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Linear Algebra

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Linear Algebra
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Hefferon, Jim
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linear systems, Gauss' method, vector spaces, linear maps and matrices, determinants, eigenvectors, eigenvalues, Linear Geometry of n-Space, Reduced Echelon Form, Linear Independence, basis and dimension, Maps Between Spaces, Determinants, Similarity, 9781616100537
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The text covers the material of a first undergraduate Linear Algebra course. You can use it either as a main text, as a supplement to another text, or for independent study. Prerequisites: A semester of calculus; students with three semesters of calculus can skip a few sections. Each chapter has three or four discussions of additional topics and applications. These are suitable for independent study or for small group work. The approach is developmental. Although the presentation is focused on covering the requisite material by proving things, it does not start with an assumption that students are already able at abstract work. Instead, it proceeds with a great deal of motivation, many computational examples, and exercises that range from routine verifications to (a few) challenges. The goal is, in the context of developing the usual material of an undergraduate linear algebra course, to help raise the level of mathematical maturity of the class. Note: both the textbook and answers to exercises are included. Author suggests you save the two files in the same directory, so that clicking on an exercise will send you to its answer and clicking on an answer will send you to its exercise. Go to: http://joshua.smcvt.edu/linalg.html/ to get the source. You need to know LaTeX and MetaPost to work with it; there is a readme file to get started and some optional material. For instructors considering adoption: the author suggests looking at the second chapter. The first chapter is necessarily computational but the second chapter shows more clearly what the book works on: bridging between lower-division mathematics with its reliance on explicitly-given algorithms, and upper division college mathematics with its emphasis on concepts and proof.
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Higher Education
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http://www.ogtp-cart.com/product.aspx?ISBN=9781616100537
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Jim Hefferon
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Diagram, Figure, Graph, Narrative text, Problem statement, Textbook
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jim@joshua.smcvt.edu
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MAS 103 - INTRODUCTORY LINEAR ALGEBRA I, MAS 105 - LINEAR ALGEBRA I (CALC. II PREREQU)
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http://florida.theorangegrove.org/og/file/b7338794-63a4-e13d-1b79-1297f6fea895/1/LinearAlgebraText.pdf

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LinearAlgebra JimHefferon )]TJ/F6 4.9813 Tf 4.566 -3.785 Td [(2 1 )]TJ/F6 4.9813 Tf 4.567 -3.786 Td [(1 3 12 31 )]TJ/F6 4.9813 Tf 4.566 -3.786 Td [(2 1 x 1 )]TJ/F6 4.9813 Tf 4.567 -3.785 Td [(1 3 x 1 12 x 1 31 )]TJ/F6 4.9813 Tf 4.567 -3.786 Td [(2 1 )]TJ/F6 4.9813 Tf 4.566 -3.786 Td [(6 8 62 81

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Notation R R + R n realnumbers,realsgreaterthan0, n -tuplesofreals N naturalnumbers: f 0 ; 1 ; 2 ;::: g C complexnumbers f ::: ::: g setof...suchthat... a::b ,[ a::b ] intervalopenorclosedofrealsbetween a and b h ::: i sequence;likeasetbutordermatters V;W;U vectorspaces ~v;~w vectors ~ 0, ~ 0 V zerovector,zerovectorof V B;D bases E n = h ~e 1 ;:::;~e n i standardbasisfor R n ~ ; ~ basisvectors Rep B ~v matrixrepresentingthevector P n setof n -thdegreepolynomials M n m setof n m matrices [ S ] spanoftheset S M N directsumofsubspaces V = W isomorphicspaces h;g homomorphisms,linearmaps H;G matrices t;s transformations;mapsfromaspacetoitself T;S squarematrices Rep B;D h matrixrepresentingthemap h h i;j matrixentryfromrow i ,column j j T j determinantofthematrix T R h ; N h rangespaceandnullspaceofthemap h R 1 h ; N 1 h generalizedrangespaceandnullspace LowercaseGreekalphabet namecharacter namecharacter namecharacter alpha iota rho beta kappa sigma gamma lambda tau delta mu upsilon epsilon nu phi zeta xi chi eta omicron o psi theta pi omega Cover. ThisisCramer'sRuleforthesystem x 1 +2 x 2 =6,3 x 1 + x 2 =8.Thesizeof therstboxisthedeterminantshowntheabsolutevalueofthesizeisthearea.The sizeofthesecondboxis x 1 timesthat,andequalsthesizeofthenalbox.Hence, x 1 isthenaldeterminantdividedbytherstdeterminant.

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Preface ThisbookhelpsstudentstomasterthematerialofastandardUSundergraduate linearalgebracourse. ThematerialisstandardinthatthetopicscoveredareGaussianreduction, vectorspaces,linearmaps,determinants,andeigenvaluesandeigenvectors.Anotherstandardaboutthebookisitsaudience:sophomoresorjuniors,usually withabackgroundofatleastonesemesterofCalculus. Thehelpthatitgivestostudentscomesfromtakingadevelopmentalapproach|thisbook'spresentationemphasizesmotivationandnaturalness,driven homebyawidevarietyofexamplesandbyextensiveandcarefulexercises.The developmentalapproachiswhatsetsthisbookapart,soIwillexpandonit below. CoursesinthebeginningofmostMathematicsprogramsfocuslessonunderstandingthetheoryandmoreoncorrectlyapplyingformulasandalgorithms. Latercoursesaskformathematicalmaturity:theabilitytofollowdierenttypes ofarguments,afamiliaritywiththethemesthatunderliemanymathematical investigationssuchaselementarysetandfunctionfacts,andacapacityforsome independentreadingandthinking.Linearalgebraisanidealspottoworkon thetransition.Itcomesearlyinaprogramsothatprogressmadeherepayso later,butalsocomeslateenoughthatstudentsareserious,oftenmajorsand minors.Thematerialiscoherent,accessible,andelegant.Thereareavarietyof argumentstyles|proofsbycontradiction,ifandonlyifstatements,andproofs byinduction,forinstance|andexamplesareplentiful. So,thisbookaimstohelpstudentsdevelopfrombeingsuccessfulattheir presentlevel,inclasseswhereamajorityofstudentsareinterestedmainlyin scienceorengineering,tobeingsuccessfulatthenextlevel,thatofserious studentsofthesubjectofmathematicsitself. Helpingstudentsmakethistransitionmeanstakingthemathematicsseriously,soalloftheresultsinthisbookareproved.Ontheotherhand,we cannotassumethatstudentshavealreadyarrived,andsoincontrastwithmore abstracttexts,wegivemanyexamplesandtheyareoftenquitedetailed. Inthepast,linearalgebratextscommonlymadethistransitionabruptly. Theybeganwithextensivecomputationsoflinearsystems,matrixmultiplications,anddeterminants.Whentheconcepts|vectorspacesandlinearmaps| nallyappeared,anddenitionsandproofsstarted,oftenthechangebrought studentstoastop.Inthisbook,whilewestartwithacomputationaltopic, linearreduction,fromtherstwedomorethancompute.Wedolinearsystems iii

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quicklybutcompletely,includingtheproofsneededtojustifywhatwearecomputing.Then,withthelinearsystemsworkasmotivationandatapointwhere thestudyoflinearcombinationsseemsnatural,thesecondchapterstartswith thedenitionofarealvectorspace.Thisoccursbytheendofthethirdweek. Anotherexampleofouremphasisonmotivationandnaturalnessisthatthe thirdchapteronlinearmapsdoesnotbeginwiththedenitionofhomomorphism,butwiththatofisomorphism.That'sbecausethisdenitioniseasily motivatedbytheobservationthatsomespacesarejustlike"others.After that,thenextsectiontakesthereasonablestepofdeninghomomorphismby isolatingtheoperation-preservationidea.Thisapproachlosesmathematical slickness,butitisagoodtradebecauseitcomesinreturnforalargegainin sensibilitytostudents. Oneaimofourdevelopmentalapproachisthatstudentsshouldfeelthroughoutthepresentationthattheycanseehowtheideasarise,andperhapspicture themselvesdoingthesametypeofwork. Theclearestexampleofthedevelopmentalapproachtakenhere|andthe featurethatmostrecommendsthisbook|istheexercises.Astudentprogresses mostwhiledoingtheexercises,sotheyhavebeenselectedwithgreatcare.Each problemsetrangesfromsimplecheckstoreasonablyinvolvedproofs.Sincean instructorusuallyassignsaboutadozenexercisesaftereachlecture,eachsection endswithabouttwicethatmany,therebyprovidingaselection.Thereare evenafewproblemsthatarechallengingpuzzlestakenfromvariousjournals, competitions,orproblemscollections.Thesearemarkedwitha` ? 'andas partofthefun,theoriginalwordinghasbeenretainedasmuchaspossible. Intotal,theexercisesareaimedtobothbuildanabilityat,andhelpstudents experiencethepleasureof, doing mathematics. Applications,andComputers. Thepointofviewtakenhere,thatlinear algebraisaboutvectorspacesandlinearmaps,isnottakentothecompleteexclusionofothers.Applicationsandtheroleofthecomputerareimportantand vitalaspectsofthesubject.Consequently,eachofthisbook'schapterscloses withafewapplicationorcomputer-relatedtopics.Someare:networkows,the speedandaccuracyofcomputerlinearreductions,LeontiefInput/Outputanalysis,dimensionalanalysis,Markovchains,votingparadoxes,analyticprojective geometry,anddierenceequations. Thesetopicsarebriefenoughtobedoneinaday'sclassortobegivenas independentprojectsforindividualsorsmallgroups.Mostsimplygiveareader atasteofthesubject,discusshowlinearalgebracomesin,pointtosomefurther reading,andgiveafewexercises.Inshort,thesetopicsinvitereaderstoseefor themselvesthatlinearalgebraisatoolthataprofessionalmusthave. TheLicense. Thisbookisfreelyavailable.Youcandownloadandreadit withoutrestriction.Ifyouareaclassinstructorthenyoucanmakepapercopies availabletoyourstudents.Pleaseseethewebpage http://joshua.smcvt.edu/linearalgebra formoreinformation. Inparticular,IprovidetheL A T E Xsourceofthetextandsomeinstructors maywishtoaddtheirownmaterial.Ifyoulike,youcansendsuchadditionsto iv

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me,andImaypossiblyincorporatethemintofutureeditions. Iamverygladforbugreports.Isavethemandtrytoperiodicallyissue updates;peoplewhocontributeinthiswayareacknowledgedinthetext'ssource les. Forpeoplereadingthisbookontheirown. Thisbook'semphasison motivationanddevelopmentmakeitagoodchoiceforself-study.But,whilea professionalinstructorcanjudgewhatpaceandtopicssuitaclass,ifyouare anindependentstudentthenperhapsyouwouldndsomeadvicehelpful. Herearetwotimetablesforasemester.Therstfocusesoncorematerial. week MondayWednesdayFriday 1 One.I.1One.I.1,2One.I.2,3 2 One.I.3One.II.1One.II.2 3 One.III.1,2One.III.2Two.I.1 4 Two.I.2Two.IITwo.III.1 5 Two.III.1,2Two.III.2 exam 6 Two.III.2,3Two.III.3Three.I.1 7 Three.I.2Three.II.1Three.II.2 8 Three.II.2Three.II.2Three.III.1 9 Three.III.1Three.III.2Three.IV.1,2 10 Three.IV.2,3,4Three.IV.4 exam 11 Three.IV.4,Three.V.1Three.V.1,2Four.I.1,2 12 Four.I.3Four.IIFour.II 13 Four.III.1Five.IFive.II.1 14 Five.II.2Five.II.3 review ThesecondtimetableismoreambitiousitsupposesthatyouknowOne.II,the elementsofvectors,usuallycoveredinthirdsemestercalculus. week MondayWednesdayFriday 1 One.I.1One.I.2One.I.3 2 One.I.3One.III.1,2One.III.2 3 Two.I.1Two.I.2Two.II 4 Two.III.1Two.III.2Two.III.3 5 Two.III.4Three.I.1 exam 6 Three.I.2Three.II.1Three.II.2 7 Three.III.1Three.III.2Three.IV.1,2 8 Three.IV.2Three.IV.3Three.IV.4 9 Three.V.1Three.V.2Three.VI.1 10 Three.VI.2Four.I.1 exam 11 Four.I.2Four.I.3Four.I.4 12 Four.IIFour.II,Four.III.1Four.III.2,3 13 Five.II.1,2Five.II.3Five.III.1 14 Five.III.2Five.IV.1,2Five.IV.2 Seethetableofcontentsforthetitlesofthesesubsections. Tohelpyoumaketimetrade-os,inthetableofcontentsIhavemarkedsubsectionsasoptionalifsomeinstructorswillpassovertheminfavorofspending v

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moretimeelsewhere.Youmightalsotrypickingoneortwotopicsthatappeal toyoufromtheendofeachchapter.You'llgetmorefromtheseifyouhave accesstocomputersoftwarethatcandothebigcalculations. Themostimportantadviceis:domanyexercises.Ihavemarkedagood samplewith X 's.Theanswersareavailable.Youshouldbeaware,however, thatfewinexperiencedpeoplecanwritecorrectproofs.Trytondaknowledgeablepersontoworkwithyouonthis. Finally,ifImay,acautionforallstudents,independentornot:Icannot overemphasizehowmuchthestatementthatIsometimeshear,Iunderstand thematerial,butit'sonlythatIhavetroublewiththeproblems"revealsalack ofunderstandingofwhatweareupto.Beingabletodothingswiththeideas istheirpoint.Thequotesbelowexpressthissentimentadmirably.Theystate whatIbelieveisthekeytoboththebeautyandthepowerofmathematicsand thesciencesingeneral,andoflinearalgebrainparticularItookthelibertyof formattingthemasverse. Iknowofnobettertactic thantheillustrationofexcitingprinciples bywell-chosenparticulars. {StephenJayGould Ifyoureallywishtolearn thenyoumustmountthemachine andbecomeacquaintedwithitstricks byactualtrial. {WilburWright JimHefferon Mathematics,SaintMichael'sCollege Colchester,VermontUSA05439 http://joshua.smcvt.edu 2008-Aug-13 Author'sNote. Inventingagoodexercise,onethatenlightensaswellastests, isacreativeact,andhardwork.Theinventordeservesrecognition.Butfor somereasontextshavetraditionallynotgivenattributionsforquestions.Ihave changedthatherewhereIwassureofthesource.Iwouldgreatlyappreciatehearingfromanyonewhocanhelpmetocorrectlyattributeothersofthe questions. vi

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Contents ChapterOne:LinearSystems1 ISolvingLinearSystems........................1 1Gauss'Method...........................2 2DescribingtheSolutionSet....................11 3General=Particular+Homogeneous..............20 IILinearGeometryof n -Space.....................32 1VectorsinSpace..........................32 2LengthandAngleMeasures ...................38 IIIReducedEchelonForm........................46 1Gauss-JordanReduction......................46 2RowEquivalence..........................52 Topic:ComputerAlgebraSystems...................62 Topic:Input-OutputAnalysis......................64 Topic:AccuracyofComputations....................68 Topic:AnalyzingNetworks........................72 ChapterTwo:VectorSpaces79 IDenitionofVectorSpace......................80 1DenitionandExamples......................80 2SubspacesandSpanningSets...................91 IILinearIndependence.........................101 1DenitionandExamples......................101 IIIBasisandDimension.........................112 1Basis.................................112 2Dimension..............................118 3VectorSpacesandLinearSystems................123 4CombiningSubspaces .......................130 Topic:Fields................................140 Topic:Crystals..............................142 Topic:VotingParadoxes.........................146 Topic:DimensionalAnalysis.......................152 vii

