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PAGE 1 AFirstCourseinLinearAlgebra PAGE 2 AFirstCourseinLinearAlgebra by RobertA.Beezer DepartmentofMathematicsandComputerScience UniversityofPugetSound Version2.02 PAGE 3 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 . PAGE 4 Tomywife,Pat. PAGE 5 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 PAGE 6 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 PAGE 7 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 PAGE 8 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 PAGE 9 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 PAGE 10 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 PAGE 11 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 PAGE 12 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 PAGE 13 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 PAGE 14 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 PAGE 15 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 PAGE 16 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 PAGE 17 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 PAGE 18 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 PAGE 19 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 PAGE 20 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 PAGE 21 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 PAGE 22 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 PAGE 23 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 PAGE 24 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 PAGE 25 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 PAGE 26 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 PAGE 27 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 PAGE 28 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 PAGE 29 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 PAGE 30 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 PAGE 31 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 PAGE 32 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 PAGE 33 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 PAGE 34 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 PAGE 35 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 PAGE 36 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 PAGE 37 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 PAGE 38 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 PAGE 39 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 PAGE 40 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 PAGE 41 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 PAGE 42 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 PAGE 43 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 PAGE 44 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 PAGE 45 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 PAGE 46 PartC Core 1 PAGE 47 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 PAGE 48 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 PAGE 49 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 PAGE 50 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 PAGE 51 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 PAGE 52 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 PAGE 53 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 PAGE 54 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 PAGE 55 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 PAGE 56 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 PAGE 57 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 PAGE 58 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 PAGE 59 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 PAGE 60 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 PAGE 61 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 PAGE 62 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 PAGE 63 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 PAGE 64 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 PAGE 65 SubsectionSSLE.EXCExercises20 answersbyusingyourdenition. ContributedbyRobertBeezerSolution[23] T20 ExplainwhythesecondequationoperationinDenitionEO[11]requiresthatthescalarbenonzero, whileinthethirdequationoperationthisrestrictiononthescalarisnotpresent. ContributedbyRobertBeezerSolution[23] Version2.02 PAGE 66 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 PAGE 67 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 PAGE 68 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 PAGE 69 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 PAGE 70 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 PAGE 71 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 PAGE 72 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 PAGE 73 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 PAGE 74 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 PAGE 75 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 PAGE 76 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 PAGE 77 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 PAGE 78 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 PAGE 79 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 PAGE 80 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 PAGE 81 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 PAGE 82 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 PAGE 83 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 PAGE 84 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 PAGE 85 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 PAGE 86 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 PAGE 87 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 PAGE 88 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 PAGE 89 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 PAGE 90 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 PAGE 91 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 PAGE 92 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 PAGE 93 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 PAGE 94 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 PAGE 95 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 PAGE 96 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 PAGE 97 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 PAGE 98 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 PAGE 99 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 PAGE 100 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 PAGE 101 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 PAGE 102 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 PAGE 103 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 PAGE 104 SubsectionTSS.EXCExercises59 constants b isacommonmultipleofthecorrespondingelementofacertaincolumnof A .Moreprecisely, thereisacomplexnumber ,andacolumnindex j ,suchthat[ b ] i = [ A ] ij forall i .Provethatthesystem isconsistent. ContributedbyRobertBeezerSolution[60] Version2.02 PAGE 105 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 PAGE 106 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 PAGE 107 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 PAGE 108 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 PAGE 109 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 PAGE 110 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 PAGE 111 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 PAGE 112 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 PAGE 113 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 PAGE 114 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 PAGE 115 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 PAGE 116 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 PAGE 117 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 PAGE 118 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 PAGE 119 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 PAGE 120 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 PAGE 121 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 PAGE 122 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 PAGE 123 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 PAGE 124 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 PAGE 125 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 PAGE 126 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 PAGE 127 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 PAGE 128 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 PAGE 129 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 PAGE 130 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 PAGE 131 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 PAGE 132 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 PAGE 133 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 PAGE 134 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 PAGE 135 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 PAGE 136 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 PAGE 137 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 PAGE 138 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 PAGE 139 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 PAGE 140 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 PAGE 141 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 PAGE 142 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 PAGE 143 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 PAGE 144 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 PAGE 145 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 PAGE 146 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 PAGE 147 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 PAGE 148 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 PAGE 149 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 PAGE 150 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 PAGE 151 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 PAGE 152 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 