneoussystem,sowehaveourpickof solutions.Themostexpedientchoicewouldbetoset x 3 =1and x 4 = x 5 = x 6 =0.However,wewillagain opttomaximizethegeneralityofourillustrationofTheoremEMHE[400]andchoose x 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(8, x 4 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3, x 5 =1and x 6 =0.Theleadstoasolutionwith x 1 =16and x 2 =12. Thisrelationoflineardependencethensaysthat 0 =16 x +12 A x )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 A 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 A 3 x + A 4 x +0 A 5 x 0 = )]TJ/F15 10.9091 Tf 5 -8.837 Td [(16+12 A )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 A 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 A 3 + A 4 x Sowedene p x =16+12 x )]TJ/F15 10.9091 Tf 10.309 0 Td [(8 x 2 )]TJ/F15 10.9091 Tf 10.308 0 Td [(3 x 3 + x 4 ,andasadvertisedintheproofofTheoremEMHE[400],we haveapolynomialofdegree m =4 > 1suchthat p A x = 0 .Nowweneedtofactor p x over C .Ifyou madeyourownchoiceof x atthestart,thisiswhereyoumighthaveafthdegreepolynomial,andwhere youmightneedtouseacomputationaltooltondrootsandfactors.Wehave p x =16+12 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(8 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x 3 + x 4 = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +1 Soweknowthat 0 = p A x = A )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 I 5 A +2 I 5 A )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 I 5 A +1 I 5 x Weapplyonefactoratatime,untilwegetthezerovector,soastodeterminethevalueof k describedin theproofofTheoremEMHE[400], A +1 I 5 x = 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1110 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 42020 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1150 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 82 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1505 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1160 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 7 7 7 7 5 2 6 6 6 6 4 3 0 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 4 3 7 7 7 7 5 = 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 7 7 5 A )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 I 5 A +1 I 5 x = 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1110 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1020 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1120 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 82 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(35 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1160 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 3 7 7 7 7 5 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 7 7 5 = 2 6 6 6 6 4 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 4 4 8 3 7 7 7 7 5 A +2 I 5 A )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 I 5 A +1 I 5 x = 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1110 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 43020 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1160 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 82 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1515 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1160 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 7 7 7 7 5 2 6 6 6 6 4 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 4 4 8 3 7 7 7 7 5 = 2 6 6 6 6 4 0 0 0 0 0 3 7 7 7 7 5 So k =3and z = A )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 I 5 A +1 I 5 x = 2 6 6 6 6 4 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 4 4 8 3 7 7 7 7 5 isaneigenvectorof A fortheeigenvalue = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2,asyoucancheckbydoingthecomputation A z .If youworkthroughthisexamplewithyourownchoiceofthevector x stronglyrecommendedthenthe eigenvalueyouwillndmaybedierent,butwillbeintheset f 3 ; 0 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 g .SeeExerciseEE.M60 [414]forasuggestedstartingvector. Version2.02

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SubsectionEE.CEEComputingEigenvaluesandEigenvectors404 SubsectionCEE ComputingEigenvaluesandEigenvectors Fortunately,weneednotrelyontheprocedureofTheoremEMHE[400]eachtimeweneedaneigenvalue. Itisthedeterminant,andspecicallyTheoremSMZD[389],thatprovidesthemaintoolforcomputing eigenvalues.Hereisaninformalsequenceofequivalencesthatisthekeytodeterminingtheeigenvalues andeigenvectorsofamatrix, A x = x A x )]TJ/F21 10.9091 Tf 10.909 0 Td [(I n x = 0 A )]TJ/F21 10.9091 Tf 10.909 0 Td [(I n x = 0 So,foraneigenvalue andassociatedeigenvector x 6 = 0 ,thevector x willbeanonzeroelementofthe nullspaceof A )]TJ/F21 10.9091 Tf 11.589 0 Td [(I n ,whilethematrix A )]TJ/F21 10.9091 Tf 11.59 0 Td [(I n willbesingularandthereforehavezerodeterminant. TheseideasaremadepreciseinTheoremEMRCP[404]andTheoremEMNS[405],butfornowthisbrief discussionshouldsuceasmotivationforthefollowingdenitionandexample. DenitionCP CharacteristicPolynomial Supposethat A isasquarematrixofsize n .Thenthe characteristicpolynomial of A isthepolynomial p A x denedby p A x =det A )]TJ/F21 10.9091 Tf 10.91 0 Td [(xI n 4 ExampleCPMS3 Characteristicpolynomialofamatrix,size3 Consider F = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 1274 24167 3 5 Then p F x =det F )]TJ/F21 10.9091 Tf 10.909 0 Td [(xI 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 127 