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ChapterThree:MapsBetweenSpaces159 IIsomorphisms.............................159 1DefinitionandExamples......................159 2DimensionCharacterizesIsomorphism..............168 IIHomomorphisms...........................176 1Denition..............................176 2RangespaceandNullspace.....................183 IIIComputingLinearMaps.......................195 1RepresentingLinearMapswithMatrices.............195 2AnyMatrixRepresentsaLinearMap ..............205 IVMatrixOperations..........................212 1SumsandScalarProducts.....................212 2MatrixMultiplication.......................214 3MechanicsofMatrixMultiplication................222 4Inverses...............................231 VChangeofBasis............................238 1ChangingRepresentationsofVectors...............238 2ChangingMapRepresentations..................242 VIProjection...............................250 1OrthogonalProjectionIntoaLine ................250 2Gram-SchmidtOrthogonalization ................254 3ProjectionIntoaSubspace ....................260 Topic:LineofBestFit..........................269 Topic:GeometryofLinearMaps....................274 Topic:MarkovChains..........................281 Topic:OrthonormalMatrices......................287 ChapterFour:Determinants293 IDefinition...............................294 1Exploration ............................294 2PropertiesofDeterminants....................299 3ThePermutationExpansion....................303 4DeterminantsExist ........................312 IIGeometryofDeterminants......................319 1DeterminantsasSizeFunctions..................319 IIIOtherFormulas............................326 1Laplace'sExpansion ........................326 Topic:Cramer'sRule...........................331 Topic:SpeedofCalculatingDeterminants...............334 Topic:ProjectiveGeometry.......................337 ChapterFive:Similarity349 IComplexVectorSpaces........................349 1FactoringandComplexNumbers;AReview ..........350 2ComplexRepresentations.....................351 IISimilarity...............................353 viii

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1DenitionandExamples......................353 2Diagonalizability..........................355 3EigenvaluesandEigenvectors...................359 IIINilpotence...............................367 1Self-Composition .........................367 2Strings ...............................370 IVJordanForm..............................381 1PolynomialsofMapsandMatrices ................381 2JordanCanonicalForm ......................388 Topic:MethodofPowers.........................401 Topic:StablePopulations........................405 Topic:LinearRecurrences........................407 AppendixA-1 Propositions...............................A-1 Quantiers...............................A-3 TechniquesofProof..........................A-5 Sets,Functions,andRelations.....................A-7 Note: starredsubsectionsareoptional. ix

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ChapterOne LinearSystems ISolvingLinearSystems Systemsoflinearequationsarecommoninscienceandmathematics.Thesetwo examplesfromhighschoolscience[Onan]giveasenseofhowtheyarise. TherstexampleisfromPhysics.Supposethatwearegiventhreeobjects, onewithamassknowntobe2kg,andareaskedtondtheunknownmasses. Supposefurtherthatexperimentationwithameterstickproducesthesetwo balances. c h 2 15 40 50 c h 2 25 50 25 Sincethesumofmomentsontheleftofeachbalanceequalsthesumofmoments ontherightthemomentofanobjectisitsmasstimesitsdistancefromthe balancepoint,thetwobalancesgivethissystemoftwoequations. 40 h +15 c =100 25 c =50+50 h ThesecondexampleofalinearsystemisfromChemistry.Wecanmix, undercontrolledconditions,tolueneC 7 H 8 andnitricacidHNO 3 toproduce trinitrotolueneC 7 H 5 O 6 N 3 alongwiththebyproductwaterconditionshaveto becontrolledverywell,indeed|trinitrotolueneisbetterknownasTNT.In whatproportionshouldthosecomponentsbemixed?Thenumberofatomsof eachelementpresentbeforethereaction x C 7 H 8 + y HNO 3 )167(! z C 7 H 5 O 6 N 3 + w H 2 O mustequalthenumberpresentafterward.Applyingthatprincipletotheele1

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2 ChapterOne.LinearSystems mentsC,H,N,andOinturngivesthissystem. 7 x =7 z 8 x +1 y =5 z +2 w 1 y =3 z 3 y =6 z +1 w Tonisheachoftheseexamplesrequiressolvingasystemofequations.In each,theequationsinvolveonlytherstpowerofthevariables.Thischapter showshowtosolveanysuchsystem. I.1Gauss'Method 1.1Denition A linearequation invariables x 1 ;x 2 ;:::;x n hastheform a 1 x 1 + a 2 x 2 + a 3 x 3 + + a n x n = d wherethenumbers a 1 ;:::;a n 2 R aretheequation's coecients and d 2 R isthe constant .An n -tuple s 1 ;s 2 ;:::;s n 2 R n isa solution of,or satises thatequationifsubstitutingthenumbers s 1 ,..., s n forthevariablesgivesa truestatement: a 1 s 1 + a 2 s 2 + ::: + a n s n = d A systemoflinearequations a 1 ; 1 x 1 + a 1 ; 2 x 2 + + a 1 ;n x n = d 1 a 2 ; 1 x 1 + a 2 ; 2 x 2 + + a 2 ;n x n = d 2 a m; 1 x 1 + a m; 2 x 2 + + a m;n x n = d m hasthesolution s 1 ;s 2 ;:::;s n ifthat n -tupleisasolutionofalloftheequationsinthesystem. 1.2Example Theorderedpair )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; 5isasolutionofthissystem. 3 x 1 +2 x 2 =7 )]TJ/F11 9.9626 Tf 7.749 0 Td [(x 1 + x 2 =6 Incontrast, ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(1isnotasolution. Findingthesetofallsolutionsis solving thesystem.Noguessworkorgood fortuneisneededtosolvealinearsystem.Thereisanalgorithmthatalways works.Thenextexampleintroducesthatalgorithm,called Gauss'method .It transformsthesystem,stepbystep,intoonewithaformthatiseasilysolved.

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SectionI.SolvingLinearSystems 3 1.3Example Tosolvethissystem 3 x 3 =9 x 1 +5 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 x 3 =2 1 3 x 1 +2 x 2 =3 werepeatedlytransformituntilitisinaformthatiseasytosolve. swaprow1withrow3 )167(! 1 3 x 1 +2 x 2 =3 x 1 +5 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x 3 =2 3 x 3 =9 multiplyrow1by3 )167(! x 1 +6 x 2 =9 x 1 +5 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x 3 =2 3 x 3 =9 add )]TJ/F7 6.9738 Tf 6.227 0 Td [(1timesrow1torow2 )167(! x 1 +6 x 2 =9 )]TJ/F11 9.9626 Tf 7.749 0 Td [(x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 x 3 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 3 x 3 =9 Thethirdstepistheonlynontrivialone.We'vementallymultipliedbothsides oftherstrowby )]TJ/F8 9.9626 Tf 7.748 0 Td [(1,mentallyaddedthattotheoldsecondrow,andwritten theresultinasthenewsecondrow. Nowwecanndthevalueofeachvariable.Thebottomequationshows that x 3 =3.Substituting3for x 3 inthemiddleequationshowsthat x 2 =1. Substitutingthosetwointothetopequationgivesthat x 1 =3andsothesystem hasauniquesolution:thesolutionsetis f ; 1 ; 3 g Mostofthissubsectionandthenextoneconsistsofexamplesofsolving linearsystemsbyGauss'method.Wewilluseitthroughoutthisbook.Itis fastandeasy.But,beforewegettothoseexamples,wewillrstshowthat thismethodisalsosafeinthatitneverlosessolutionsorpicksupextraneous solutions. 1.4TheoremGauss'method Ifalinearsystemischangedtoanother byoneoftheseoperations anequationisswappedwithanother anequationhasbothsidesmultipliedbyanonzeroconstant anequationisreplacedbythesumofitselfandamultipleofanother thenthetwosystemshavethesamesetofsolutions. Eachofthosethreeoperationshasarestriction.Multiplyingarowby0is notallowedbecauseobviouslythatcanchangethesolutionsetofthesystem. Similarly,addingamultipleofarowtoitselfisnotallowedbecauseadding )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 timestherowtoitselfhastheeectofmultiplyingtherowby0.Finally,swappingarowwithitselfisdisallowedtomakesomeresultsinthefourthchapter easiertostateandrememberandbesides,self-swappingdoesn'taccomplish anything.

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4 ChapterOne.LinearSystems Proof Wewillcovertheequationswapoperationhereandsavetheothertwo casesforExercise29. Considerthisswapofrow i withrow j a 1 ; 1 x 1 + a 1 ; 2 x 2 + a 1 ;n x n = d 1 a i; 1 x 1 + a i; 2 x 2 + a i;n x n = d i a j; 1 x 1 + a j; 2 x 2 + a j;n x n = d j a m; 1 x 1 + a m; 2 x 2 + a m;n x n = d m )167(! a 1 ; 1 x 1 + a 1 ; 2 x 2 + a 1 ;n x n = d 1 a j; 1 x 1 + a j; 2 x 2 + a j;n x n = d j a i; 1 x 1 + a i; 2 x 2 + a i;n x n = d i a m; 1 x 1 + a m; 2 x 2 + a m;n x n = d m The n -tuple s 1 ;:::;s n satisesthesystembeforetheswapifandonlyif substitutingthevalues,the s 's,forthevariables,the x 's,givestruestatements: a 1 ; 1 s 1 + a 1 ; 2 s 2 + + a 1 ;n s n = d 1 and... a i; 1 s 1 + a i; 2 s 2 + + a i;n s n = d i and... a j; 1 s 1 + a j; 2 s 2 + + a j;n s n = d j and... a m; 1 s 1 + a m; 2 s 2 + + a m;n s n = d m Inarequirementconsistingofstatementsand-edtogetherwecanrearrange theorderofthestatements,sothatthisrequirementismetifandonlyif a 1 ; 1 s 1 + a 1 ; 2 s 2 + + a 1 ;n s n = d 1 and... a j; 1 s 1 + a j; 2 s 2 + + a j;n s n = d j and... a i; 1 s 1 + a i; 2 s 2 + + a i;n s n = d i and... a m; 1 s 1 + a m; 2 s 2 + + a m;n s n = d m Thisisexactlytherequirementthat s 1 ;:::;s n solvesthesystemaftertherow swap. QED 1.5Denition ThethreeoperationsfromTheorem1.4arethe elementary reductionoperations ,or rowoperations ,or Gaussianoperations .Theyare swapping multiplyingbyascalar or rescaling ,and pivoting Whenwritingoutthecalculations,wewillabbreviate`row i 'by` i '.For instance,wewilldenoteapivotoperationby k i + j ,withtherowthatis changedwrittensecond.Wewillalso,tosavewriting,oftenlistpivotsteps togetherwhentheyusethesame i 1.6Example AtypicaluseofGauss'methodistosolvethissystem. x + y =0 2 x )]TJ/F11 9.9626 Tf 14.943 0 Td [(y +3 z =3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 y )]TJ/F11 9.9626 Tf 14.944 0 Td [(z =3 Thersttransformationofthesysteminvolvesusingtherstrowtoeliminate the x inthesecondrowandthe x inthethird.Togetridofthesecondrow's 2 x ,wemultiplytheentirerstrowby )]TJ/F8 9.9626 Tf 7.749 0 Td [(2,addthattothesecondrow,and writetheresultinasthenewsecondrow.Togetridofthethirdrow's x ,we multiplytherstrowby )]TJ/F8 9.9626 Tf 7.749 0 Td [(1,addthattothethirdrow,andwritetheresultin asthenewthirdrow. )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 1 + 2 )167(! )]TJ/F10 6.9738 Tf 6.227 0 Td [( 1 + 3 x + y =0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 y +3 z =3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 y )]TJ/F11 9.9626 Tf 14.944 0 Td [(z =3

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SectionI.SolvingLinearSystems 5 Notethatthetwo 1 steps )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 1 + 2 and )]TJ/F11 9.9626 Tf 7.748 0 Td [( 1 + 3 arewrittenasoneoperation.Inthissecondsystem,thelasttwoequationsinvolveonlytwounknowns. Tonishwetransformthesecondsystemintoathirdsystem,wherethelast equationinvolvesonlyoneunknown.Thistransformationusesthesecondrow toeliminate y fromthethirdrow. )]TJ/F10 6.9738 Tf 6.227 0 Td [( 2 + 3 )167(! x + y =0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 y +3 z =3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 z =0 Nowwearesetupforthesolution.Thethirdrowshowsthat z =0.Substitute thatbackintothesecondrowtoget y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(1,andthensubstitutebackintothe rstrowtoget x =1. 1.7Example ForthePhysicsproblemfromthestartofthischapter,Gauss' methodgivesthis. 40 h +15 c =100 )]TJ/F8 9.9626 Tf 7.749 0 Td [(50 h +25 c =50 5 = 4 1 + 2 )167(! 40 h +15 c =100 = 4 c =175 So c =4,andback-substitutiongivesthat h =1.TheChemistryproblemis solvedlater. 1.8Example Thereduction x + y + z =9 2 x +4 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 z =1 3 x +6 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 z =0 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 1 + 2 )167(! )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 1 + 3 x + y + z =9 2 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(5 z = )]TJ/F8 9.9626 Tf 7.749 0 Td [(17 3 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(8 z = )]TJ/F8 9.9626 Tf 7.749 0 Td [(27 )]TJ/F7 6.9738 Tf 6.226 0 Td [( = 2 2 + 3 )167(! x + y + z =9 2 y )]TJ/F8 9.9626 Tf 35.422 0 Td [(5 z = )]TJ/F8 9.9626 Tf 7.749 0 Td [(17 )]TJ/F8 9.9626 Tf 7.749 0 Td [( = 2 z = )]TJ/F8 9.9626 Tf 7.749 0 Td [( = 2 showsthat z =3, y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1,and x =7. Astheseexamplesillustrate,Gauss'methodusestheelementaryreduction operationstosetupback-substitution. 1.9Denition Ineachrow,therstvariablewithanonzerocoecientisthe row's leadingvariable .Asystemisin echelonform ifeachleadingvariableis totherightoftheleadingvariableintherowaboveitexceptfortheleading variableintherstrow. 1.10Example Theonlyoperationneededintheexamplesaboveispivoting. Hereisalinearsystemthatrequirestheoperationofswappingequations.After therstpivot x )]TJ/F11 9.9626 Tf 14.944 0 Td [(y =0 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y + z +2 w =4 y + w =0 2 z + w =5 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 1 + 2 )167(! x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y =0 z +2 w =4 y + w =0 2 z + w =5

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6 ChapterOne.LinearSystems thesecondequationhasnoleading y .Togetone,welooklowerdowninthe systemforarowthathasaleading y andswapitin. 2 $ 3 )167(! x )]TJ/F11 9.9626 Tf 9.962 0 Td [(y =0 y + w =0 z +2 w =4 2 z + w =5 Hadtherebeenmorethanonerowbelowthesecondwithaleading y thenwe couldhaveswappedinanyone.TherestofGauss'methodgoesasbefore. )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 3 + 4 )167(! x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y =0 y + w =0 z +2 w =4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 w = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 Back-substitutiongives w =1, z =2, y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1,and x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1. Strictlyspeaking,theoperationofrescalingrowsisnotneededtosolvelinear systems.Wehaveincludeditbecausewewilluseitlaterinthischapteraspart ofavariationonGauss'method,theGauss-Jordanmethod. Allofthesystemsseensofarhavethesamenumberofequationsasunknowns.Allofthemhaveasolution,andforallofthemthereisonlyone solution.Wenishthissubsectionbyseeingforcontrastsomeotherthingsthat canhappen. 1.11Example Linearsystemsneednothavethesamenumberofequations asunknowns.Thissystem x +3 y =1 2 x + y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 2 x +2 y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 hasmoreequationsthanvariables.Gauss'methodhelpsusunderstandthis systemalso,sincethis )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 1 + 2 )167(! )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 1 + 3 x +3 y =1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 showsthatoneoftheequationsisredundant.Echelonform )]TJ/F7 6.9738 Tf 6.227 0 Td [( = 5 2 + 3 )167(! x +3 y =1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 0=0 gives y =1and x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2.The`0=0'isderivedfromtheredundancy. Thatexample'ssystemhasmoreequationsthanvariables.Gauss'method isalsousefulonsystemswithmorevariablesthanequations.Manyexamples areinthenextsubsection.