PAGE 153 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 PAGE 154 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 PAGE 155 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 PAGE 156 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 PAGE 157 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 PAGE 158 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 PAGE 159 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 PAGE 160 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 PAGE 161 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 PAGE 162 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 PAGE 163 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 PAGE 164 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 PAGE 165 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 PAGE 166 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 PAGE 167 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 PAGE 168 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 PAGE 169 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 PAGE 170 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 PAGE 171 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 PAGE 172 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 PAGE 173 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 PAGE 174 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 PAGE 175 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 PAGE 176 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 PAGE 177 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 PAGE 178 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 PAGE 179 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 PAGE 180 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 PAGE 181 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 PAGE 182 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 PAGE 183 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 PAGE 184 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 PAGE 185 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 PAGE 186 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 PAGE 187 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 PAGE 188 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 PAGE 189 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 PAGE 190 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 PAGE 191 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 PAGE 192 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 PAGE 193 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 PAGE 194 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 PAGE 195 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 PAGE 196 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 PAGE 197 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 PAGE 198 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 PAGE 199 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 PAGE 200 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 PAGE 201 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 PAGE 202 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 PAGE 203 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 PAGE 204 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 PAGE 205 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 PAGE 206 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 PAGE 207 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 PAGE 208 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 PAGE 209 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 PAGE 210 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 PAGE 211 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 PAGE 212 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 PAGE 213 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 PAGE 214 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 PAGE 215 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 PAGE 216 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 PAGE 217 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 PAGE 218 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 PAGE 219 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 PAGE 220 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 PAGE 221 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 PAGE 222 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 PAGE 223 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 PAGE 224 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 PAGE 225 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 PAGE 226 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 PAGE 227 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 PAGE 228 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 PAGE 229 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 PAGE 230 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 PAGE 231 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 PAGE 232 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 PAGE 233 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 PAGE 234 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 PAGE 235 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 PAGE 236 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 PAGE 237 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 PAGE 238 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 PAGE 239 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 PAGE 240 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 PAGE 241 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 PAGE 242 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 PAGE 243 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 PAGE 244 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 PAGE 245 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 PAGE 246 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 PAGE 247 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 PAGE 248 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 PAGE 249 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 PAGE 250 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 PAGE 251 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 PAGE 252 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 PAGE 253 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 PAGE 254 SubsectionMM.EXCExercises209 ofthethreesolutionsasthehypothesisandaninterestingstatementabout LS A; b astheconclusion, andthengiveaproof. ContributedbyRobertBeezerSolution[210] Version2.02 PAGE 255 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 PAGE 256 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 PAGE 257 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 PAGE 258 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 PAGE 259 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 PAGE 260 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 PAGE 261 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 PAGE 262 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 PAGE 263 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 PAGE 264 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 PAGE 265 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 PAGE 266 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 PAGE 267 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 PAGE 268 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 PAGE 269 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 PAGE 270 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 PAGE 271 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 PAGE 272 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 PAGE 273 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 PAGE 274 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 PAGE 275 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 PAGE 276 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 PAGE 277 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 PAGE 278 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 PAGE 279 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 PAGE 280 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 PAGE 281 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 PAGE 282 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 PAGE 283 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 PAGE 284 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 PAGE 285 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 PAGE 286 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 PAGE 287 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 PAGE 288 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 PAGE 289 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 PAGE 290 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 PAGE 291 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 PAGE 292 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 PAGE 293 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 PAGE 294 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 PAGE 295 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 PAGE 296 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 PAGE 297 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 PAGE 298 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 PAGE 299 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 PAGE 300 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 PAGE 301 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 PAGE 302 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 PAGE 303 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 PAGE 304 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 PAGE 305 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 PAGE 306 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 PAGE 307 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 PAGE 308 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 