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 4 24167 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x DenitionCP[403] = )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 7 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 4 167 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x + )]TJ/F15 10.9091 Tf 8.484 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 124 247 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x DenitionDM[375] + )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 127 )]TJ/F21 10.9091 Tf 10.91 0 Td [(x 2416 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x )]TJ/F21 10.9091 Tf 10.909 0 Td [(x )]TJ/F21 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4TheoremDMST[376] + )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 + )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F21 10.9091 Tf 10.91 0 Td [(x =3+5 x + x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 3 = )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x +1 2 Thecharacteristicpolynomialisourmaincomputationaltoolforndingeigenvalues,andwillsometimes beusedtoaidusindeterminingthepropertiesofeigenvalues. Version2.02

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SubsectionEE.CEEComputingEigenvaluesandEigenvectors405 TheoremEMRCP EigenvaluesofaMatrixareRootsofCharacteristicPolynomials Suppose A isasquarematrix.Then isaneigenvalueof A ifandonlyif p A =0. Proof Suppose A hassize n isaneigenvalueof A thereexists x 6 = 0 sothat A x = x DenitionEEM[396] thereexists x 6 = 0 sothat A x )]TJ/F21 10.9091 Tf 10.909 0 Td [( x = 0 thereexists x 6 = 0 sothat A x )]TJ/F21 10.9091 Tf 10.909 0 Td [(I n x = 0 TheoremMMIM[200] thereexists x 6 = 0 sothat A )]TJ/F21 10.9091 Tf 10.909 0 Td [(I n x = 0 TheoremMMDAA[201] A )]TJ/F21 10.9091 Tf 10.909 0 Td [(I n issingularDenitionNM[71] det A )]TJ/F21 10.9091 Tf 10.909 0 Td [(I n =0TheoremSMZD[389] p A =0DenitionCP[403] ExampleEMS3 Eigenvaluesofamatrix,size3 InExampleCPMS3[403]wefoundthecharacteristicpolynomialof F = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 1274 24167 3 5 tobe p F x = )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F15 10.9091 Tf 9.671 0 Td [(3 x +1 2 .Factored,wecanndallofitsrootseasily,theyare x =3and x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1.By TheoremEMRCP[404], =3and = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1arebotheigenvaluesof F ,andthesearetheonlyeigenvalues of F .We'vefoundthemall. Letusnowturnourattentiontothecomputationofeigenvectors. DenitionEM EigenspaceofaMatrix Supposethat A isasquarematrixand isaneigenvalueof A .Thenthe eigenspace of A for E A isthesetofalltheeigenvectorsof A for ,togetherwiththeinclusionofthezerovector. 4 ExampleSEE[396]hintedthatthesetofeigenvectorsforasingleeigenvaluemighthavesomeclosure properties,andwiththeadditionofthenon-eigenvector, 0 ,weindeedgetawholesubspace. TheoremEMS EigenspaceforaMatrixisaSubspace Suppose A isasquarematrixofsize n and isaneigenvalueof A .Thentheeigenspace E A isasubspace ofthevectorspace C n Proof WewillcheckthethreeconditionsofTheoremTSS[293].First,DenitionEM[404]explicitly includesthezerovectorin E A ,sothesetisnon-empty. Supposethat x ; y 2E A ,thatis, x and y aretwoeigenvectorsof A for .Then A x + y = A x + A y TheoremMMDAA[201] = x + yx ; y eigenvectorsof A = x + y PropertyDVAC[87] Soeither x + y = 0 ,or x + y isaneigenvectorof A for DenitionEEM[396].So,ineitherevent, x + y 2E A ,andwehaveadditiveclosure. Version2.02

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SubsectionEE.CEEComputingEigenvaluesandEigenvectors406 Supposethat 2 C ,andthat x 2E A ,thatis, x isaneigenvectorof A for .Then A x = A x TheoremMMSMM[201] = xx aneigenvectorof A = x PropertySMAC[86] Soeither x = 0 ,or x isaneigenvectorof A for DenitionEEM[396].So,ineitherevent, x 2E A andwehavescalarclosure. WiththethreeconditionsofTheoremTSS[293]met,weknow E A isasubspace. TheoremEMS[404]tellsusthataneigenspaceisasubspaceandhenceavectorspaceinitsown right.Ournexttheoremtellsushowtoquicklyconstructthissubspace. TheoremEMNS EigenspaceofaMatrixisaNullSpace Suppose A isasquarematrixofsize n and isaneigenvalueof A .Then E A = N A )]TJ/F21 10.9091 Tf 10.909 0 Td [(I n Proof Theconclusionofthistheoremisanequalityofsets,sonormallywewouldfollowtheadviceof DenitionSE[684].However,inthiscasewecanconstructasequenceofequivalenceswhichwilltogether providethetwosubsetinclusionsweneed.First,noticethat 0 2E A byDenitionEM[404]and 0 2N A )]TJ/F21 10.9091 Tf 10.909 0 Td [(I n byTheoremHSC[62].Nowconsideranynonzerovector x 2 C n x 2E A A x = x DenitionEM[404] A x )]TJ/F21 10.9091 Tf 10.909 0 Td [( x = 0 A x )]TJ/F21 10.9091 Tf 10.909 0 Td [(I n x = 0 TheoremMMIM[200] A )]TJ/F21 10.9091 Tf 10.909 0 Td [(I n x = 0 TheoremMMDAA[201] x 2N A )]TJ/F21 10.9091 Tf 10.91 0 Td [(I n DenitionNSM[64] YoumightnoticethecloseparallelsanddierencesbetweentheproofsofTheoremEMRCP[404] andTheoremEMNS[405].SinceTheoremEMNS[405]describesthesetofalltheeigenvectorsof A asa nullspacewecanusetechniquessuchasTheoremBNS[139]toprovideconcisedescriptionsofeigenspaces. TheoremEMNS[405]alsoprovidesatrivialproofforTheoremEMS[404]. ExampleESMS3 Eigenspacesofamatrix,size3 ExampleCPMS3[403]andExampleEMS3[404]describethecharacteristicpolynomialandeigenvaluesof the3 3matrix F = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(13 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 1274 24167 3 5 Wewillnowtaketheeacheigenvalueinturnandcomputeitseigenspace.Todothis,werow-reduce thematrix F )]TJ/F21 10.9091 Tf 10.996 0 Td [(I 3 inordertodeterminesolutionstothehomogeneoussystem LS F )]TJ/F21 10.9091 Tf 10.909 0 Td [(I 3 ; 0 andthen expresstheeigenspaceasthenullspaceof F )]TJ/F21 10.9091 Tf 10.379 0 Td [(I 3 TheoremEMNS[405].TheoremBNS[139]thentells ushowtowritethenullspaceasthespanofabasis. =3 F )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 I 3 = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(16 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 1244 24164 3 5 RREF )443()223()222()443(! 2 4 1 0 1 2 0 1 )]TJ/F19 7.9701 Tf 9.68 4.295 Td [(1 2 000 3 5 Version2.02