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SectionI.SolvingLinearSystems 7 Anotherwaythatlinearsystemscandierfromtheexamplesshownearlier isthatsomelinearsystemsdonothaveauniquesolution.Thiscanhappenin twoways. Therstisthatitcanfailtohaveanysolutionatall. 1.12Example Contrastthesysteminthelastexamplewiththisone. x +3 y =1 2 x + y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 2 x +2 y =0 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 1 + 2 )167(! )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 1 + 3 x +3 y =1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 y = )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 Herethesystemisinconsistent:nopairofnumberssatisesalloftheequations simultaneously.Echelonformmakesthisinconsistencyobvious. )]TJ/F7 6.9738 Tf 6.227 0 Td [( = 5 2 + 3 )167(! x +3 y =1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 0=2 Thesolutionsetisempty. 1.13Example Thepriorsystemhasmoreequationsthanunknowns,butthat isnotwhatcausestheinconsistency|Example1.11hasmoreequationsthan unknownsandyetisconsistent.Norishavingmoreequationsthanunknowns necessaryforinconsistency,asisillustratedbythisinconsistentsystemwiththe samenumberofequationsasunknowns. x +2 y =8 2 x +4 y =8 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 1 + 2 )167(! x +2 y =8 0= )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 Theotherwaythatalinearsystemcanfailtohaveauniquesolutionisto havemanysolutions. 1.14Example Inthissystem x + y =4 2 x +2 y =8 anypairofnumberssatisfyingtherstequationautomaticallysatisesthesecond.Thesolutionset f x;y x + y =4 g isinnite;someofitsmembers are ; 4, )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; 5,and : 5 ; 1 : 5.TheresultofapplyingGauss'methodhere contrastswiththepriorexamplebecausewedonotgetacontradictoryequation. )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 1 + 2 )167(! x + y =4 0=0 Don'tbefooledbythe`0=0'equationinthatexample.Itisnotthesignal thatasystemhasmanysolutions.

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8 ChapterOne.LinearSystems 1.15Example Theabsenceofa`0=0'doesnotkeepasystemfromhaving manydierentsolutions.Thissystemisinechelonform x + y + z =0 y + z =0 hasno`0=0',andyethasinnitelymanysolutions.Forinstance,eachof theseisasolution: ; 1 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(1, ; 1 = 2 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 = 2, ; 0 ; 0,and ; )]TJ/F11 9.9626 Tf 7.748 0 Td [(; .Thereare innitelymanysolutionsbecauseanytriplewhoserstcomponentis0and whosesecondcomponentisthenegativeofthethirdisasolution. Nordoesthepresenceofa`0=0'meanthatthesystemmusthavemany solutions.Example1.11showsthat.Sodoesthissystem,whichdoesnothave manysolutions|infactithasnone|despitethatwhenitisbroughttoechelon formithasa`0=0'row. 2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 z =6 y + z =1 2 x + y )]TJ/F11 9.9626 Tf 14.944 0 Td [(z =7 3 y +3 z =0 )]TJ/F10 6.9738 Tf 6.226 0 Td [( 1 + 3 )167(! 2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 z =6 y + z =1 y + z =1 3 y +3 z =0 )]TJ/F10 6.9738 Tf 6.227 0 Td [( 2 + 3 )167(! )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 2 + 4 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 z =6 y + z =1 0=0 0= )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 Wewillnishthissubsectionwithasummaryofwhatwe'veseensofar aboutGauss'method. Gauss'methodusesthethreerowoperationstosetasystemupforback substitution.Ifanystepshowsacontradictoryequationthenwecanstop withtheconclusionthatthesystemhasnosolutions.Ifwereachechelonform withoutacontradictoryequation,andeachvariableisaleadingvariableinits row,thenthesystemhasauniquesolutionandwenditbybacksubstitution. Finally,ifwereachechelonformwithoutacontradictoryequation,andthereis notauniquesolutionatleastonevariableisnotaleadingvariablethenthe systemhasmanysolutions. Thenextsubsectiondealswiththethirdcase|wewillseehowtodescribe thesolutionsetofasystemwithmanysolutions. Exercises X 1.16 UseGauss'methodtondtheuniquesolutionforeachsystem. a 2 x +3 y =13 x )]TJ/F32 8.9664 Tf 13.823 0 Td [(y = )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 b x )]TJ/F32 8.9664 Tf 9.215 0 Td [(z =0 3 x + y =1 )]TJ/F32 8.9664 Tf 7.168 0 Td [(x + y + z =4 X 1.17 UseGauss'methodtosolveeachsystemorconclude`manysolutions'or`no solutions'.

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SectionI.SolvingLinearSystems 9 a 2 x +2 y =5 x )]TJ/F29 8.9664 Tf 9.215 0 Td [(4 y =0 b )]TJ/F32 8.9664 Tf 7.167 0 Td [(x + y =1 x + y =2 c x )]TJ/F29 8.9664 Tf 9.215 0 Td [(3 y + z =1 x + y +2 z =14 d )]TJ/F32 8.9664 Tf 7.167 0 Td [(x )]TJ/F32 8.9664 Tf 13.823 0 Td [(y =1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(3 x )]TJ/F29 8.9664 Tf 9.215 0 Td [(3 y =2 e 4 y + z =20 2 x )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 y + z =0 x + z =5 x + y )]TJ/F32 8.9664 Tf 9.215 0 Td [(z =10 f 2 x + z + w =5 y )]TJ/F32 8.9664 Tf 9.215 0 Td [(w = )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 3 x )]TJ/F32 8.9664 Tf 13.823 0 Td [(z )]TJ/F32 8.9664 Tf 9.215 0 Td [(w =0 4 x + y +2 z + w =9 X 1.18 TherearemethodsforsolvinglinearsystemsotherthanGauss'method.One oftentaughtinhighschoolistosolveoneoftheequationsforavariable,then substitutetheresultingexpressionintootherequations.Thatstepisrepeated untilthereisanequationwithonlyonevariable.Fromthat,therstnumber inthesolutionisderived,andthenback-substitutioncanbedone.Thismethod takeslongerthanGauss'method,sinceitinvolvesmorearithmeticoperations, andisalsomorelikelytoleadtoerrors.Toillustratehowitcanleadtowrong conclusions,wewillusethesystem x +3 y =1 2 x + y = )]TJ/F29 8.9664 Tf 7.168 0 Td [(3 2 x +2 y =0 fromExample1.12. a Solvetherstequationfor x andsubstitutethatexpressionintothesecond equation.Findtheresulting y b Againsolvetherstequationfor x ,butthistimesubstitutethatexpression intothethirdequation.Findthis y Whatextrastepmustauserofthismethodtaketoavoiderroneouslyconcluding asystemhasasolution? X 1.19 Forwhichvaluesof k aretherenosolutions,manysolutions,oraunique solutiontothissystem? x )]TJ/F32 8.9664 Tf 13.823 0 Td [(y =1 3 x )]TJ/F29 8.9664 Tf 9.216 0 Td [(3 y = k X 1.20 Thissystemisnotlinear,insomesense, 2sin )]TJ/F29 8.9664 Tf 15.359 0 Td [(cos +3tan =3 4sin +2cos )]TJ/F29 8.9664 Tf 9.215 0 Td [(2tan =10 6sin )]TJ/F29 8.9664 Tf 9.215 0 Td [(3cos +tan =9 andyetwecannonethelessapplyGauss'method.Doso.Doesthesystemhavea solution? X 1.21 Whatconditionsmusttheconstants,the b 's,satisfysothateachofthese systemshasasolution? Hint. ApplyGauss'methodandseewhathappenstothe rightside.[Anton] a x )]TJ/F29 8.9664 Tf 9.215 0 Td [(3 y = b 1 3 x + y = b 2 x +7 y = b 3 2 x +4 y = b 4 b x 1 +2 x 2 +3 x 3 = b 1 2 x 1 +5 x 2 +3 x 3 = b 2 x 1 +8 x 3 = b 3 1.22 Trueorfalse:asystemwithmoreunknownsthanequationshasatleastone solution.Asalways,tosay`true'youmustproveit,whiletosay`false'youmust produceacounterexample. 1.23 MustanyChemistryproblemliketheonethatstartsthissubsection|abalancethereactionproblem|haveinnitelymanysolutions? X 1.24 Findthecoecients a b ,and c sothatthegraphof f x = ax 2 + bx + c passes throughthepoints ; 2, )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 ; 6,and ; 3.

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10 ChapterOne.LinearSystems 1.25 Gauss'methodworksbycombiningtheequationsinasystemtomakenew equations. a Cantheequation3 x )]TJ/F29 8.9664 Tf 7.673 0 Td [(2 y =5bederived,byasequenceofGaussianreduction steps,fromtheequationsinthissystem? x + y =1 4 x )]TJ/F32 8.9664 Tf 9.215 0 Td [(y =6 b Cantheequation5 x )]TJ/F29 8.9664 Tf 7.573 0 Td [(3 y =2bederived,byasequenceofGaussianreduction steps,fromtheequationsinthissystem? 2 x +2 y =5 3 x + y =4 c Cantheequation6 x )]TJ/F29 8.9664 Tf 9.31 0 Td [(9 y +5 z = )]TJ/F29 8.9664 Tf 7.167 0 Td [(2bederived,byasequenceofGaussian reductionsteps,fromtheequationsinthesystem? 2 x + y )]TJ/F32 8.9664 Tf 9.215 0 Td [(z =4 6 x )]TJ/F29 8.9664 Tf 9.215 0 Td [(3 y + z =5 1.26 Provethat,where a;b;:::;e arerealnumbersand a 6 =0,if ax + by = c hasthesamesolutionsetas ax + dy = e thentheyarethesameequation.Whatif a =0? X 1.27 Showthatif ad )]TJ/F32 8.9664 Tf 9.215 0 Td [(bc 6 =0then ax + by = j cx + dy = k hasauniquesolution. X 1.28 Inthesystem ax + by = c dx + ey = f eachoftheequationsdescribesalineinthe xy -plane.Bygeometricalreasoning, showthattherearethreepossibilities:thereisauniquesolution,thereisno solution,andthereareinnitelymanysolutions. 1.29 FinishtheproofofTheorem1.4. 1.30 Isthereatwo-unknownslinearsystemwhosesolutionsetisallof R 2 ? X 1.31 AreanyoftheoperationsusedinGauss'methodredundant?Thatis,can anyoftheoperationsbesynthesizedfromtheothers? 1.32 ProvethateachoperationofGauss'methodisreversible.Thatis,showthatif twosystemsarerelatedbyarowoperation S 1 S 2 thenthereisarowoperation togoback S 2 S 1 ? 1.33 Aboxholdingpennies,nickelsanddimescontainsthirteencoinswithatotal valueof83cents.Howmanycoinsofeachtypeareinthebox?[Anton] ? 1.34 Fourpositiveintegersaregiven.Selectanythreeoftheintegers,ndtheir arithmeticaverage,andaddthisresulttothefourthinteger.Thusthenumbers 29,23,21,and17areobtained.Oneoftheoriginalintegersis:

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SectionI.SolvingLinearSystems 11 a 19 b 21 c 23 d 29 e 17 [Con.Prob.1955] ? X 1.35 Laughatthis:AHAHA+TEHE=TEHAW.Itresultedfromsubstituting acodeletterforeachdigitofasimpleexampleinaddition,anditisrequiredto identifythelettersandprovethesolutionunique.[Am.Math.Mon.,Jan.1935] ? 1.36 TheWohascumCountyBoardofCommissioners,whichhas20members,recentlyhadtoelectaPresident.Therewerethreecandidates A B ,and C ;on eachballotthethreecandidatesweretobelistedinorderofpreference,withno abstentions.Itwasfoundthat11members,amajority,preferred A over B thus theother9preferred B over A .Similarly,itwasfoundthat12memberspreferred C over A .Giventheseresults,itwassuggestedthat B shouldwithdraw,toenable arunoelectionbetween A and C .However, B protested,anditwasthenfound that14memberspreferred B over C !TheBoardhasnotyetrecoveredfromtheresultingconfusion.Giventhateverypossibleorderof A B C appearedonatleast oneballot,howmanymembersvotedfor B astheirrstchoice?[Wohascumno.2] ? 1.37 Thissystemof n linearequationswith n unknowns,"saidtheGreatMathematician,hasacuriousproperty." Goodheavens!"saidthePoorNut,Whatisit?" Note,"saidtheGreatMathematician,thattheconstantsareinarithmetic progression." It'sallsoclearwhenyouexplainit!"saidthePoorNut.Doyoumeanlike 6 x +9 y =12and15 x +18 y =21?" Quiteso,"saidtheGreatMathematician,pullingouthisbassoon.Indeed, thesystemhasauniquesolution.Canyoundit?" Goodheavens!"criedthePoorNut,Iambaed." Areyou?[Am.Math.Mon.,Jan.1963] I.2DescribingtheSolutionSet Alinearsystemwithauniquesolutionhasasolutionsetwithoneelement.A linearsystemwithnosolutionhasasolutionsetthatisempty.Inthesecases thesolutionsetiseasytodescribe.Solutionsetsareachallengetodescribe onlywhentheycontainmanyelements. 2.1Example Thissystemhasmanysolutionsbecauseinechelonform 2 x + z =3 x )]TJ/F11 9.9626 Tf 9.962 0 Td [(y )]TJ/F11 9.9626 Tf 9.963 0 Td [(z =1 3 x )]TJ/F11 9.9626 Tf 9.962 0 Td [(y =4 )]TJ/F7 6.9738 Tf 6.227 0 Td [( = 2 1 + 2 )167(! )]TJ/F7 6.9738 Tf 6.227 0 Td [( = 2 1 + 3 2 x + z =3 )]TJ/F11 9.9626 Tf 7.749 0 Td [(y )]TJ/F8 9.9626 Tf 9.962 0 Td [( = 2 z = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 )]TJ/F11 9.9626 Tf 7.749 0 Td [(y )]TJ/F8 9.9626 Tf 9.962 0 Td [( = 2 z = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 )]TJ/F10 6.9738 Tf 6.227 0 Td [( 2 + 3 )167(! 2 x + z =3 )]TJ/F11 9.9626 Tf 7.749 0 Td [(y )]TJ/F8 9.9626 Tf 9.962 0 Td [( = 2 z = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 0=0 notallofthevariablesareleadingvariables.TheGauss'methodtheorem showedthatatriplesatisestherstsystemifandonlyifitsatisesthethird. Thus,thesolutionset f x;y;z 2 x + z =3and x )]TJ/F11 9.9626 Tf 9.962 0 Td [(y )]TJ/F11 9.9626 Tf 9.963 0 Td [(z =1and3 x )]TJ/F11 9.9626 Tf 9.962 0 Td [(y =4 g