PAGE 309 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 PAGE 310 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 PAGE 311 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 PAGE 312 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 PAGE 313 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 PAGE 314 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 PAGE 315 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 PAGE 316 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 PAGE 317 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 PAGE 318 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 PAGE 319 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 PAGE 320 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 PAGE 321 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 PAGE 322 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 PAGE 323 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 PAGE 324 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 PAGE 325 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 PAGE 326 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 PAGE 327 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 PAGE 328 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 PAGE 329 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 PAGE 330 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 PAGE 331 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 PAGE 332 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 PAGE 333 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 PAGE 334 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 PAGE 335 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 PAGE 336 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 PAGE 337 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 PAGE 338 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 PAGE 339 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 PAGE 340 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 PAGE 341 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 PAGE 342 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 PAGE 343 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 PAGE 344 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 PAGE 345 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 PAGE 346 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 PAGE 347 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 PAGE 348 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 PAGE 349 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 PAGE 350 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 PAGE 351 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 PAGE 352 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 PAGE 353 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 PAGE 354 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 PAGE 355 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 PAGE 356 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 PAGE 357 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 PAGE 358 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 PAGE 359 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 PAGE 360 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 PAGE 361 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 PAGE 362 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 PAGE 363 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 PAGE 364 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 PAGE 365 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 PAGE 366 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 PAGE 367 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 PAGE 368 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 PAGE 369 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 PAGE 370 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 PAGE 371 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 PAGE 372 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 PAGE 373 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 PAGE 374 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 PAGE 375 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 PAGE 376 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 PAGE 377 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 PAGE 378 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 PAGE 379 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 PAGE 380 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 PAGE 381 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 PAGE 382 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 PAGE 383 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 PAGE 384 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 PAGE 385 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 PAGE 386 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 PAGE 387 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 PAGE 388 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 PAGE 389 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 PAGE 390 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 PAGE 391 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 PAGE 392 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 PAGE 393 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 PAGE 394 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 PAGE 395 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 PAGE 396 SubsectionD.READReadingQuestions351 SubsectionREAD ReadingQuestions 1.Whatisthedimensionofthevectorspace P 6 ,thesetofallpolynomialsofdegree6orless? 2.Howaretherankandnullityofamatrixrelated? 3.Explainwhywemightsaythatanonsingularmatrixhasfullrank." Version2.02 PAGE 397 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 PAGE 398 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 PAGE 399 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 PAGE 400 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 PAGE 401 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 PAGE 402 SubsectionPD.GTGoldilocks'Theorem357 1.If m>t ,then S islinearlydependent. 2.If m PAGE 403 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 PAGE 404 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 PAGE 405 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 PAGE 406 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 PAGE 407 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 PAGE 408 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 PAGE 409 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 PAGE 410 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 PAGE 411 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 PAGE 412 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 PAGE 413 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 PAGE 414 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 PAGE 415 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 PAGE 416 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 PAGE 417 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 PAGE 418 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 PAGE 419 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 PAGE 420 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 PAGE 421 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 PAGE 422 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 PAGE 423 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 ` PAGE 424 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 PAGE 425 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 PAGE 426 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 PAGE 427 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 PAGE 428 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 PAGE 429 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 PAGE 430 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 PAGE 431 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 PAGE 432 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 PAGE 433 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 PAGE 434 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 PAGE 435 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 PAGE 436 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 PAGE 437 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 PAGE 438 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 PAGE 439 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 PAGE 440 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 PAGE 441 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 PAGE 442 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 PAGE 443 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 PAGE 444 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 PAGE 445 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 PAGE 446 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 PAGE 447 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 PAGE 448 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 PAGE 449 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 PAGE 450 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 PAGE 451 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 PAGE 452 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 PAGE 453 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 PAGE 454 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 PAGE 455 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 PAGE 456 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 PAGE 457 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 PAGE 458 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 PAGE 459 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 PAGE 460 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 PAGE 461 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 = | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||