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SubsectionEE.ECEEExamplesofComputingEigenvaluesandEigenvectors407 E F = N F )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 I 3 = 8 < : 2 4 )]TJ/F19 7.9701 Tf 9.681 4.295 Td [(1 2 1 2 1 3 5 9 = ; + = 8 < : 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 2 3 5 9 = ; + = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 F +1 I 3 = 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 1284 24168 3 5 RREF )443()223()222()443(! 2 4 1 2 3 1 3 000 000 3 5 E F )]TJ/F15 10.9091 Tf 8.485 0 Td [(1= N F +1 I 3 = 8 < : 2 4 )]TJ/F19 7.9701 Tf 9.681 4.295 Td [(2 3 1 0 3 5 ; 2 4 )]TJ/F19 7.9701 Tf 9.68 4.295 Td [(1 3 0 1 3 5 9 = ; + = 8 < : 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 0 3 5 ; 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 3 3 5 9 = ; + Eigenspacesinhand,wecaneasilycomputeeigenvectorsbyformingnontriviallinearcombinationsof thebasisvectorsdescribingeacheigenspace.Inparticular,noticethatwecanprettyup"ourbasis vectorsbyusingscalarmultiplestoclearoutfractions.Morepowerfulscienticcalculators,andmost everymathematicalsoftwarepackage,willcomputeeigenvaluesofamatrixalongwithbasisvectorsofthe eigenspaces.Besuretounderstandhowyourdeviceoutputscomplexnumbers,sincetheyarelikelyto occur.Also,thebasisvectorswillnotnecessarilylookliketheresultsofanapplicationofTheoremBNS [139].DuplicatingtheresultsofthenextsectionSubsectionEE.ECEE[406]withyourdevicewouldbe verygoodpractice.See:ComputationE.SAGE[677]. SubsectionECEE ExamplesofComputingEigenvaluesandEigenvectors Notheoremsinthissection,justaselectionofexamplesmeanttoillustratetherangeofpossibilitiesforthe eigenvaluesandeigenvectorsofamatrix.Theseexamplescanallbedonebyhand,thoughthecomputation ofthecharacteristicpolynomialwouldbeverytime-consuminganderror-prone.Itcanalsobedicult tofactoranarbitrarypolynomial,thoughifweweretosuggestthatmostofoureigenvaluesaregoing tobeintegers,thenitcanbeeasiertohuntforroots.Theseexamplesaremeanttolooksimilartoa concatenationofExampleCPMS3[403],ExampleEMS3[404]andExampleESMS3[405].First,wewill sneakinapairofdenitionssowecanillustratethemthroughoutthissequenceofexamples. DenitionAME AlgebraicMultiplicityofanEigenvalue Supposethat A isasquarematrixand isaneigenvalueof A .Thenthe algebraicmultiplicity of A ,isthehighestpowerof x )]TJ/F21 10.9091 Tf 10.909 0 Td [( thatdividesthecharacteristicpolynomial, p A x ThisdenitioncontainsNotationAME. 4 Sinceaneigenvalue isarootofthecharacteristicpolynomial,thereisalwaysafactorof x )]TJ/F21 10.9091 Tf 11.49 0 Td [( andthealgebraicmultiplicityisjustthepowerofthisfactorinafactorizationof p A x .Soinparticular, A 1.Comparethedenitionofalgebraicmultiplicitywiththenextdenition. DenitionGME GeometricMultiplicityofanEigenvalue Supposethat A isasquarematrixand isaneigenvalueof A .Thenthe geometricmultiplicity of A ,isthedimensionoftheeigenspace E A ThisdenitioncontainsNotationGME. 4 Sinceeveryeigenvaluemusthaveatleastoneeigenvector,theassociatedeigenspacecannotbetrivial, andso A 1. ExampleEMMS4 Eigenvaluemultiplicities,matrixofsize4 Version2.02

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SubsectionEE.ECEEExamplesofComputingEigenvaluesandEigenvectors408 Considerthematrix B = 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 12149 65 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4510 3 7 7 5 then p B x =8 )]TJ/F15 10.9091 Tf 10.909 0 Td [(20 x +18 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(7 x 3 + x 4 = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 3 Sotheeigenvaluesare =1 ; 2withalgebraicmultiplicities B =1and B =3. Computingeigenvectors, =1 B )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 I 4 = 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(31 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 12049 65 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(459 3 7 7 5 RREF )443()223()222()443(! 2 6 6 4 1 0 1 3 0 0 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 000 1 0000 3 7 7 5 E B = N B )]TJ/F15 10.9091 Tf 10.91 0 Td [(1 I 4 = 8 > > < > > : 2 6 6 4 )]TJ/F19 7.9701 Tf 9.681 4.296 Td [(1 3 1 1 0 3 7 7 5 9 > > = > > ; + = 8 > > < > > : 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 3 0 3 7 7 5 9 > > = > > ; + =2 B )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 I 4 = 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(41 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(149 65 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(458 3 7 7 5 RREF )443()223()222()443(! 