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12 ChapterOne.LinearSystems canalsobedescribedas f x;y;z 2 x + z =3and )]TJ/F11 9.9626 Tf 7.748 0 Td [(y )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 z= 2= )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 = 2 g .However,thisseconddescriptionisnotmuchofanimprovement.Ithastwoequationsinsteadofthree,butitstillinvolvessomehard-to-understandinteraction amongthevariables. Togetadescriptionthatisfreeofanysuchinteraction,wetakethevariablethatdoesnotleadanyequation, z ,anduseittodescribethevariables thatdolead, x and y .Thesecondequationgives y = = 2 )]TJ/F8 9.9626 Tf 10.908 0 Td [( = 2 z and therstequationgives x = = 2 )]TJ/F8 9.9626 Tf 10.282 0 Td [( = 2 z .Thus,thesolutionsetcanbedescribedas f x;y;z = = 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [( = 2 z; = 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [( = 2 z;z z 2 R g .Forinstance, = 2 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 = 2 ; 2isasolutionbecausetaking z =2givesarstcomponentof1 = 2 andasecondcomponentof )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 = 2. Theadvantageofthisdescriptionovertheonesaboveisthattheonlyvariable appearing, z ,isunrestricted|itcanbeanyrealnumber. 2.2Denition Thenon-leadingvariablesinanechelon-formlinearsystem are freevariables Intheechelonformsystemderivedintheaboveexample, x and y areleading variablesand z isfree. 2.3Example Alinearsystemcanendwithmorethanonevariablefree.This rowreduction x + y + z )]TJ/F11 9.9626 Tf 14.944 0 Td [(w =1 y )]TJ/F11 9.9626 Tf 14.944 0 Td [(z + w = )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 3 x +6 z )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 w =6 )]TJ/F11 9.9626 Tf 7.749 0 Td [(y + z )]TJ/F11 9.9626 Tf 14.944 0 Td [(w =1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 1 + 3 )167(! x + y + z )]TJ/F11 9.9626 Tf 14.944 0 Td [(w =1 y )]TJ/F11 9.9626 Tf 14.944 0 Td [(z + w = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 y +3 z )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 w =3 )]TJ/F11 9.9626 Tf 7.749 0 Td [(y + z )]TJ/F11 9.9626 Tf 14.944 0 Td [(w =1 3 2 + 3 )167(! 2 + 4 x + y + z )]TJ/F11 9.9626 Tf 9.963 0 Td [(w =1 y )]TJ/F11 9.9626 Tf 9.963 0 Td [(z + w = )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 0=0 0=0 endswith x and y leading,andwithboth z and w free.Togetthedescription thatwepreferwewillstartatthebottom.Werstexpress y intermsof thefreevariables z and w with y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1+ z )]TJ/F11 9.9626 Tf 10.86 0 Td [(w .Next,movinguptothe topequation,substitutingfor y intherstequation x + )]TJ/F8 9.9626 Tf 7.749 0 Td [(1+ z )]TJ/F11 9.9626 Tf 10.437 0 Td [(w + z )]TJ/F11 9.9626 Tf -335.962 -11.956 Td [(w =1andsolvingfor x yields x =2 )]TJ/F8 9.9626 Tf 10.744 0 Td [(2 z +2 w .Thus,thesolutionsetis f 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 z +2 w; )]TJ/F8 9.9626 Tf 7.748 0 Td [(1+ z )]TJ/F11 9.9626 Tf 9.963 0 Td [(w;z;w z;w 2 R g Wepreferthisdescriptionbecausetheonlyvariablesthatappear, z and w areunrestricted.Thismakesthejobofdecidingwhichfour-tuplesaresystem solutionsintoaneasyone.Forinstance,taking z =1and w =2givesthe solution ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 ; 1 ; 2.Incontrast, ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; 1 ; 2isnotasolution,sincetherst componentofanysolutionmustbe2minustwicethethirdcomponentplus twicethefourth.

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SectionI.SolvingLinearSystems 13 2.4Example Afterthisreduction 2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 y =0 z +3 w =2 3 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 y =0 x )]TJ/F11 9.9626 Tf 14.944 0 Td [(y +2 z +6 w =4 )]TJ/F7 6.9738 Tf 6.227 0 Td [( = 2 1 + 3 )167(! )]TJ/F7 6.9738 Tf 6.227 0 Td [( = 2 1 + 4 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y =0 z +3 w =2 0=0 2 z +6 w =4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 2 + 4 )167(! 2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y =0 z +3 w =2 0=0 0=0 x and z lead, y and w arefree.Thesolutionsetis f y;y; 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 w;w y;w 2 R g Forinstance, ; 1 ; 2 ; 0satisesthesystem|take y =1and w =0.Thefourtuple ; 0 ; 5 ; 4isnotasolutionsinceitsrstcoordinatedoesnotequalits second. Werefertoavariableusedtodescribeafamilyofsolutionsasa parameter andwesaythatthesetaboveis parametrized with y and w .Theterms `parameter'and`freevariable'donotmeanthesamething.Above, y and w arefreebecauseintheechelonformsystemtheydonotleadanyrow.They areparametersbecausetheyareusedinthesolutionsetdescription.Wecould haveinsteadparametrizedwith y and z byrewritingthesecondequationas w =2 = 3 )]TJ/F8 9.9626 Tf 10.487 0 Td [( = 3 z .Inthatcase,thefreevariablesarestill y and w ,butthe parametersare y and z .Noticethatwecouldnothaveparametrizedwith x and y ,sothereissometimesarestrictiononthechoiceofparameters.Theterms `parameter'and`free'arerelatedbecause,asweshallshowlaterinthischapter, thesolutionsetofasystemcanalwaysbeparametrizedwiththefreevariables. Consequently,weshallparametrizeallofourdescriptionsinthisway. 2.5Example Thisisanothersystemwithinnitelymanysolutions. x +2 y =1 2 x + z =2 3 x +2 y + z )]TJ/F11 9.9626 Tf 9.963 0 Td [(w =4 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 1 + 2 )167(! )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 1 + 3 x +2 y =1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 y + z =0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 y + z )]TJ/F11 9.9626 Tf 9.963 0 Td [(w =1 )]TJ/F10 6.9738 Tf 6.226 0 Td [( 2 + 3 )167(! x +2 y =1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 y + z =0 )]TJ/F11 9.9626 Tf 7.749 0 Td [(w =1 Theleadingvariablesare x y ,and w .Thevariable z isfree.Noticeherethat, althoughthereareinnitelymanysolutions,thevalueofoneofthevariablesis xed| w = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1.Write w intermsof z with w = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1+0 z .Then y = = 4 z Toexpress x intermsof z ,substitutefor y intotherstequationtoget x = 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [( = 2 z .Thesolutionsetis f )]TJ/F8 9.9626 Tf 9.962 0 Td [( = 2 z; = 4 z;z; )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 z 2 R g Wenishthissubsectionbydevelopingthenotationforlinearsystemsand theirsolutionsetsthatweshalluseintherestofthisbook. 2.6Denition An m n matrix isarectangulararrayofnumberswith m rows and n columns .Eachnumberinthematrixisan entry

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14 ChapterOne.LinearSystems Matricesareusuallynamedbyuppercaseromanletters,e.g. A .Eachentryis denotedbythecorrespondinglower-caseletter,e.g. a i;j isthenumberinrow i andcolumn j ofthearray.Forinstance, A = 12 : 25 34 )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 hastworowsandthreecolumns,andsoisa2 3matrix.Readthattwoby-three";thenumberofrowsisalwaysstatedrst.Theentryinthesecond rowandrstcolumnis a 2 ; 1 =3.Notethattheorderofthesubscriptsmatters: a 1 ; 2 6 = a 2 ; 1 since a 1 ; 2 =2 : 2.Theparenthesesaroundthearrayareatypographicdevicesothatwhentwomatricesaresidebysidewecantellwhereone endsandtheotherstarts. Matricesoccurthroughoutthisbook.Weshalluse M n m todenotethe collectionof n m matrices. 2.7Example Wecanabbreviatethislinearsystem x 1 +2 x 2 =4 x 2 )]TJ/F11 9.9626 Tf 14.943 0 Td [(x 3 =0 x 1 +2 x 3 =4 withthismatrix. 0 @ 120 4 01 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 0 102 4 1 A Theverticalbarjustremindsareaderofthedierencebetweenthecoecients onthesystems'slefthandsideandtheconstantsontheright.Whenabar isusedtodivideamatrixintoparts,wecallitan augmented matrix.Inthis notation,Gauss'methodgoesthisway. 0 @ 120 4 01 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 0 102 4 1 A )]TJ/F10 6.9738 Tf 6.227 0 Td [( 1 + 3 )167(! 0 @ 120 4 01 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 0 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(22 0 1 A 2 2 + 3 )167(! 0 @ 120 4 01 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 0 000 0 1 A Thesecondrowstandsfor y )]TJ/F11 9.9626 Tf 9.973 0 Td [(z =0andtherstrowstandsfor x +2 y =4so thesolutionsetis f )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 z;z;z z 2 R g .Oneadvantageofthenewnotationis thattheclericalloadofGauss'method|thecopyingofvariables,thewriting of+'sand='s,etc.|islighter. Wewillalsousethearraynotationtoclarifythedescriptionsofsolution sets.Adescriptionlike f )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 z +2 w; )]TJ/F8 9.9626 Tf 7.748 0 Td [(1+ z )]TJ/F11 9.9626 Tf 9.963 0 Td [(w;z;w z;w 2 R g fromExample2.3ishardtoread.Wewillrewriteittogroupalltheconstantstogether, allthecoecientsof z together,andallthecoecientsof w together.Wewill writethemvertically,inone-columnwidematrices. f 0 B B @ 2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 0 0 1 C C A + 0 B B @ )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 1 1 0 1 C C A z + 0 B B @ 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 0 1 1 C C A w z;w 2 R g

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SectionI.SolvingLinearSystems 15 Forinstance,thetoplinesaysthat x =2 )]TJ/F8 9.9626 Tf 10.106 0 Td [(2 z +2 w .Thenextsectiongivesa geometricinterpretationthatwillhelpuspicturethesolutionsetswhenthey arewritteninthisway. 2.8Denition A vector or columnvector isamatrixwithasinglecolumn. Amatrixwithasinglerowisa rowvector .Theentriesofavectorareits components Vectorsareanexceptiontotheconventionofrepresentingmatriceswith capitalromanletters.Weuselower-caseromanorgreeklettersoverlinedwith anarrow: ~a ~ b ,...or ~ ~ ,...boldfaceisalsocommon: a or .Forinstance, thisisacolumnvectorwithathirdcomponentof7. ~v = 0 @ 1 3 7 1 A 2.9Denition Thelinearequation a 1 x 1 + a 2 x 2 + + a n x n = d with unknowns x 1 ;:::;x n is satised by ~s = 0 B @ s 1 s n 1 C A if a 1 s 1 + a 2 s 2 + + a n s n = d .Avectorsatisesalinearsystemifitsatises eachequationinthesystem. Thestyleofdescriptionofsolutionsetsthatweuseinvolvesaddingthe vectors,andalsomultiplyingthembyrealnumbers,suchasthe z and w .We needtodenetheseoperations. 2.10Denition The vectorsum of ~u and ~v isthis. ~u + ~v = 0 B @ u 1 u n 1 C A + 0 B @ v 1 v n 1 C A = 0 B @ u 1 + v 1 u n + v n 1 C A Ingeneral,twomatriceswiththesamenumberofrowsandthesamenumber ofcolumnsaddinthisway,entry-by-entry. 2.11Denition The scalarmultiplication oftherealnumber r andthevector ~v isthis. r ~v = r 0 B @ v 1 v n 1 C A = 0 B @ rv 1 rv n 1 C A Ingeneral,anymatrixismultipliedbyarealnumberinthisentry-by-entry way.

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16 ChapterOne.LinearSystems Scalarmultiplicationcanbewrittenineitherorder: r ~v or ~v r ,orwithout the` 'symbol: r~v .Donotrefertoscalarmultiplicationas`scalarproduct' becausethatnameisusedforadierentoperation. 2.12Example 0 @ 2 3 1 1 A + 0 @ 3 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 4 1 A = 0 @ 2+3 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 1+4 1 A = 0 @ 5 2 5 1 A 7 0 B B @ 1 4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 1 C C A = 0 B B @ 7 28 )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 )]TJ/F8 9.9626 Tf 7.749 0 Td [(21 1 C C A Noticethatthedenitionsofvectoradditionandscalarmultiplicationagree wheretheyoverlap,forinstance, ~v + ~v =2 ~v Withthenotationdened,wecannowsolvesystemsinthewaythatwewill usethroughoutthisbook. 2.13Example Thissystem 2 x + y )]TJ/F11 9.9626 Tf 14.944 0 Td [(w =4 y + w + u =4 x )]TJ/F11 9.9626 Tf 9.963 0 Td [(z +2 w =0 reducesinthisway. 0 @ 210 )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 4 01011 4 10 )]TJ/F8 9.9626 Tf 7.749 0 Td [(120 0 1 A )]TJ/F7 6.9738 Tf 6.227 0 Td [( = 2 1 + 3 )167(! 0 @ 210 )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 4 01011 4 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(15 = 20 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 1 A = 2 2 + 3 )167(! 0 @ 210 )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 4 01011 4 00 )]TJ/F8 9.9626 Tf 7.749 0 Td [(131 = 2 0 1 A Thesolutionsetis f w + = 2 u; 4 )]TJ/F11 9.9626 Tf 9.963 0 Td [(w )]TJ/F11 9.9626 Tf 9.963 0 Td [(u; 3 w + = 2 u;w;u w;u 2 R g .We writethatinvectorform. f 0 B B B B @ x y z w u 1 C C C C A = 0 B B B B @ 0 4 0 0 0 1 C C C C A + 0 B B B B @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 3 1 0 1 C C C C A w + 0 B B B B @ 1 = 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 = 2 0 1 1 C C C C A u w;u 2 R g Noteagainhowwellvectornotationsetsothecoecientsofeachparameter. Forinstance,thethirdrowofthevectorformshowsplainlythatif u isheld xedthen z increasesthreetimesasfastas w Thatformatalsoshowsplainlythatthereareinnitelymanysolutions.For example,wecanx u as0,let w rangeovertherealnumbers,andconsiderthe rstcomponent x .Wegetinnitelymanyrstcomponentsandhenceinnitely manysolutions.

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SectionI.SolvingLinearSystems 17 Anotherthingshownplainlyisthatsettingboth w and u tozerogivesthat this 0 B B B B @ x y z w u 1 C C C C A = 0 B B B B @ 0 4 0 0 0 1 C C C C A isaparticularsolutionofthelinearsystem. 2.14Example Inthesameway,thissystem x )]TJ/F11 9.9626 Tf 14.944 0 Td [(y + z =1 3 x + z =3 5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y +3 z =5 reduces 0 @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 1 301 3 5 )]TJ/F8 9.9626 Tf 7.749 0 Td [(23 5 1 A )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 1 + 2 )167(! )]TJ/F7 6.9738 Tf 6.227 0 Td [(5 1 + 3 0 @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 1 03 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 0 03 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 0 1 A )]TJ/F10 6.9738 Tf 6.227 0 Td [( 2 + 3 )167(! 0 @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 1 03 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 0 000 0 1 A toaone-parametersolutionset. f 0 @ 1 0 0 1 A + 0 @ )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 = 3 2 = 3 1 1 A z z 2 R g Beforetheexercises,wepausetopointoutsomethingsthatwehaveyetto do. ThersttwosubsectionshavebeenonthemechanicsofGauss'method. Exceptforoneresult,Theorem1.4|withoutwhichdevelopingthemethod doesn'tmakesensesinceitsaysthatthemethodgivestherightanswers|we havenotstoppedtoconsideranyoftheinterestingquestionsthatarise. Forexample,canwealwaysdescribesolutionsetsasabove,withaparticular solutionvectoraddedtoanunrestrictedlinearcombinationofsomeothervectors?Thesolutionsetswedescribedwithunrestrictedparameterswereeasily seentohaveinnitelymanysolutionssoananswertothisquestioncouldtell ussomethingaboutthesizeofsolutionsets.Ananswertothatquestioncould alsohelpuspicturethesolutionsets,in R 2 ,orin R 3 ,etc. ManyquestionsarisefromtheobservationthatGauss'methodcanbedone inmorethanonewayforinstance,whenswappingrows,wemayhaveachoice ofwhichrowtoswapwith.Theorem1.4saysthatwemustgetthesame solutionsetnomatterhowweproceed,butifwedoGauss'methodintwo dierentwaysmustwegetthesamenumberoffreevariablesbothtimes,so thatanytwosolutionsetdescriptionshavethesamenumberofparameters? Mustthosebethesamevariablese.g.,isitimpossibletosolveaproblemone wayandget y and w freeorsolveitanotherwayandget y and z free?