2 6 6 4 1 001 = 2 0 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 00 1 1 = 2 0000 3 7 7 5 E B = N B )]TJ/F15 10.9091 Tf 10.91 0 Td [(2 I 4 = 8 > > < > > : 2 6 6 4 )]TJ/F19 7.9701 Tf 9.681 4.295 Td [(1 2 1 )]TJ/F19 7.9701 Tf 9.681 4.295 Td [(1 2 1 3 7 7 5 9 > > = > > ; + = 8 > > < > > : 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 7 7 5 9 > > = > > ; + Soeacheigenspacehasdimension1andso B =1and B =1.Thisexampleisofinterestbecause ofthediscrepancybetweenthetwomultiplicitiesfor =2.Inmanyofourexamplesthealgebraicand geometricmultiplicitieswillbeequalforalloftheeigenvaluesasitwasfor =1inthisexample,sokeep thisexampleinmind.WewillhavesomeexplanationsforthisphenomenonlaterseeExampleNDMS4 [440]. ExampleESMS4 Eigenvalues,symmetricmatrixofsize4 Considerthematrix C = 2 6 6 4 1011 0111 1110 1101 3 7 7 5 then p C x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3+4 x +2 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x 3 + x 4 = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 2 x +1 Sotheeigenvaluesare =3 ; 1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1withalgebraicmultiplicities C =1, C =2and C )]TJ/F15 10.9091 Tf 8.485 0 Td [(1=1. Computingeigenvectors, =3 C )]TJ/F15 10.9091 Tf 10.91 0 Td [(3 I 4 = 2 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2011 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(211 11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(20 110 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 5 RREF )443()223()222()443(! 2 6 6 4 1 00 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 00 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0000 3 7 7 5 Version2.02

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SubsectionEE.ECEEExamplesofComputingEigenvaluesandEigenvectors409 E C = N C )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 I 4 = 8 > > < > > : 2 6 6 4 1 1 1 1 3 7 7 5 9 > > = > > ; + =1 C )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 I 4 = 2 6 6 4 0011 0011 1100 1100 3 7 7 5 RREF )443()223()222()443(! 2 6 6 4 1 100 00 1 1 0000 0000 3 7 7 5 E C = N C )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 I 4 = 8 > > < > > : 2 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 1 0 0 3 7 7 5 ; 2 6 6 4 0 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 3 7 7 5 9 > > = > > ; + = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 C +1 I 4 = 2 6 6 4 2011 0211 1120 1102 3 7 7 5 RREF )443()223()222()443(! 2 6 6 4 1 001 0 1 01 00 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0000 3 7 7 5 E C )]TJ/F15 10.9091 Tf 8.485 0 Td [(1= N C +1 I 4 = 8 > > < > > : 2 6 6 4 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 1 1 3 7 7 5 9 > > = > > ; + Sotheeigenspacedimensionsyieldgeometricmultiplicities C =1, C =2and C )]TJ/F15 10.9091 Tf 8.485 0 Td [(1=1,the sameasforthealgebraicmultiplicities.Thisexampleisofinterestbecause A isasymmetricmatrix,and willbethesubjectofTheoremHMRE[427]. ExampleHMEM5 Highmultiplicityeigenvalues,matrixofsize5 Considerthematrix E = 2 6 6 6 6 4 291426 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(47 )]TJ/F15 10.9091 Tf 8.484 0 Td [(22 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1113 191054 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(19 )]TJ/F15 10.9091 Tf 8.484 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(28 7431 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 3 7 7 7 7 5 then p E x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(16+16 x +8 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(16 x 3 +7 x 4 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 5 = )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 4 x +1 Sotheeigenvaluesare =2 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1withalgebraicmultiplicities E =4and E )]TJ/F15 10.9091 Tf 8.485 0 Td [(1=1. Computingeigenvectors, =2 E )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 I 5 = 2 6 6 6 6 4 271426 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(47 )]TJ/F15 10.9091 Tf 8.485 0 Td [(24 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1113 191034 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(19 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(48 7431 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 3 7 7 7 7 5 RREF )443()223()222()443(! 2 6 6 6 6 6 4 1 0010 0 1 0 )]TJ/F19 7.9701 Tf 9.68 4.295 Td [(3 2 )]TJ/F19 7.9701 Tf 9.68 4.295 Td [(1 2 00 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 00000 00000 3 7 7 7 7 7 5 E E = N E )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 I 5 = 8 > > > > < > > > > : 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 2 0 1 0 3 7 7 7 7 5 ; 2 6 6 6 6 4 0 1 2 1 0 1 3 7 7 7 7 5 9 > > > > = > > > > ; + = 8 > > > > < > > > > : 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 0 2 0 3 7 7 7 7 5 ; 2 6 6 6 6 4 0 1 2 0 2 3 7 7 7 7 5 9 > > > > = > > > > ; + Version2.02

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SubsectionEE.ECEEExamplesofComputingEigenvaluesandEigenvectors410 = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 E +1 I 5 = 2 6 6 6 6 4 301426 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 )]TJ/F15 10.9091 Tf 8.485 0 Td [(47 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1113 191064 )]TJ/F15 10.9091 Tf 8.485 0 Td [(8 )]TJ/F15 10.9091 Tf 8.485 0 Td [(19 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(18 7431 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 7 7 5 RREF )443()223()222()443(! 2 6 6 6 6 6 4 1 0020 0 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(40 00 1 10 0000 1 00000 3 7 7 7 7 7 5 E E )]TJ/F15 10.9091 Tf 8.485 0 Td [(1= N E +1 I 5 = 8 > > > > < > > > > : 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 0 3 7 7 7 7 5 9 > > > > = > > > > ; + Sotheeigenspacedimensionsyieldgeometricmultiplicities E =2and E )]TJ/F15 10.9091 Tf 8.485 0 Td [(1=1.Thisexampleis ofinterestbecause =2hassuchalargealgebraicmultiplicity,whichisalsonotequaltoitsgeometric multiplicity. ExampleCEMS6 Complexeigenvalues,matrixofsize6 Considerthematrix F = 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(59 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3441122530 17 )]TJ/F15 10.9091 Tf 8.485 0 Td [(46 )]TJ/F15 10.9091 Tf 8.485 0 Td [(36 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(29 )]TJ/F15 10.9091 Tf 8.485 0 Td [(233 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11958 )]TJ/F15 10.9091 Tf 8.485 0 Td [(357554 15781 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4321 )]TJ/F15 10.9091 Tf 8.485 0 Td [(51 )]TJ/F15 10.9091 Tf 8.485 0 Td [(39 )]TJ/F15 10.9091 Tf 8.485 0 Td [(91 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4832 )]TJ/F15 10.9091 Tf 8.485 0 Td [(53226 209107 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5528 )]TJ/F15 10.9091 Tf 8.485 0 Td [(69 )]TJ/F15 10.9091 Tf 8.485 0 Td [(50 3 7 7 7 7 7 7 5 then p F x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(50+55 x +13 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(50 x 3 +32 x 4 )]TJ/F15 10.9091 Tf 10.909 0 Td [(9 x 5 + x 6 = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +1 x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(4 x +5 2 = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +1 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(+ i x )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 2 = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x +1 x )]TJ/F15 10.9091 Tf 10.91 0 Td [(+ i 2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 2 Sotheeigenvaluesare =2 ; )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 ; 2+ i; 2 )]TJ/F21 10.9091 Tf 11.954 0 Td [(i withalgebraicmultiplicities F =1, F )]TJ/F15 10.9091 Tf 8.485 0 Td [(1=1, F + i =2and F )]TJ/F21 10.9091 Tf 10.909 0 Td [(i =2. Computingeigenvectors, =2 F )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 I 6 = 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(61 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3441122530 15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(46 )]TJ/F15 10.9091 Tf 8.485 0 Td [(36 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(29 )]TJ/F15 10.9091 Tf 8.485 0 Td [(233 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11956 )]TJ/F15 10.9091 Tf 8.485 0 Td [(357554 15781 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4319 )]TJ/F15 10.9091 Tf 8.485 0 Td [(51 )]TJ/F15 10.9091 Tf 8.485 0 Td [(39 )]TJ/F15 10.9091 Tf 8.485 0 Td [(91 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4832 )]TJ/F15 10.9091 Tf 8.485 0 Td [(53026 209107 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5528 )]TJ/F15 10.9091 Tf 8.485 0 Td [(69 )]TJ/F15 10.9091 Tf 8.485 0 Td [(52 3 7 7 7 7 7 7 5 RREF )443()223()222()443(! 2 6 6 6 6 6 6 6 4 1 0000 1 5 0 1 0000 00 1 00 3 5 000 1 0 )]TJ/F19 7.9701 Tf 9.681 4.295 Td [(1 5 0000 1 4 5 000000 3 7 7 7 7 7 7 7 5 E F = N F )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 I 6 = 8 > > > > > > < > > > > > > : 2 6 6 6 6 6 6 4 )]TJ/F19 7.9701 Tf 9.681 4.295 Td [(1 5 0 )]TJ/F19 7.9701 Tf 9.681 4.295 Td [(3 5 1 5 )]TJ/F19 7.9701 Tf 9.681 4.295 Td [(4 5 1 3 7 7 7 7 7 7 5 9 > > > > > > = > > > > > > ; + = 8 > > > > > > < > > > > > > : 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 5 3 7 7 7 7 7 7 5 9 > > > > > > = > > > > > > ; + Version2.02

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SubsectionEE.ECEEExamplesofComputingEigenvaluesandEigenvectors411 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 F +1 I 6 = 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(58 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3441122530 18 )]TJ/F15 10.9091 Tf 8.485 0 Td [(46 )]TJ/F15 10.9091 Tf 8.485 0 Td [(36 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(29 )]TJ/F15 10.9091 Tf 8.485 0 Td [(233 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11959 )]TJ/F15 10.9091 Tf 8.485 0 Td [(357554 15781 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4322 )]TJ/F15 10.9091 Tf 8.485 0 Td [(51 )]TJ/F15 10.9091 Tf 8.485 0 Td [(39 )]TJ/F15 10.9091 Tf 8.485 0 Td [(91 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4832 )]TJ/F15 10.9091 Tf 8.485 0 Td [(53326 209107 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5528 )]TJ/F15 10.9091 Tf 8.485 0 Td [(69 )]TJ/F15 10.9091 Tf 8.485 0 Td [(49 3 7 7 7 7 7 7 5 RREF )443()223()222()443(! 