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18 ChapterOne.LinearSystems Intherestofthischapterweanswerthesequestions.Theanswertoeach is`yes'.Therstquestionisansweredinthelastsubsectionofthissection.In thesecondsectionwegiveageometricdescriptionofsolutionsets.Inthenal sectionofthischapterwetacklethelastsetofquestions.Consequently,bythe endoftherstchapterwewillnotonlyhaveasolidgroundinginthepractice ofGauss'method,wewillalsohaveasolidgroundinginthetheory.Wewillbe sureofwhatcanandcannothappeninareduction. Exercises X 2.15 Findtheindicatedentryofthematrix,ifitisdened. A = 131 2 )]TJ/F29 8.9664 Tf 7.168 0 Td [(14 a a 2 ; 1 b a 1 ; 2 c a 2 ; 2 d a 3 ; 1 X 2.16 Givethesizeofeachmatrix. a 104 215 b 11 )]TJ/F29 8.9664 Tf 7.168 0 Td [(11 3 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 c 510 105 X 2.17 Dotheindicatedvectoroperation,ifitisdened. a 2 1 1 + 3 0 4 b 5 4 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 c 1 5 1 )]TJ/F1 9.9626 Tf 9.227 16.787 Td [( 3 1 1 d 7 2 1 +9 3 5 e 1 2 + 1 2 3 f 6 3 1 1 )]TJ/F29 8.9664 Tf 9.216 0 Td [(4 2 0 3 +2 1 1 5 X 2.18 Solveeachsystemusingmatrixnotation.Expressthesolutionusingvectors. a 3 x +6 y =18 x +2 y =6 b x + y =1 x )]TJ/F32 8.9664 Tf 9.215 0 Td [(y = )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 c x 1 + x 3 =4 x 1 )]TJ/F32 8.9664 Tf 9.215 0 Td [(x 2 +2 x 3 =5 4 x 1 )]TJ/F32 8.9664 Tf 9.215 0 Td [(x 2 +5 x 3 =17 d 2 a + b )]TJ/F32 8.9664 Tf 9.216 0 Td [(c =2 2 a + c =3 a )]TJ/F32 8.9664 Tf 9.215 0 Td [(b =0 e x +2 y )]TJ/F32 8.9664 Tf 9.215 0 Td [(z =3 2 x + y + w =4 x )]TJ/F32 8.9664 Tf 13.822 0 Td [(y + z + w =1 f x + z + w =4 2 x + y )]TJ/F32 8.9664 Tf 9.215 0 Td [(w =2 3 x + y + z =7 X 2.19 Solveeachsystemusingmatrixnotation.Giveeachsolutionsetinvector notation. a 2 x + y )]TJ/F32 8.9664 Tf 9.215 0 Td [(z =1 4 x )]TJ/F32 8.9664 Tf 9.215 0 Td [(y =3 b x )]TJ/F32 8.9664 Tf 13.823 0 Td [(z =1 y +2 z )]TJ/F32 8.9664 Tf 9.215 0 Td [(w =3 x +2 y +3 z )]TJ/F32 8.9664 Tf 9.215 0 Td [(w =7 c x )]TJ/F32 8.9664 Tf 16.383 0 Td [(y + z =0 y + w =0 3 x )]TJ/F29 8.9664 Tf 11.775 0 Td [(2 y +3 z + w =0 )]TJ/F32 8.9664 Tf 7.168 0 Td [(y )]TJ/F32 8.9664 Tf 9.215 0 Td [(w =0 d a +2 b +3 c + d )]TJ/F32 8.9664 Tf 9.215 0 Td [(e =1 3 a )]TJ/F32 8.9664 Tf 13.822 0 Td [(b + c + d + e =3 X 2.20 Thevectorisintheset.Whatvalueoftheparametersproducesthatvector? a 5 )]TJ/F29 8.9664 Tf 7.168 0 Td [(5 f 1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 k k 2 R g b )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 2 1 f )]TJ/F29 8.9664 Tf 7.168 0 Td [(2 1 0 i + 3 0 1 j i;j 2 R g

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SectionI.SolvingLinearSystems 19 c 0 )]TJ/F29 8.9664 Tf 7.167 0 Td [(4 2 f 1 1 0 m + 2 0 1 n m;n 2 R g 2.21 Decideifthevectorisintheset. a 3 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 f )]TJ/F29 8.9664 Tf 7.167 0 Td [(6 2 k k 2 R g b 5 4 f 5 )]TJ/F29 8.9664 Tf 7.168 0 Td [(4 j j 2 R g c 2 1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 f 0 3 )]TJ/F29 8.9664 Tf 7.168 0 Td [(7 + 1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 3 r r 2 R g d 1 0 1 f 2 0 1 j + )]TJ/F29 8.9664 Tf 7.167 0 Td [(3 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 1 k j;k 2 R g 2.22 Parametrizethesolutionsetofthisone-equationsystem. x 1 + x 2 + + x n =0 X 2.23a ApplyGauss'methodtotheleft-handsidetosolve x +2 y )]TJ/F32 8.9664 Tf 13.823 0 Td [(w = a 2 x + z = b x + y +2 w = c for x y z ,and w ,intermsoftheconstants a b ,and c b Useyouranswerfromthepriorparttosolvethis. x +2 y )]TJ/F32 8.9664 Tf 13.823 0 Td [(w =3 2 x + z =1 x + y +2 w = )]TJ/F29 8.9664 Tf 7.168 0 Td [(2 X 2.24 Whyisthecommaneededinthenotation` a i;j 'formatrixentries? X 2.25 Givethe4 4matrixwhose i;j -thentryis a i + j ; b )]TJ/F29 8.9664 Tf 7.168 0 Td [(1tothe i + j power. 2.26 Foranymatrix A ,the transpose of A ,written A trans ,isthematrixwhose columnsaretherowsof A .Findthetransposeofeachofthese. a 123 456 b 2 )]TJ/F29 8.9664 Tf 7.168 0 Td [(3 11 c 510 105 d 1 1 0 X 2.27a Describeallfunctions f x = ax 2 + bx + c suchthat f =2and f )]TJ/F29 8.9664 Tf 7.167 0 Td [(1=6. b Describeallfunctions f x = ax 2 + bx + c suchthat f =2. 2.28 Showthatanysetofvepointsfromtheplane R 2 lieonacommonconic section,thatis,theyallsatisfysomeequationoftheform ax 2 + by 2 + cxy + dx + ey + f =0wheresomeof a;:::;f arenonzero. 2.29 Makeupafourequations/fourunknownssystemhaving a aone-parametersolutionset; b atwo-parametersolutionset; c athree-parametersolutionset. ? 2.30a Solvethesystemofequations. ax + y = a 2 x + ay =1 Forwhatvaluesof a doesthesystemfailtohavesolutions,andforwhatvalues of a arethereinnitelymanysolutions?

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20 ChapterOne.LinearSystems b Answertheabovequestionforthesystem. ax + y = a 3 x + ay =1 [USSROlympiadno.174] ? 2.31 Inairagold-surfacedsphereweighs7588grams.Itisknownthatitmay containoneormoreofthemetalsaluminum,copper,silver,orlead.Whenweighed successivelyunderstandardconditionsinwater,benzene,alcohol,andglycerine itsrespectiveweightsare6588,6688,6778,and6328grams.Howmuch,ifany, oftheforenamedmetalsdoesitcontainifthespecicgravitiesofthedesignated substancesaretakentobeasfollows? Aluminum2 : 7Alcohol0.81 Copper8 : 9Benzene0 : 90 Gold19 : 3Glycerine1 : 26 Lead11 : 3Water1 : 00 Silver10 : 8 [Math.Mag.,Sept.1952] I.3General=Particular+Homogeneous Thepriorsubsectionhasmanydescriptionsofsolutionsets.Theyallta pattern.Theyhaveavectorthatisaparticularsolutionofthesystemadded toanunrestrictedcombinationofsomeothervectors.Thesolutionsetfrom Example2.13illustrates. f 0 B B B B @ 0 4 0 0 0 1 C C C C A | {z } particular solution + w 0 B B B B @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 3 1 0 1 C C C C A + u 0 B B B B @ 1 = 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 = 2 0 1 1 C C C C A | {z } unrestricted combination w;u 2 R g Thecombinationisunrestrictedinthat w and u canbeanyrealnumbers| thereisnoconditionlikesuchthat2 w )]TJ/F11 9.9626 Tf 8.94 0 Td [(u =0"thatwouldrestrictwhichpairs w;u canbeusedtoformcombinations. Thatexampleshowsaninnitesolutionsetconformingtothepattern.We canthinkoftheothertwokindsofsolutionsetsasalsottingthesamepattern.Aone-elementsolutionsettsinthatithasaparticularsolution,and theunrestrictedcombinationpartisatrivialsumthatis,insteadofbeinga combinationoftwovectors,asabove,oracombinationofonevector,itisa combinationofnovectors.Azero-elementsolutionsettsthepatternsince thereisnoparticularsolution,andsothesetofsumsofthatformisempty. Wewillshowthattheexamplesfromthepriorsubsectionarerepresentative, inthatthedescriptionpatterndiscussedaboveholdsforeverysolutionset.

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SectionI.SolvingLinearSystems 21 3.1Theorem Foranylinearsystemtherearevectors ~ 1 ,..., ~ k suchthat thesolutionsetcanbedescribedas f ~p + c 1 ~ 1 + + c k ~ k c 1 ;:::;c k 2 R g where ~p isanyparticularsolution,andwherethesystemhas k freevariables. Thisdescriptionhastwoparts,theparticularsolution ~p andalsotheunrestrictedlinearcombinationofthe ~ 's.Weshallprovethetheoremintwo correspondingparts,withtwolemmas. Wewillfocusrstontheunrestrictedcombinationpart.Todothat,we considersystemsthathavethevectorofzeroesasoneoftheparticularsolutions, sothat ~p + c 1 ~ 1 + + c k ~ k canbeshortenedto c 1 ~ 1 + + c k ~ k 3.2Denition Alinearequationis homogeneous ifithasaconstantofzero, thatis,ifitcanbeputintheform a 1 x 1 + a 2 x 2 + + a n x n =0. Theseare`homogeneous'becauseallofthetermsinvolvethesamepowerof theirvariable|therstpower|includinga`0 x 0 'thatwecanimagineison therightside. 3.3Example Withanylinearsystemlike 3 x +4 y =3 2 x )]TJ/F11 9.9626 Tf 14.944 0 Td [(y =1 weassociateasystemofhomogeneousequationsbysettingtherightsideto zeros. 3 x +4 y =0 2 x )]TJ/F11 9.9626 Tf 14.944 0 Td [(y =0 Ourinterestinthehomogeneoussystemassociatedwithalinearsystemcanbe understoodbycomparingthereductionofthesystem 3 x +4 y =3 2 x )]TJ/F11 9.9626 Tf 14.944 0 Td [(y =1 )]TJ/F7 6.9738 Tf 6.227 0 Td [( = 3 1 + 2 )167(! 3 x +4 y =3 )]TJ/F8 9.9626 Tf 7.749 0 Td [( = 3 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 withthereductionoftheassociatedhomogeneoussystem. 3 x +4 y =0 2 x )]TJ/F11 9.9626 Tf 14.944 0 Td [(y =0 )]TJ/F7 6.9738 Tf 6.227 0 Td [( = 3 1 + 2 )167(! 3 x +4 y =0 )]TJ/F8 9.9626 Tf 7.748 0 Td [( = 3 y =0 Obviouslythetworeductionsgointhesameway.Wecanstudyhowlinearsystemsarereducedbyinsteadstudyinghowtheassociatedhomogeneoussystems arereduced. Studyingtheassociatedhomogeneoussystemhasagreatadvantageover studyingtheoriginalsystem.Nonhomogeneoussystemscanbeinconsistent. Butahomogeneoussystemmustbeconsistentsincethereisalwaysatleastone solution,thevectorofzeros.

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22 ChapterOne.LinearSystems 3.4Denition Acolumnorrowvectorofallzerosisa zerovector ,denoted ~ 0. Therearemanydierentzerovectors,e.g.,theone-tallzerovector,thetwo-tall zerovector,etc.Nonetheless,peopleoftenrefertothe"zerovector,expecting thatthesizeoftheonebeingdiscussedwillbeclearfromthecontext. 3.5Example Somehomogeneoussystemshavethezerovectorastheironly solution. 3 x +2 y + z =0 6 x +4 y =0 y + z =0 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 1 + 2 )167(! 3 x +2 y + z =0 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 z =0 y + z =0 2 $ 3 )167(! 3 x +2 y + z =0 y + z =0 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 z =0 3.6Example Somehomogeneoussystemshavemanysolutions.Oneexample istheChemistryproblemfromtherstpageofthisbook. 7 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(7 z =0 8 x + y )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 z )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 w =0 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 z =0 3 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 z )]TJ/F11 9.9626 Tf 14.944 0 Td [(w =0 )]TJ/F7 6.9738 Tf 6.227 0 Td [( = 7 1 + 2 )167(! 7 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(7 z =0 y +3 z )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 w =0 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 z =0 3 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(6 z )]TJ/F11 9.9626 Tf 14.944 0 Td [(w =0 )]TJ/F10 6.9738 Tf 6.227 0 Td [( 2 + 3 )167(! )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 2 + 4 7 x )]TJ/F8 9.9626 Tf 22.692 0 Td [(7 z =0 y +3 z )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 w =0 )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 z +2 w =0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(15 z +5 w =0 )]TJ/F7 6.9738 Tf 6.226 0 Td [( = 2 3 + 4 )167(! 7 x )]TJ/F8 9.9626 Tf 17.711 0 Td [(7 z =0 y +3 z )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 w =0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 z +2 w =0 0=0 Thesolutionset: f 0 B B @ 1 = 3 1 1 = 3 1 1 C C A w w 2 R g hasmanyvectorsbesidesthezerovectorifweinterpret w asanumberof moleculesthensolutionsmakesenseonlywhen w isanonnegativemultipleof 3. WenowhavetheterminologytoprovethetwopartsofTheorem3.1.The rstlemmadealswithunrestrictedcombinations. 3.7Lemma Foranyhomogeneouslinearsystemthereexistvectors ~ 1 ,..., ~ k suchthatthesolutionsetofthesystemis f c 1 ~ 1 + + c k ~ k c 1 ;:::;c k 2 R g where k isthenumberoffreevariablesinanechelonformversionofthesystem.