2 6 6 6 6 6 6 6 4 1 0000 1 2 0 1 000 )]TJ/F19 7.9701 Tf 9.68 4.295 Td [(3 2 00 1 00 1 2 000 1 00 0000 1 )]TJ/F19 7.9701 Tf 9.68 4.295 Td [(1 2 000000 3 7 7 7 7 7 7 7 5 E F )]TJ/F15 10.9091 Tf 8.485 0 Td [(1= N F + I 6 = 8 > > > > > > < > > > > > > : 2 6 6 6 6 6 6 4 )]TJ/F19 7.9701 Tf 9.68 4.295 Td [(1 2 3 2 )]TJ/F19 7.9701 Tf 9.68 4.295 Td [(1 2 0 1 2 1 3 7 7 7 7 7 7 5 9 > > > > > > = > > > > > > ; + = 8 > > > > > > < > > > > > > : 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 3 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 1 2 3 7 7 7 7 7 7 5 9 > > > > > > = > > > > > > ; + =2+ i F )]TJ/F15 10.9091 Tf 10.909 0 Td [(+ i I 6 = 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(61 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i )]TJ/F15 10.9091 Tf 8.484 0 Td [(3441122530 15 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i )]TJ/F15 10.9091 Tf 8.485 0 Td [(46 )]TJ/F15 10.9091 Tf 8.485 0 Td [(36 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(29 )]TJ/F15 10.9091 Tf 8.484 0 Td [(233 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11956 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i )]TJ/F15 10.9091 Tf 8.485 0 Td [(357554 15781 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4319 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i )]TJ/F15 10.9091 Tf 8.485 0 Td [(51 )]TJ/F15 10.9091 Tf 8.485 0 Td [(39 )]TJ/F15 10.9091 Tf 8.485 0 Td [(91 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4832 )]TJ/F15 10.9091 Tf 8.485 0 Td [(530 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 26 209107 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5528 )]TJ/F15 10.9091 Tf 8.485 0 Td [(69 )]TJ/F15 10.9091 Tf 8.485 0 Td [(52 )]TJ/F21 10.9091 Tf 10.91 0 Td [(i 3 7 7 7 7 7 7 5 RREF )443()223()222()443(! 2 6 6 6 6 6 6 6 4 1 0000 1 5 + i 0 1 000 1 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i 00 1 001 000 1 0 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 0000 1 1 000000 3 7 7 7 7 7 7 7 5 E F + i = N F )]TJ/F15 10.9091 Tf 10.909 0 Td [(+ i I 6 = 8 > > > > > > < > > > > > > : 2 6 6 6 6 6 6 4 )]TJ/F19 7.9701 Tf 9.68 4.295 Td [(1 5 + i 1 5 +2 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 3 7 7 7 7 7 7 5 9 > > > > > > = > > > > > > ; + = 8 > > > > > > < > > > > > > : 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 9+2 i )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 5 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 5 3 7 7 7 7 7 7 5 9 > > > > > > = > > > > > > ; + =2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i F )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F21 10.9091 Tf 10.909 0 Td [(i I 6 = 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(61+ i )]TJ/F15 10.9091 Tf 8.484 0 Td [(3441122530 15+ i )]TJ/F15 10.9091 Tf 8.485 0 Td [(46 )]TJ/F15 10.9091 Tf 8.485 0 Td [(36 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 )]TJ/F15 10.9091 Tf 8.485 0 Td [(29 )]TJ/F15 10.9091 Tf 8.484 0 Td [(233 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11956+ i )]TJ/F15 10.9091 Tf 8.485 0 Td [(357554 15781 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4319+ i )]TJ/F15 10.9091 Tf 8.485 0 Td [(51 )]TJ/F15 10.9091 Tf 8.485 0 Td [(39 )]TJ/F15 10.9091 Tf 8.485 0 Td [(91 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4832 )]TJ/F15 10.9091 Tf 8.485 0 Td [(530+ i 26 209107 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5528 )]TJ/F15 10.9091 Tf 8.485 0 Td [(69 )]TJ/F15 10.9091 Tf 8.485 0 Td [(52+ i 3 7 7 7 7 7 7 5 Version2.02

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SubsectionEE.ECEEExamplesofComputingEigenvaluesandEigenvectors412 RREF )443()223()222()443(! 2 6 6 6 6 6 6 6 4 1 0000 1 5 )]TJ/F21 10.9091 Tf 10.909 0 Td [(i 0 1 000 1 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9+2 i 00 1 001 000 1 0 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 0000 1 1 000000 3 7 7 7 7 7 7 7 5 E F )]TJ/F21 10.9091 Tf 10.909 0 Td [(i = N F )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F21 10.9091 Tf 10.909 0 Td [(i I 6 = 8 > > > > > > < > > > > > > : 2 6 6 6 6 6 6 4 1 5 )]TJ/F15 10.9091 Tf 8.484 0 Td [(7+ i 1 5 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 3 7 7 7 7 7 7 5 9 > > > > > > = > > > > > > ; + = 8 > > > > > > < > > > > > > : 2 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7+ i 9 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 i )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 5 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 5 3 7 7 7 7 7 7 5 9 > > > > > > = > > > > > > ; + Sotheeigenspacedimensionsyieldgeometricmultiplicities F =1, F )]TJ/F15 10.9091 Tf 8.485 0 Td [(1=1, F + i =1 and F )]TJ/F21 10.9091 Tf 10.909 0 Td [(i =1.Thisexampledemonstratessomeofthepossibilitiesfortheappearanceofcomplex eigenvalues,evenwhenalltheentriesofthematrixarereal.Noticehowallthenumbersintheanalysisof =2 )]TJ/F21 10.9091 Tf 11.041 0 Td [(i areconjugatesofthecorrespondingnumberintheanalysisof =2+ i .Thisisthecontentof theupcomingTheoremERMCP[423]. ExampleDEMS5 Distincteigenvalues,matrixofsize5 Considerthematrix H = 2 6 6 6 6 4 1518 )]TJ/F15 10.9091 Tf 8.485 0 Td [(86 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 531 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 )]TJ/F15 10.9091 Tf 8.484 0 Td [(45 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(43 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4617 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1415 2630 )]TJ/F15 10.9091 Tf 8.485 0 Td [(128 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 3 7 7 7 7 5 then p H x = )]TJ/F15 10.9091 Tf 8.484 0 Td [(6 x + x 2 +7 x 3 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 4 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 5 = x x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 x +1 x +3 Sotheeigenvaluesare =2 ; 1 ; 0 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(3withalgebraicmultiplicities H =1, H =1, H =1, H )]TJ/F15 10.9091 Tf 8.485 0 Td [(1=1and H )]TJ/F15 10.9091 Tf 8.485 0 Td [(3=1. Computingeigenvectors, =2 H )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 I 5 = 2 6 6 6 6 4 1318 )]TJ/F15 10.9091 Tf 8.485 0 Td [(86 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 511 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(43 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(43 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4617 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1615 2630 )]TJ/F15 10.9091 Tf 8.485 0 Td [(128 )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 3 7 7 7 7 5 RREF )443()223()222()443(! 2 6 6 6 6 6 4 1 000 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 1 001 00 1 02 000 1 1 00000 3 7 7 7 7 7 5 E H = N H )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 I 5 = 8 > > > > < > > > > : 2 6 6 6 6 4 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 1 3 7 7 7 7 5 9 > > > > = > > > > ; + =1 H )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 I 5 = 2 6 6 6 6 4 1418 )]TJ/F15 10.9091 Tf 8.485 0 Td [(86 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 521 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(44 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(43 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4617 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1515 2630 )]TJ/F15 10.9091 Tf 8.484 0 Td [(128 )]TJ/F15 10.9091 Tf 8.485 0 Td [(11 3 7 7 7 7 5 RREF )443()223()222()443(! 2 6 6 6 6 6 4 1 000 )]TJ/F19 7.9701 Tf 9.681 4.295 Td [(1 2 0 1 000 00 1 0 1 2 000 1 1 00000 3 7 7 7 7 7 5 Version2.02

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SubsectionEE.ECEEExamplesofComputingEigenvaluesandEigenvectors413 E H = N H )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 I 5 = 8 > > > > < > > > > : 2 6 6 6 6 4 1 2 0 )]TJ/F19 7.9701 Tf 9.68 4.295 Td [(1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 1 3 7 7 7 7 5 9 > > > > = > > > > ; + = 8 > > > > < > > > > : 2 6 6 6 6 4 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 2 3 7 7 7 7 5 9 > > > > = > > > > ; + =0 H )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 I 5 = 2 6 6 6 6 4 1518 )]TJ/F15 10.9091 Tf 8.485 0 Td [(86 )]TJ/F15 10.9091 Tf 8.484 0 Td [(5 531 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(3 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(45 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(43 )]TJ/F15 10.9091 Tf 8.484 0 Td [(4617 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1415 2630 )]TJ/F15 10.9091 Tf 8.485 0 Td [(128 )]TJ/F15 10.9091 Tf 8.485 0 Td [(10 3 7 7 7 7 5 RREF )443()223()222()443(! 2 6 6 6 6 6 4 1 0001 0 1 00 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 00 1 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 000 1 0 00000 3 7 7 7 7 7 5 E H = N H )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 I 5 = 8 > > > > < > > > > : 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 2 0 1 3 7 7 7 7 5 9 > > > > = > > > > ; + = )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 H +1 I 5 = 2 6 6 6 6 4 1618 )]TJ/F15 10.9091 Tf 8.485 0 Td [(86 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 541 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(46 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(43 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4617 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1315 2630 )]TJ/F15 10.9091 Tf 8.485 0 Td [(128 )]TJ/F15 10.9091 Tf 8.485 0 Td [(9 3 7 7 7 7 5 RREF )443()223()222()443(! 2 6 6 6 6 6 4 1 000 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 = 2 0 1 000 00 1 00 000 1 1 = 2 00000 3 7 7 7 7 7 5 E H )]TJ/F15 10.9091 Tf 8.485 0 Td [(1= N H +1 I 5 = 8 > > > > < > > > > : 2 6 6 6 6 4 1 2 0 0 )]TJ/F19 7.9701 Tf 9.681 4.295 Td [(1 2 1 3 7 7 7 7 5 9 > > > > = > > > > ; + = 8 > > > > < > > > > : 2 6 6 6 6 4 1 0 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 2 3 7 7 7 7 5 9 > > > > = > > > > ; + = )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 H +3 I 5 = 2 6 6 6 6 4 1818 )]TJ/F15 10.9091 Tf 8.485 0 Td [(86 )]TJ/F15 10.9091 Tf 8.485 0 Td [(5 561 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 0 )]TJ/F15 10.9091 Tf 8.485 0 Td [(48 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(43 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4617 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1115 2630 )]TJ/F15 10.9091 Tf 8.485 0 Td [(128 )]TJ/F15 10.9091 Tf 8.485 0 Td [(7 3 7 7 7 7 5 RREF )443()223()222()443(! 2 6 6 6 6 6 4 1 000 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 0 1 00 1 2 00 1 01 000 1 2 00000 3 7 7 7 7 7 5 E H )]TJ/F15 10.9091 Tf 8.485 0 Td [(3= N H +3 I 5 = 8 > > > > < > > > > : 2 6 6 6 6 4 1 )]TJ/F19 7.9701 Tf 9.68 4.296 Td [(1 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 3 7 7 7 7 5 9 > > > > = > > > > ; + = 8 > > > > < > > > > : 2 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 1 2 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 7 7 5 9 > > > > = > > > > ; + Sotheeigenspacedimensionsyieldgeometricmultiplicities H =1, H =1, H =1, H )]TJ/F15 10.9091 Tf 8.485 0 Td [(1=1 and H )]TJ/F15 10.9091 Tf 8.485 0 Td [(3=1,identicaltothealgebraicmultiplicities.Thisexampleisofinterestfortworeasons.First, =0isaneigenvalue,illustratingtheupcomingTheoremSMZE[420].Second,alltheeigenvaluesare distinct,yieldingalgebraicandgeometricmultiplicitiesof1foreacheigenvalue,illustratingTheoremDED [440]. Version2.02