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SectionI.SolvingLinearSystems 23 Beforetheproof,wewillrecallthebacksubstitutioncalculationsthatwere doneinthepriorsubsection.Imaginethatwehavebroughtasystemtothis echelonform. x +2 y )]TJ/F11 9.9626 Tf 9.963 0 Td [(z +2 w =0 )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 y + z =0 )]TJ/F11 9.9626 Tf 7.749 0 Td [(w =0 Wenextperformback-substitutiontoexpresseachvariableintermsofthe freevariable z .Workingfromthebottomup,wegetrstthat w is0 z nextthat y is = 3 z ,andthensubstitutingthosetwointothetopequation x +2 = 3 z )]TJ/F11 9.9626 Tf 10.446 0 Td [(z +2=0gives x = = 3 z .So,backsubstitutiongives aparametrizationofthesolutionsetbystartingatthebottomequationand usingthefreevariablesastheparameterstoworkrow-by-rowtothetop.The proofbelowfollowsthispattern. Comment: Thatis,thisproofjustdoesavericationofthebookkeepingin backsubstitutiontoshowthatwehaven'toverlookedanyobscurecaseswhere thisprocedurefails,say,byleadingtoadivisionbyzero.Sothisargument, whilequitedetailed,doesn'tgiveusanynewinsights.Nevertheless,wehave writtenitoutfortworeasons.Therstreasonisthatweneedtheresult|the computationalprocedurethatweemploymustbeveriedtoworkaspromised. Thesecondreasonisthattherow-by-rownatureofbacksubstitutionleadstoa proofthatusesthetechniqueofmathematicalinduction. Thisisanimportant, andnon-obvious,prooftechniquethatweshalluseanumberoftimesinthis book.Doinganinductionargumentheregivesusachancetoseeoneinasetting wheretheproofmaterialiseasytofollow,andsothetechniquecanbestudied. Readerswhoareunfamiliarwithinductionargumentsshouldbesuretomaster thisoneandtheoneslaterinthischapterbeforegoingontothesecondchapter. Proof FirstuseGauss'methodtoreducethehomogeneoussystemtoechelon form.Wewillshowthateachleadingvariablecanbeexpressedintermsoffree variables.Thatwillnishtheargumentbecausethenwecanusethosefree variablesastheparameters.Thatis,the ~ 'sarethevectorsofcoecientsof thefreevariablesasinExample3.6,wherethesolutionis x = = 3 w y = w z = = 3 w ,and w = w Wewillproceedbymathematicalinduction,whichhastwosteps.Thebase stepoftheargumentwillbetofocusonthebottom-mostnon-`0=0'equation andwriteitsleadingvariableintermsofthefreevariables.Theinductivestep oftheargumentwillbetoarguethatifwecanexpresstheleadingvariablesfrom thebottom t rowsintermsoffreevariables,thenwecanexpresstheleading variableofthenextrowup|the t +1-throwupfromthebottom|interms offreevariables.Withthosetwosteps,thetheoremwillbeprovedbecauseby thebasestepitistrueforthebottomequation,andbytheinductivestepthe factthatitistrueforthebottomequationshowsthatitistrueforthenext oneup,andthenanotherapplicationoftheinductivestepimpliesitistruefor thirdequationup,etc. Moreinformationonmathematicalinductionisintheappendix.

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24 ChapterOne.LinearSystems Forthebasestep,considerthebottom-mostnon-`0=0'equationthecase wherealltheequationsare`0=0'istrivial.Wecallthatthe m -throw: a m;` m x ` m + a m;` m +1 x ` m +1 + + a m;n x n =0 where a m;` m 6 =0.Thenotationherehas` ` 'standfor`leading',so a m;` m means thecoecient,fromtherow m ofthevariableleadingrow m ".Eitherthere arevariablesinthisequationotherthantheleadingone x ` m orelsethereare not.Ifthereareothervariables x ` m +1 ,etc.,thentheymustbefreevariables becausethisisthebottomnon-`0=0'row.Movethemtotherightanddivide by a m;` m x ` m = )]TJ/F11 9.9626 Tf 7.749 0 Td [(a m;` m +1 =a m;` m x ` m +1 + + )]TJ/F11 9.9626 Tf 7.748 0 Td [(a m;n =a m;` m x n toexpressthisleadingvariableintermsoffreevariables.Iftherearenofree variablesinthisequationthen x ` m =0seethetrickypoint"notedfollowing thisproof. Fortheinductivestep,weassumethatforthe m -thequation,andforthe m )]TJ/F8 9.9626 Tf 9.963 0 Td [(1-thequation,...,andforthe m )]TJ/F11 9.9626 Tf 9.962 0 Td [(t -thequation,wecanexpressthe leadingvariableintermsoffreevariableswhere0 t
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SectionI.SolvingLinearSystems 25 Proof Wewillshowmutualsetinclusion,thatanysolutiontothesystemis intheabovesetandthatanythinginthesetisasolutiontothesystem. Forsetinclusiontherstway,thatifavectorsolvesthesystemthenitis inthesetdescribedabove,assumethat ~s solvesthesystem.Then ~s )]TJ/F11 9.9626 Tf 10.109 0 Td [(~p solves theassociatedhomogeneoussystemsinceforeachequationindex i a i; 1 s 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(p 1 + + a i;n s n )]TJ/F11 9.9626 Tf 9.962 0 Td [(p n = a i; 1 s 1 + + a i;n s n )]TJ/F8 9.9626 Tf 9.962 0 Td [( a i; 1 p 1 + + a i;n p n = d i )]TJ/F11 9.9626 Tf 9.963 0 Td [(d i =0 where p j and s j arethe j -thcomponentsof ~p and ~s .Wecanwrite ~s )]TJ/F11 9.9626 Tf 10.163 0 Td [(~p as ~ h where ~ h solvestheassociatedhomogeneoussystem,toexpress ~s intherequired ~p + ~ h form. Forsetinclusiontheotherway,takeavectoroftheform ~p + ~ h ,where ~p solvesthesystemand ~ h solvestheassociatedhomogeneoussystem,andnote thatitsolvesthegivensystem:foranyequationindex i a i; 1 p 1 + h 1 + + a i;n p n + h n = a i; 1 p 1 + + a i;n p n + a i; 1 h 1 + + a i;n h n = d i +0 = d i where h j isthe j -thcomponentof ~ h QED ThetwolemmasabovetogetherestablishTheorem3.1.Werememberthat theoremwiththesloganGeneral=Particular+Homogeneous". 3.9Example ThissystemillustratesTheorem3.1. x +2 y )]TJ/F11 9.9626 Tf 14.943 0 Td [(z =1 2 x +4 y =2 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 z =0 Gauss'method )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 1 + 2 )167(! x +2 y )]TJ/F11 9.9626 Tf 14.943 0 Td [(z =1 2 z =0 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 z =0 2 $ 3 )167(! x +2 y )]TJ/F11 9.9626 Tf 14.944 0 Td [(z =1 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 z =0 2 z =0 showsthatthegeneralsolutionisasingletonset. f 0 @ 1 0 0 1 A g Moreinformationonequalityofsetsisintheappendix.

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26 ChapterOne.LinearSystems Thatsinglevectoris,ofcourse,aparticularsolution.Theassociatedhomogeneoussystemreducesviathesamerowoperations x +2 y )]TJ/F11 9.9626 Tf 14.944 0 Td [(z =0 2 x +4 y =0 y )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 z =0 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 1 + 2 )167(! 2 $ 3 )167(! x +2 y )]TJ/F11 9.9626 Tf 14.943 0 Td [(z =0 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 z =0 2 z =0 toalsogiveasingletonset. f 0 @ 0 0 0 1 A g Asthetheoremstates,andasdiscussedatthestartofthissubsection,inthis single-solutioncasethegeneralsolutionresultsfromtakingtheparticularsolutionandaddingtoittheuniquesolutionoftheassociatedhomogeneoussystem. 3.10Example Alsodiscussedthereisthatthecasewherethegeneralsolution setisemptytsthe`General=Particular+Homogeneous'pattern.Thissystem illustrates.Gauss'method x + z + w = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 2 x )]TJ/F11 9.9626 Tf 9.962 0 Td [(y + w =3 x + y +3 z +2 w =1 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 1 + 2 )167(! )]TJ/F10 6.9738 Tf 6.227 0 Td [( 1 + 3 x + z + w = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F11 9.9626 Tf 7.749 0 Td [(y )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 z )]TJ/F11 9.9626 Tf 9.963 0 Td [(w =5 y +2 z + w =2 showsthatithasnosolutions.Theassociatedhomogeneoussystem,ofcourse, hasasolution. x + z + w =0 2 x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y + w =0 x + y +3 z +2 w =0 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 1 + 2 )167(! )]TJ/F10 6.9738 Tf 6.227 0 Td [( 1 + 3 2 + 3 )167(! x + z + w =0 )]TJ/F11 9.9626 Tf 7.749 0 Td [(y )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 z )]TJ/F11 9.9626 Tf 9.962 0 Td [(w =0 0=0 Infact,thesolutionsetofthehomogeneoussystemisinnite. f 0 B B @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 1 0 1 C C A z + 0 B B @ )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 0 1 1 C C A w z;w 2 R g However,becausenoparticularsolutionoftheoriginalsystemexists,thegeneral solutionsetisempty|therearenovectorsoftheform ~p + ~ h becausethereare no ~p 's. 3.11Corollary Solutionsetsoflinearsystemsareeitherempty,haveone element,orhaveinnitelymanyelements. Proof We'veseenexamplesofallthreehappeningsoweneedonlyprovethat thosearetheonlypossibilities. First,noticeahomogeneoussystemwithatleastonenon~ 0solution ~v has innitelymanysolutionsbecausethesetofmultiples s~v isinnite|if s 6 =1 then s~v )]TJ/F11 9.9626 Tf 9.576 0 Td [(~v = s )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ~v iseasilyseentobenon~ 0,andso s~v 6 = ~v .

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SectionI.SolvingLinearSystems 27 Now,applyLemma3.8toconcludethatasolutionset f ~p + ~ h ~ h solvestheassociatedhomogeneoussystem g iseitheremptyifthereisnoparticularsolution ~p ,orhasoneelementifthere isa ~p andthehomogeneoussystemhastheuniquesolution ~ 0,orisinniteif thereisa ~p andthehomogeneoussystemhasanon~ 0solution,andthusbythe priorparagraphhasinnitelymanysolutions. QED Thistablesummarizesthefactorsaectingthesizeofageneralsolution. numberofsolutionsofthe associatedhomogeneoussystem particular solution exists? one innitelymany yes unique solution innitelymany solutions no no solutions no solutions Thefactoronthetopofthetableisthesimplerone.Whenweperform Gauss'methodonalinearsystem,ignoringtheconstantsontherightsideand sopayingattentiononlytothecoecientsontheleft-handside,weeitherend witheveryvariableleadingsomeroworelsewendthatsomevariabledoesnot leadarow,thatis,thatsomevariableisfree.Ofcourse,ignoringtheconstants ontheright"isformalizedbyconsideringtheassociatedhomogeneoussystem. Wearesimplyputtingasideforthemomentthepossibilityofacontradictory equation. Aniceinsightintothefactoronthetopofthistableatworkcomesfromconsideringthecaseofasystemhavingthesamenumberofequationsasvariables. Thissystemwillhaveasolution,andthesolutionwillbeunique,ifandonlyifit reducestoanechelonformsystemwhereeveryvariableleadsitsrow,whichwill happenifandonlyiftheassociatedhomogeneoussystemhasauniquesolution. Thus,thequestionofuniquenessofsolutionisespeciallyinterestingwhenthe systemhasthesamenumberofequationsasvariables. 3.12Denition Asquarematrixis nonsingular ifitisthematrixofcoecientsofahomogeneoussystemwithauniquesolution.Itis singular otherwise, thatis,ifitisthematrixofcoecientsofahomogeneoussystemwithinnitely manysolutions. 3.13Example ThesystemsfromExample3.3,Example3.5,andExample3.9 eachhaveanassociatedhomogeneoussystemwithauniquesolution.Thusthese matricesarenonsingular. 34 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 0 @ 321 6 )]TJ/F8 9.9626 Tf 7.749 0 Td [(40 011 1 A 0 @ 12 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 240 01 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 1 A

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28 ChapterOne.LinearSystems TheChemistryproblemfromExample3.6isahomogeneoussystemwithmore thanonesolutionsoitsmatrixissingular. 0 B B @ 70 )]TJ/F8 9.9626 Tf 7.748 0 Td [(70 81 )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 01 )]TJ/F8 9.9626 Tf 7.748 0 Td [(30 03 )]TJ/F8 9.9626 Tf 7.748 0 Td [(6 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 C C A 3.14Example Therstofthesematricesisnonsingularwhilethesecondis singular 12 34 12 36 becausetherstofthesehomogeneoussystemshasauniquesolutionwhilethe secondhasinnitelymanysolutions. x +2 y =0 3 x +4 y =0 x +2 y =0 3 x +6 y =0 Wehavemadethedistinctioninthedenitionbecauseasystemwiththesame numberofequationsasvariablesbehavesinoneoftwoways,dependingon whetheritsmatrixofcoecientsisnonsingularorsingular.Asystemwhere thematrixofcoecientsisnonsingularhasauniquesolutionforanyconstants ontherightside:forinstance,Gauss'methodshowsthatthissystem x +2 y = a 3 x +4 y = b hastheuniquesolution x = b )]TJ/F8 9.9626 Tf 10.191 0 Td [(2 a and y = a )]TJ/F11 9.9626 Tf 10.192 0 Td [(b = 2.Ontheotherhand,a systemwherethematrixofcoecientsissingularneverhasauniquesolution| ithaseithernosolutionsorelsehasinnitelymany,aswiththese. x +2 y =1 3 x +6 y =2 x +2 y =1 3 x +6 y =3 Thus,`singular'canbethoughtofasconnotingtroublesome",oratleastnot ideal". Theabovetablehastwofactors.Wehavealreadyconsideredthefactor alongthetop:wecantellwhichcolumnagivenlinearsystemgoesinsolelyby consideringthesystem'sleft-handside|theconstantsontheright-handside playnoroleinthisfactor.Thetable'sotherfactor,determiningwhethera particularsolutionexists,istougher.Considerthesetwo 3 x +2 y =5 3 x +2 y =5 3 x +2 y =5 3 x +2 y =4 withthesameleftsidesbutdierentrightsides.Obviously,thersthasa solutionwhiletheseconddoesnot,soheretheconstantsontherightside

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SectionI.SolvingLinearSystems 29 decideifthesystemhasasolution.Wecouldconjecturethattheleftsideofa linearsystemdeterminesthenumberofsolutionswhiletherightsidedetermines ifsolutionsexist,butthatguessisnotcorrect.Comparethesetwosystems 3 x +2 y =5 4 x +2 y =4 3 x +2 y =5 3 x +2 y =4 withthesamerightsidesbutdierentleftsides.Thersthasasolutionbut theseconddoesnot.Thustheconstantsontherightsideofasystemdon't decidealonewhetherasolutionexists;rather,itdependsonsomeinteraction betweentheleftandrightsides. Forsomeintuitionaboutthatinteraction,considerthissystemwithoneof thecoecientsleftastheparameter c x +2 y +3 z =1 x + y + z =1 cx +3 y +4 z =0 If c =2thissystemhasnosolutionbecausetheleft-handsidehasthethirdrow asasumofthersttwo,whiletheright-handdoesnot.If c 6 =2thissystemhas auniquesolutiontryitwith c =1.Forasystemtohaveasolution,ifonerow ofthematrixofcoecientsontheleftisalinearcombinationofotherrows, thenontherighttheconstantfromthatrowmustbethesamecombinationof constantsfromthesamerows. Moreintuitionabouttheinteractioncomesfromstudyinglinearcombinations.Thatwillbeourfocusinthesecondchapter,afterwenishthestudyof Gauss'methoditselfintherestofthischapter. Exercises X 3.15 Solveeachsystem.Expressthesolutionsetusingvectors.Identifytheparticularsolutionandthesolutionsetofthehomogeneoussystem. a 3 x +6 y =18 x +2 y =6 b x + y =1 x )]TJ/F32 8.9664 Tf 9.215 0 Td [(y = )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 c x 1 + x 3 =4 x 1 )]TJ/F32 8.9664 Tf 9.215 0 Td [(x 2 +2 x 3 =5 4 x 1 )]TJ/F32 8.9664 Tf 9.215 0 Td [(x 2 +5 x 3 =17 d 2 a + b )]TJ/F32 8.9664 Tf 9.215 0 Td [(c =2 2 a + c =3 a )]TJ/F32 8.9664 Tf 9.215 0 Td [(b =0 e x +2 y )]TJ/F32 8.9664 Tf 9.215 0 Td [(z =3 2 x + y + w =4 x )]TJ/F32 8.9664 Tf 13.823 0 Td [(y + z + w =1 f x + z + w =4 2 x + y )]TJ/F32 8.9664 Tf 9.215 0 Td [(w =2 3 x + y + z =7 3.16 Solveeachsystem,givingthesolutionsetinvectornotation.Identifythe particularsolutionandthesolutionofthehomogeneoussystem. a 2 x + y )]TJ/F32 8.9664 Tf 9.215 0 Td [(z =1 4 x )]TJ/F32 8.9664 Tf 9.215 0 Td [(y =3 b x )]TJ/F32 8.9664 Tf 13.823 0 Td [(z =1 y +2 z )]TJ/F32 8.9664 Tf 9.215 0 Td [(w =3 x +2 y +3 z )]TJ/F32 8.9664 Tf 9.215 0 Td [(w =7 c x )]TJ/F32 8.9664 Tf 16.383 0 Td [(y + z =0 y + w =0 3 x )]TJ/F29 8.9664 Tf 11.775 0 Td [(2 y +3 z + w =0 )]TJ/F32 8.9664 Tf 7.168 0 Td [(y )]TJ/F32 8.9664 Tf 9.216 0 Td [(w =0 d a +2 b +3 c + d )]TJ/F32 8.9664 Tf 9.215 0 Td [(e =1 3 a )]TJ/F32 8.9664 Tf 13.823 0 Td [(b + c + d + e =3 X 3.17 Forthesystem 2 x )]TJ/F32 8.9664 Tf 13.823 0 Td [(y )]TJ/F32 8.9664 Tf 13.823 0 Td [(w =3 y + z +2 w =2 x )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 y )]TJ/F32 8.9664 Tf 9.215 0 Td [(z = )]TJ/F29 8.9664 Tf 7.167 0 Td [(1

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30 ChapterOne.LinearSystems whichofthesecanbeusedastheparticularsolutionpartofsomegeneralsolution? a 0 B @ 0 )]TJ/F29 8.9664 Tf 7.167 0 Td [(3 5 0 1 C A b 0 B @ 2 1 1 0 1 C A c 0 B @ )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(4 8 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 1 C A X 3.18 Lemma3.8saysthatanyparticularsolutionmaybeusedfor ~p .Find,if possible,ageneralsolutiontothissystem x )]TJ/F32 8.9664 Tf 13.823 0 Td [(y + w =4 2 x +3 y )]TJ/F32 8.9664 Tf 9.215 0 Td [(z =0 y + z + w =4 thatusesthegivenvectorasitsparticularsolution. a 0 B @ 0 0 0 4 1 C A b 0 B @ )]TJ/F29 8.9664 Tf 7.168 0 Td [(5 1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(7 10 1 C A c 0 B @ 2 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 1 1 1 C A 3.19 Oneoftheseisnonsingularwhiletheotherissingular.Whichiswhich? a 13 4 )]TJ/F29 8.9664 Tf 7.168 0 Td [(12 b 13 412 X 3.20 Singularornonsingular? a 12 13 b 12 )]TJ/F29 8.9664 Tf 7.168 0 Td [(3 )]TJ/F29 8.9664 Tf 7.168 0 Td [(6 c 121 131 Careful! d 121 113 347 e 221 105 )]TJ/F29 8.9664 Tf 7.168 0 Td [(114 X 3.21 Isthegivenvectorinthesetgeneratedbythegivenset? a 2 3 ; f 1 4 ; 1 5 g b )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 0 1 ; f 2 1 0 ; 1 0 1 g c 1 3 0 ; f 1 0 4 ; 2 1 5 ; 3 3 0 ; 4 2 1 g d 0 B @ 1 0 1 1 1 C A ; f 0 B @ 2 1 0 1 1 C A ; 0 B @ 3 0 0 2 1 C A g 3.22 Provethatanylinearsystemwithanonsingularmatrixofcoecientshasa solution,andthatthesolutionisunique. 3.23 Totellthewholetruth,thereisanothertrickypointtotheproofofLemma3.7. Whathappensiftherearenonon-`0=0'equations?Therearen'tanymoretricky pointsafterthisone. X 3.24 Provethatif ~s and ~ t satisfyahomogeneoussystemthensodothesevectors. a ~s + ~ t b 3 ~s c k~s + m ~ t for k;m 2 R What'swrongwith:Thesethreeshowthatifahomogeneoussystemhasone solutionthenithasmanysolutions|anymultipleofasolutionisanothersolution,

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SectionI.SolvingLinearSystems 31 andanysumofsolutionsisasolutionalso|sotherearenohomogeneoussystems withexactlyonesolution."? 3.25 Provethatifasystemwithonlyrationalcoecientsandconstantshasa solutionthenithasatleastoneall-rationalsolution.Mustithaveinnitelymany?

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32 ChapterOne.LinearSystems IILinearGeometryof n -Space Forreaderswhohaveseentheelementsofvectorsbefore,incalculusorphysics, thissectionisanoptionalreview.However,laterworkwillrefertothismaterial soitisnotoptionalifitisnotareview. Intherstsection,wehadtodoabitofworktoshowthatthereareonly threetypesofsolutionsets|singleton,empty,andinnite.Butinthespecial caseofsystemswithtwoequationsandtwounknownsthisiseasytosee.Draw eachtwo-unknownsequationasalineintheplaneandthenthetwolinescould haveauniqueintersection,beparallel,orbethesameline. Uniquesolution 3 x +2 y =7 x )]TJ/F11 9.9626 Tf 14.944 0 Td [(y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Nosolutions 3 x +2 y =7 3 x +2 y =4 Innitelymany solutions 3 x +2 y =7 6 x +4 y =14 Thesepicturesdon'tprovetheresultsfromthepriorsection,whichapplyto anynumberoflinearequationsandanynumberofunknowns,butnonetheless theydohelpustounderstandthoseresults.Thissectiondevelopstheideas thatweneedtoexpressourresultsfromthepriorsection,andfromsomefuture sections,geometrically.Inparticular,whilethetwo-dimensionalcaseisfamiliar, toextendtosystemswithmorethantwounknownsweshallneedsomehigherdimensionalgeometry. II.1VectorsinSpace Higher-dimensionalgeometry"soundsexotic.Itisexotic|interestingand eye-opening.Butitisn'tdistantorunreachable. Webeginbydeningone-dimensionalspacetobetheset R 1 .Toseethat denitionisreasonable,drawaone-dimensionalspace andmaketheusualcorrespondencewith R :pickapointtolabel0andanother tolabel1. 01 Now,withascaleandadirection,ndingthepointcorrespondingto,say+2 : 17, iseasy|startat0andheadinthedirectionof1i.e.,thepositivedirection, butdon'tstopthere,go2 : 17timesasfar.

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SectionII.LinearGeometryof n -Space 33 Thebasicideahere,combiningmagnitudewithdirection,isthekeytoextendingtohigherdimensions. Anobjectcomprisedofamagnitudeandadirectionisa vector wewilluse thesamewordasintheprevioussectionbecauseweshallshowbelowhowto describesuchanobjectwithacolumnvector.Wecandrawavectorashaving somelength,andpointingsomewhere. Thereisasubtletyhere|thesevectors areequal,eventhoughtheystartindierentplaces,becausetheyhaveequal lengthsandequaldirections.Again:thosevectorsarenotjustalike,theyare equal. Howcanthingsthatareindierentplacesbeequal?Thinkofavectoras representingadisplacement`vector'isLatinforcarrier"ortraveler".These squaresundergothesamedisplacement,despitethatthosedisplacementsstart indierentplaces. Sometimes,toemphasizethispropertyvectorshaveofnotbeinganchored,they arereferredtoas free vectors.Thus,thesefreevectorsareequalaseachisa displacementofoneoverandtwoup. Moregenerally,vectorsintheplanearethesameifandonlyiftheyhavethe samechangeinrstcomponentsandthesamechangeinsecondcomponents:the vectorextendingfrom a 1 ;a 2 to b 1 ;b 2 equalsthevectorfrom c 1 ;c 2 to d 1 ;d 2 ifandonlyif b 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(a 1 = d 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(c 1 and b 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(a 2 = d 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(c 2 Anexpressionlike`thevectorthat,wereittostartat a 1 ;a 2 ,wouldextend to b 1 ;b 2 'isawkward.Weinsteaddescribesuchavectoras b 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(a 1 b 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(a 2 sothat,forinstance,the`oneoverandtwoup'arrowsshownabovepicturethis vector. 1 2

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34 ChapterOne.LinearSystems Weoftendrawthearrowasstartingattheorigin,andwethensayitisinthe canonicalposition or naturalposition .Whenthevector b 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(a 1 b 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(a 2 isinitscanonicalpositionthenitextendstotheendpoint b 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(a 1 ;b 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(a 2 Wetypicallyjustrefertothepoint 1 2 ratherthantheendpointofthecanonicalpositionof"thatvector.Thus,we willcallbothofthesesets R 2 f x 1 ;x 2 x 1 ;x 2 2 R gf x 1 x 2 x 1 ;x 2 2 R g Inthepriorsectionwedenedvectorsandvectoroperationswithanalgebraicmotivation; r v 1 v 2 = rv 1 rv 2 v 1 v 2 + w 1 w 2 = v 1 + w 1 v 2 + w 2 wecannowinterpretthoseoperationsgeometrically.Forinstance,if ~v representsadisplacementthen3 ~v representsadisplacementinthesamedirection butthreetimesasfar,and )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ~v representsadisplacementofthesamedistance as ~v butintheoppositedirection. ~v )]TJ/F32 8.9664 Tf 6.829 0 Td [(~v 3 ~v And,where ~v and ~w representdisplacements, ~v + ~w representsthosedisplacementscombined. ~v ~w ~v + ~w Thelongarrowisthecombineddisplacementinthissense:if,inoneminute,a ship'smotiongivesitthedisplacementrelativetotheearthof ~v andapassenger'smotiongivesadisplacementrelativetotheship'sdeckof ~w ,then ~v + ~w is thedisplacementofthepassengerrelativetotheearth. Anotherwaytounderstandthevectorsumiswiththe parallelogramrule Drawtheparallelogramformedbythevectors ~v 1 ;~v 2 andthenthesum ~v 1 + ~v 2 extendsalongthediagonaltothefarcorner.

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SectionII.LinearGeometryof n -Space 35 ~v + ~w ~v ~w Theabovedrawingsshowhowvectorsandvectoroperationsbehavein R 2 Wecanextendto R 3 ,ortoevenhigher-dimensionalspaceswherewehaveno pictures,withtheobviousgeneralization:thefreevectorthat,ifitstartsat a 1 ;:::;a n ,endsat b 1 ;:::;b n ,isrepresentedbythiscolumn 0 B @ b 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(a 1 b n )]TJ/F11 9.9626 Tf 9.963 0 Td [(a n 1 C A vectorsareequaliftheyhavethesamerepresentation,wearen'ttoocareful todistinguishbetweenapointandthevectorwhosecanonicalrepresentation endsatthatpoint, R n = f 0 B @ v 1 v n 1 C A v 1 ;:::;v n 2 R g andadditionandscalarmultiplicationarecomponent-wise. Havingconsideredpoints,wenowturntothelines.In R 2 ,thelinethrough ; 2and ; 1iscomprisedoftheendpointsofthevectorsinthisset f 1 2 + t 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 t 2 R g Thatdescriptionexpressesthispicture. 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = 3 1 )]TJ/F1 9.9626 Tf 8.041 10.312 Td [( 1 2 Thevectorassociatedwiththeparameter t hasitswholebodyintheline|it isa directionvector fortheline.Notethatpointsonthelinetotheleftof x =1 aredescribedusingnegativevaluesof t In R 3 ,thelinethrough ; 2 ; 1and ; 3 ; 2isthesetofendpointsof vectorsofthisform f 1 2 1 + t 1 1 1 t 2 R g

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36 ChapterOne.LinearSystems andlinesinevenhigher-dimensionalspacesworkinthesameway. Ifalineusesoneparameter,sothatthereisfreedomtomovebackand forthinonedimension,thenaplanemustinvolvetwo.Forexample,theplane throughthepoints ; 0 ; 5, ; 1 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(3,and )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; 4 ; 0 : 5consistsofendpointsof thevectorsin f 0 @ 1 0 5 1 A + t 0 @ 1 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 1 A + s 0 @ )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 : 5 1 A t;s 2 R g thecolumnvectorsassociatedwiththeparameters 0 @ 1 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 1 A = 0 @ 2 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 1 A )]TJ/F1 9.9626 Tf 9.963 20.025 Td [(0 @ 1 0 5 1 A 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 : 5 1 A = 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 4 0 : 5 1 A )]TJ/F1 9.9626 Tf 9.963 20.025 Td [(0 @ 1 0 5 1 A aretwovectorswhosewholebodieslieintheplane.Aswiththeline,notethat somepointsinthisplanearedescribedwithnegative t 'sornegative s 'sorboth. Adescriptionofplanesthatisoftenencounteredinalgebraandcalculus usesasingleequationastheconditionthatdescribestherelationshipamong therst,second,andthirdcoordinatesofpointsinaplane. P = f x y z 2 x + y + z =4 g Thetranslationfromsuchadescriptiontothevectordescriptionthatwefavor inthisbookistothinkoftheconditionasaone-equationlinearsystemand parametrize x = = 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(y )]TJ/F11 9.9626 Tf 9.962 0 Td [(z P = f 2 0 0 + )]TJ/F7 6.9738 Tf 6.227 0 Td [(0 : 5 1 0 y + )]TJ/F7 6.9738 Tf 6.227 0 Td [(0 : 5 0 1 z y;z 2 R g Generalizingfromlinesandplanes,wedenea k -dimensionallinearsurface or k -at in R n tobe f ~p + t 1 ~v 1 + t 2 ~v 2 + + t k ~v k t 1 ;:::;t k 2 R g where ~v 1 ;:::;~v k 2 R n .Forexample,in R 4 f 0 B B @ 2 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(0 : 5 1 C C A + t 0 B B @ 1 0 0 0 1 C C A t 2 R g

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SectionII.LinearGeometryof n -Space 37 isaline, f 0 B B @ 0 0 0 0 1 C C A + t 0 B B @ 1 1 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 C C A + s 0 B B @ 2 0 1 0 1 C C A t;s 2 R g isaplane,and f 0 B B @ 3 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 0 : 5 1 C C A + r 0 B B @ 0 0 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 C C A + s 0 B B @ 1 0 1 0 1 C C A + t 0 B B @ 2 0 1 0 1 C C A r;s;t 2 R g isathree-dimensionallinearsurface.Again,theintuitionisthatalinepermitsmotioninonedirection,aplanepermitsmotionincombinationsoftwo directions,etc. Alinearsurfacedescriptioncanbemisleadingaboutthedimension|this L = f 0 B B @ 1 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 1 C C A + t 0 B B @ 1 1 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 C C A + s 0 B B @ 2 2 0 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 1 C C A t;s 2 R g isa degenerate planebecauseitisactuallyaline. L = f 0 B B @ 1 0 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 1 C C A + r 0 B B @ 1 1 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 C C A r 2 R g WeshallseeintheLinearIndependencesectionofChapterTwowhatrelationshipsamongvectorscausesthelinearsurfacetheygeneratetobedegenerate. Wenishthissubsectionbyrestatingourconclusionsfromtherstsection ingeometricterms.First,thesolutionsetofalinearsystemwith n unknowns isalinearsurfacein R n .Specically,itisa k -dimensionallinearsurface,where k isthenumberoffreevariablesinanechelonformversionofthesystem. Second,thesolutionsetofahomogeneouslinearsystemisalinearsurface passingthroughtheorigin.Finally,wecanviewthegeneralsolutionsetofany linearsystemasbeingthesolutionsetofitsassociatedhomogeneoussystem osetfromtheoriginbyavector,namelybyanyparticularsolution. Exercises X 1.1 Findthecanonicalnameforeachvector. a thevectorfrom ; 1to ; 2in R 2 b thevectorfrom ; 3to ; 5in R 2 c thevectorfrom ; 0 ; 6to ; 0 ; 3in R 3 d thevectorfrom ; 8 ; 8to ; 8 ; 8in R 3 X 1.2 Decideifthetwovectorsareequal. a thevectorfrom ; 3to ; 2andthevectorfrom ; )]TJ/F29 8.9664 Tf 7.167 0 Td [(2to ; 1