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SubsectionEE.READReadingQuestions414 SubsectionREAD ReadingQuestions Suppose A isthe2 2matrix A = )]TJ/F15 10.9091 Tf 8.485 0 Td [(58 )]TJ/F15 10.9091 Tf 8.485 0 Td [(47 1.Findtheeigenvaluesof A 2.Findtheeigenspacesof A 3.Forthepolynomial p x =3 x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x +2,compute p A Version2.02

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SubsectionEE.EXCExercises415 SubsectionEXC Exercises C19 Findtheeigenvalues,eigenspaces,algebraicmultiplicitiesandgeometricmultiplicitiesforthematrix below.Itispossibletodoallthesecomputationsbyhand,anditwouldbeinstructivetodoso. C = )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F15 10.9091 Tf 8.485 0 Td [(66 ContributedbyRobertBeezerSolution[415] C20 Findtheeigenvalues,eigenspaces,algebraicmultiplicitiesandgeometricmultiplicitiesforthematrix below.Itispossibletodoallthesecomputationsbyhand,anditwouldbeinstructivetodoso. B = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1230 )]TJ/F15 10.9091 Tf 8.485 0 Td [(513 ContributedbyRobertBeezerSolution[415] C21 Thematrix A belowhas =2asaneigenvalue.Findthegeometricmultiplicityof =2using yourcalculatoronlyforrow-reducingmatrices. A = 2 6 6 4 18 )]TJ/F15 10.9091 Tf 8.485 0 Td [(1533 )]TJ/F15 10.9091 Tf 8.485 0 Td [(15 )]TJ/F15 10.9091 Tf 8.485 0 Td [(48 )]TJ/F15 10.9091 Tf 8.485 0 Td [(66 )]TJ/F15 10.9091 Tf 8.485 0 Td [(99 )]TJ/F15 10.9091 Tf 8.485 0 Td [(169 5 )]TJ/F15 10.9091 Tf 8.485 0 Td [(69 )]TJ/F15 10.9091 Tf 8.485 0 Td [(4 3 7 7 5 ContributedbyRobertBeezerSolution[416] C22 Withoutusingacalculator,ndtheeigenvaluesofthematrix B B = 2 )]TJ/F15 10.9091 Tf 8.484 0 Td [(1 11 ContributedbyRobertBeezerSolution[416] M60 RepeatExampleCAEHW[401]bychoosing x = 2 6 6 6 6 4 0 8 2 1 2 3 7 7 7 7 5 andthenarriveataneigenvalueandeigenvectorofthematrix A .Thehardway. ContributedbyRobertBeezerSolution[416] T10 Amatrix A isidempotentif A 2 = A .Showthattheonlypossibleeigenvaluesofanidempotent matrixare =0and =1.Thengiveanexampleofamatrixthatisidempotentandhasbothofthese twovaluesaseigenvalues. ContributedbyRobertBeezerSolution[417] T20 Supposethat and aretwodierenteigenvaluesofthesquarematrix A .Provethattheintersection oftheeigenspacesforthesetwoeigenvaluesistrivial.Thatis, E A E A = f 0 g ContributedbyRobertBeezerSolution[417] Version2.02

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SubsectionEE.SOLSolutions416 SubsectionSOL Solutions C19 ContributedbyRobertBeezerStatement[414] Firstcomputethecharacteristicpolynomial, p C x =det C )]TJ/F21 10.9091 Tf 10.909 0 Td [(xI 2 DenitionCP[403] = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 2 )]TJ/F15 10.9091 Tf 8.485 0 Td [(66 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x )]TJ/F21 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(6 = x 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [(5 x +6 = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 Sotheeigenvaluesof C arethesolutionsto p C x =0,namely, =2and =3. Toobtaintheeigenspaces,constructtheappropriatesingularmatricesandndexpressionsforthenull spacesofthesematrices. =2 C )]TJ/F15 10.9091 Tf 10.909 0 Td [( I 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(32 )]TJ/F15 10.9091 Tf 8.485 0 Td [(64 RREF )443()223()222()443(! 1 )]TJ/F19 7.9701 Tf 9.681 4.295 Td [(2 3 00 E C = N C )]TJ/F15 10.9091 Tf 10.909 0 Td [( I 2 = 2 3 1 = 2 3 =3 C )]TJ/F15 10.9091 Tf 10.909 0 Td [( I 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(42 )]TJ/F15 10.9091 Tf 8.485 0 Td [(63 RREF )443()223()222()443(! 1 )]TJ/F19 7.9701 Tf 9.681 4.295 Td [(1 2 00 E C = N C )]TJ/F15 10.9091 Tf 10.909 0 Td [( I 2 = 1 2 1 = 1 2 C20 ContributedbyRobertBeezerStatement[414] Thecharacteristicpolynomialof B is p B x =det B )]TJ/F21 10.9091 Tf 10.909 0 Td [(xI 2 DenitionCP[403] = )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x 30 )]TJ/F15 10.9091 Tf 8.485 0 Td [(513 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x = )]TJ/F15 10.9091 Tf 8.485 0 Td [(12 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x )]TJ/F21 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [( )]TJ/F15 10.9091 Tf 8.485 0 Td [(5TheoremDMST[376] = x 2 )]TJ/F21 10.9091 Tf 10.909 0 Td [(x )]TJ/F15 10.9091 Tf 10.909 0 Td [(6 = x )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 x +2 Fromthiswendeigenvalues =3 ; )]TJ/F15 10.9091 Tf 8.485 0 Td [(2withalgebraicmultiplicities B =1and B )]TJ/F15 10.9091 Tf 8.485 0 Td [(2=1. Foreigenvectorsandgeometricmultiplicities,westudythenullspacesof B )]TJ/F21 10.9091 Tf 11.427 0 Td [(I 2 TheoremEMNS [405]. =3 B )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 I 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1530 )]TJ/F15 10.9091 Tf 8.485 0 Td [(510 RREF )443()223()222()443(! 1 )]TJ/F15 10.9091 Tf 8.484 0 Td [(2 00 E B = N B )]TJ/F15 10.9091 Tf 10.909 0 Td [(3 I 2 = 2 1 Version2.02

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SubsectionEE.SOLSolutions417 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 B +2 I 2 = )]TJ/F15 10.9091 Tf 8.485 0 Td [(1030 )]TJ/F15 10.9091 Tf 8.485 0 Td [(515 RREF )443()223()222()443(! 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(3 00 E B )]TJ/F15 10.9091 Tf 8.485 0 Td [(2= N B +2 I 2 = 3 1 Eacheigenspacehasdimensionone,sowehavegeometricmultiplicities B =1and B )]TJ/F15 10.9091 Tf 8.485 0 Td [(2=1. C21 ContributedbyRobertBeezerStatement[414] If =2isaneigenvalueof A ,thematrix A )]TJ/F15 10.9091 Tf 10.256 0 Td [(2 I 4 willbesingular,anditsnullspacewillbetheeigenspace of A .