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38 ChapterOne.LinearSystems b thevectorfrom ; 1 ; 1to ; 0 ; 4andthevectorfrom ; 1 ; 4to ; 0 ; 7 X 1.3 Does ; 0 ; 2 ; 1lieonthelinethrough )]TJ/F29 8.9664 Tf 7.168 0 Td [(2 ; 1 ; 1 ; 0and ; 10 ; )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 ; 4? X 1.4a Describetheplanethrough ; 1 ; 5 ; )]TJ/F29 8.9664 Tf 7.168 0 Td [(1, ; 2 ; 2 ; 0,and ; 1 ; 0 ; 4. b Istheorigininthatplane? 1.5 Describetheplanethatcontainsthispointandline. 2 0 3 f )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 0 )]TJ/F29 8.9664 Tf 7.168 0 Td [(4 + 1 1 2 t t 2 R g X 1.6 Intersecttheseplanes. f 1 1 1 t + 0 1 3 s t;s 2 R gf 1 1 0 + 0 3 0 k + 2 0 4 m k;m 2 R g X 1.7 Intersecteachpair,ifpossible. a f 1 1 2 + t 0 1 1 t 2 R g f 1 3 )]TJ/F29 8.9664 Tf 7.168 0 Td [(2 + s 0 1 2 s 2 R g b f 2 0 1 + t 1 1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 t 2 R g f s 0 1 2 + w 0 4 1 s;w 2 R g 1.8 Whenaplanedoesnotpassthroughtheorigin,performingoperationsonvectorswhosebodieslieinitismorecomplicatedthanwhentheplanepassesthrough theorigin.Considerthepictureinthissubsectionoftheplane f 2 0 0 + )]TJ/F29 8.9664 Tf 7.168 0 Td [(0 : 5 1 0 y + )]TJ/F29 8.9664 Tf 7.168 0 Td [(0 : 5 0 1 z y;z 2 R g andthethreevectorsitshows,withendpoints ; 0 ; 0, : 5 ; 1 ; 0,and : 5 ; 0 ; 1. a Redrawthepicture,includingthevectorintheplanethatistwiceaslong astheonewithendpoint : 5 ; 1 ; 0.Theendpointofyourvectorisnot ; 2 ; 0; whatisit? b Redrawthepicture,includingtheparallelogramintheplanethatshowsthe sumofthevectorsendingat : 5 ; 0 ; 1and : 5 ; 1 ; 0.Theendpointofthesum, onthediagonal,isnot ; 1 ; 1;whatisit? 1.9 Showthatthelinesegments a 1 ;a 2 b 1 ;b 2 and c 1 ;c 2 d 1 ;d 2 havethesame lengthsandslopesif b 1 )]TJ/F32 8.9664 Tf 9.216 0 Td [(a 1 = d 1 )]TJ/F32 8.9664 Tf 9.215 0 Td [(c 1 and b 2 )]TJ/F32 8.9664 Tf 9.215 0 Td [(a 2 = d 2 )]TJ/F32 8.9664 Tf 9.216 0 Td [(c 2 .Isthatonlyif? 1.10 Howshould R 0 bedened? ? X 1.11 Apersontravelingeastwardatarateof3milesperhourndsthatthewind appearstoblowdirectlyfromthenorth.Ondoublinghisspeeditappearstocome fromthenortheast.Whatwasthewind'svelocity?[Math.Mag.,Jan.1957] 1.12 Eucliddescribesaplaneasasurfacewhichliesevenlywiththestraightlines onitself".Commentatorse.g.,HeronhaveinterpretedthistomeanAplane surfaceissuchthat,ifastraightlinepassthroughtwopointsonit,theline coincideswhollywithitateveryspot,allways".Translationsfrom[Heath],pp. 171-172.Doplanes,asdescribedinthissection,havethatproperty?Doesthis descriptionadequatelydeneplanes?

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SectionII.LinearGeometryof n -Space 39 II.2LengthandAngleMeasures We'vetranslatedtherstsection'sresultsaboutsolutionsetsintogeometric termsforinsightintohowthosesetslook.Butwemustwatchoutnottobe misleadbyourownterms;labelingsubsetsof R k oftheforms f ~p + t~v t 2 R g and f ~p + t~v + s~w t;s 2 R g aslines"andplanes"doesn'tmakethemactlike thelinesandplanesofourpriorexperience.Rather,wemustensurethatthe namessuitthesets.Whilewecan'tprovethatthesetssatisfyourintuition| wecan'tproveanythingaboutintuition|inthissubsectionwe'llobservethat aresultfamiliarfrom R 2 and R 3 ,whengeneralizedtoarbitrary R k ,supports theideathatalineisstraightandaplaneisat.Specically,we'llseehowto doEuclideangeometryinaplane"bygivingadenitionoftheanglebetween two R n vectorsintheplanethattheygenerate. 2.1Denition The length ofavector ~v 2 R n isthis. k ~v k = q v 2 1 + + v 2 n 2.2Remark ThisisanaturalgeneralizationofthePythagoreanTheorem.A classicdiscussionisin[Polya]. Wecanusethatdenitiontoderiveaformulafortheanglebetweentwo vectors.Foramodelofwhattodo,considertwovectorsin R 3 ~v ~u Putthemincanonicalpositionand,intheplanethattheydetermine,consider thetriangleformedby ~u ~v ,and ~u )]TJ/F11 9.9626 Tf 9.576 0 Td [(~v ApplytheLawofCosines, k ~u )]TJ/F11 9.9626 Tf 9.632 0 Td [(~v k 2 = k ~u k 2 + k ~v k 2 )]TJ/F8 9.9626 Tf 10.018 0 Td [(2 k ~u kk ~v k cos ,where istheanglebetweenthevectors.Expandbothsides u 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(v 1 2 + u 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(v 2 2 + u 3 )]TJ/F11 9.9626 Tf 9.963 0 Td [(v 3 2 = u 2 1 + u 2 2 + u 2 3 + v 2 1 + v 2 2 + v 2 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 k ~u kk ~v k cos andsimplify. =arccos u 1 v 1 + u 2 v 2 + u 3 v 3 k ~u kk ~v k

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40 ChapterOne.LinearSystems Inhigherdimensionsnopicturesucesbutwecanmakethesameargument analytically.First,theformofthenumeratorisclear|itcomesfromthemiddle termsofthesquares u 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(v 1 2 u 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(v 2 2 ,etc. 2.3Denition The dotproduct or innerproduct ,or scalarproduct oftwo n -componentrealvectorsisthelinearcombinationoftheircomponents. ~u ~v = u 1 v 1 + u 2 v 2 + + u n v n Notethatthedotproductoftwovectorsisarealnumber,notavector,andthat thedotproductofavectorfrom R n withavectorfrom R m isdenedonlywhen n equals m .Notealsothisrelationshipbetweendotproductandlength:dotting avectorwithitselfgivesitslengthsquared ~u ~u = u 1 u 1 + + u n u n = k ~u k 2 2.4Remark Thewordinginthatdenitionallowsoneorbothofthetwoto bearowvectorinsteadofacolumnvector.Somebooksrequirethattherst vectorbearowvectorandthatthesecondvectorbeacolumnvector.Weshall notbethatstrict. Stillreasoningwithletters,butguidedbythepictures,weusethenext theoremtoarguethatthetriangleformedby ~u ~v ,and ~u )]TJ/F11 9.9626 Tf 9.721 0 Td [(~v in R n liesinthe planarsubsetof R n generatedby ~u and ~v 2.5TheoremTriangleInequality Forany ~u;~v 2 R n k ~u + ~v kk ~u k + k ~v k withequalityifandonlyifoneofthevectorsisanonnegativescalarmultiple oftheotherone. Thisinequalityisthesourceofthefamiliarsaying,Theshortestdistance betweentwopointsisinastraightline." ~u ~v ~u + ~v start nish Proof We'llusesomealgebraicpropertiesofdotproductthatwehavenot yetchecked,forinstancethat ~u ~a + ~ b = ~u ~a + ~u ~ b andthat ~u ~v = ~v ~u .See Exercise17.Thedesiredinequalityholdsifandonlyifitssquareholds. k ~u + ~v k 2 k ~u k + k ~v k 2 ~u + ~v ~u + ~v k ~u k 2 +2 k ~u kk ~v k + k ~v k 2 ~u ~u + ~u ~v + ~v ~u + ~v ~v ~u ~u +2 k ~u kk ~v k + ~v ~v 2 ~u ~v 2 k ~u kk ~v k

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SectionII.LinearGeometryof n -Space 41 That,inturn,holdsifandonlyiftherelationshipobtainedbymultiplyingboth sidesbythenonnegativenumbers k ~u k and k ~v k 2 k ~v k ~u k ~u k ~v 2 k ~u k 2 k ~v k 2 andrewriting 0 k ~u k 2 k ~v k 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 k ~v k ~u k ~u k ~v + k ~u k 2 k ~v k 2 istrue.Butfactoring 0 k ~u k ~v )-222(k ~v k ~u k ~u k ~v )-222(k ~v k ~u showsthatthiscertainlyistruesinceitonlysaysthatthesquareofthelength ofthevector k ~u k ~v )-222(k ~v k ~u isnotnegative. Asforequality,itholdswhen,andonlywhen, k ~u k ~v )-147(k ~v k ~u is ~ 0.Thecheck that k ~u k ~v = k ~v k ~u ifandonlyifonevectorisanonnegativerealscalarmultiple oftheotheriseasy. QED Thisresultsupportstheintuitionthateveninhigher-dimensionalspaces, linesarestraightandplanesareat.Foranytwopointsinalinearsurface,the linesegmentconnectingthemiscontainedinthatsurfacethisiseasilychecked fromthedenition.Butifthesurfacehasabendthenthatwouldallowfora shortcutshownheregrayed,whilethesegmentfrom P to Q thatiscontained inthesurfaceissolid. P Q BecausetheTriangleInequalitysaysthatinany R n ,theshortestcutbetween twoendpointsissimplythelinesegmentconnectingthem,linearsurfaceshave nosuchbends. Backtothedenitionofanglemeasure.TheheartoftheTriangleInequality'sproofisthe` ~u ~v k ~u kk ~v k 'line.Atrstglance,areadermightwonder ifsomepairsofvectorssatisfytheinequalityinthisway:while ~u ~v isalarge number,withabsolutevaluebiggerthantheright-handside,itisanegative largenumber.Thenextresultsaysthatnosuchpairofvectorsexists. 2.6CorollaryCauchy-SchwartzInequality Forany ~u;~v 2 R n j ~u ~v jk ~u kk ~v k withequalityifandonlyifonevectorisascalarmultipleoftheother. Proof TheTriangleInequality'sproofshowsthat ~u ~v k ~u kk ~v k soif ~u ~v is positiveorzerothenwearedone.If ~u ~v isnegativethenthisholds. j ~u ~v j = )]TJ/F8 9.9626 Tf 7.749 0 Td [( ~u ~v = )]TJ/F11 9.9626 Tf 7.62 0 Td [(~u ~v k)]TJ/F11 9.9626 Tf 27.545 0 Td [(~u kk ~v k = k ~u kk ~v k TheequalityconditionisExercise18. QED

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42 ChapterOne.LinearSystems TheCauchy-Schwartzinequalityassuresusthatthenextdenitionmakes sensebecausethefractionhasabsolutevaluelessthanorequaltoone. 2.7Denition The angle betweentwononzerovectors ~u;~v 2 R n is =arccos ~u ~v k ~u kk ~v k theanglebetweenthezerovectorandanyothervectorisdenedtobearight angle. Thusvectorsfrom R n areorthogonalifandonlyiftheirdotproductiszero. 2.8Example Thesevectorsareorthogonal. 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 1 =0 Thearrowsareshownawayfromcanonicalpositionbutneverthelessthevectors areorthogonal. 2.9Example The R 3 angleformulagivenatthestartofthissubsectionisa specialcaseofthedenition.Betweenthesetwo 0 3 2 1 1 0 theangleis arccos ++ p 1 2 +1 2 +0 2 p 0 2 +3 2 +2 2 =arccos 3 p 2 p 13 approximately0 : 94radians.Noticethatthesevectorsarenotorthogonal.Althoughthe yz -planemayappeartobeperpendiculartothe xy -plane,infact thetwoplanesarethatwayonlyintheweaksensethattherearevectorsineach orthogonaltoallvectorsintheother.Noteveryvectorineachisorthogonalto allvectorsintheother. Exercises X 2.10 Findthelengthofeachvector.

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SectionII.LinearGeometryof n -Space 43 a 3 1 b )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 2 c 4 1 1 d 0 0 0 e 0 B @ 1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 1 0 1 C A X 2.11 Findtheanglebetweeneachtwo,ifitisdened. a 1 2 ; 1 4 b 1 2 0 ; 0 4 1 c 1 2 ; 1 4 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 X 2.12 DuringmaneuversprecedingtheBattleofJutland,theBritishbattlecruiser Lion movedasfollowsinnauticalmiles:1 : 2milesnorth,6 : 1miles38degrees eastofsouth,4 : 0milesat89degreeseastofnorth,and6 : 5milesat31degrees eastofnorth.Findthedistancebetweenstartingandendingpositions.[Ohanian] 2.13 Find k sothatthesetwovectorsareperpendicular. k 1 4 3 2.14 Describethesetofvectorsin R 3 orthogonaltothisone. 1 3 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 X 2.15a Findtheanglebetweenthediagonaloftheunitsquarein R 2 andoneof theaxes. b Findtheanglebetweenthediagonaloftheunitcubein R 3 andoneofthe axes. c Findtheanglebetweenthediagonaloftheunitcubein R n andoneofthe axes. d Whatisthelimit,as n goesto 1 ,oftheanglebetweenthediagonalofthe unitcubein R n andoneoftheaxes? 2.16 Isanyvectorperpendiculartoitself? X 2.17 Describethealgebraicpropertiesofdotproduct. a Isitright-distributiveoveraddition: ~u + ~v ~w = ~u ~w + ~v ~w ? b Isisleft-distributiveoveraddition? c Doesitcommute? d Associate? e Howdoesitinteractwithscalarmultiplication? Asalways,anyassertionmustbebackedbyeitheraprooforanexample. 2.18 VerifytheequalityconditioninCorollary2.6,theCauchy-SchwartzInequality. a Showthatif ~u isanegativescalarmultipleof ~v then ~u ~v and ~v ~u areless thanorequaltozero. b Showthat j ~u ~v j = k ~u kk ~v k ifandonlyifonevectorisascalarmultipleof theother. 2.19 Supposethat ~u ~v = ~u ~w and ~u 6 = ~ 0.Must ~v = ~w ? X 2.20 Doesanyvectorhavelengthzeroexceptazerovector?Ifyes",producean example.Ifno",proveit. X 2.21 Findthemidpointofthelinesegmentconnecting x 1 ;y 1 with x 2 ;y 2 in R 2 Generalizeto R n 2.22 Showthatif ~v 6 = ~ 0then ~v= k ~v k haslengthone.Whatif ~v = ~ 0? 2.23 Showthatif r 0then r~v is r timesaslongas ~v .Whatif r< 0?

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44 ChapterOne.LinearSystems X 2.24 Avector ~v 2 R n oflengthoneisa unit vector.Showthatthedotproduct oftwounitvectorshasabsolutevaluelessthanorequaltoone.Can`lessthan' happen?Can`equalto'? 2.25 Provethat k ~u + ~v k 2 + k ~u )]TJ/F32 8.9664 Tf 8.877 0 Td [(~v k 2 =2 k ~u k 2 +2 k ~v k 2 : 2.26 Showthatif ~x ~y =0forevery ~y then ~x = ~ 0. 2.27 Is k ~u 1 + + ~u n kk ~u 1 k + + k ~u n k ?Ifitistruethenitwouldgeneralize theTriangleInequality. 2.28 WhatistheratiobetweenthesidesintheCauchy-Schwartzinequality? 2.29 Whyisthezerovectordenedtobeperpendiculartoeveryvector? 2.30 Describetheanglebetweentwovectorsin R 1 2.31 Giveasimplenecessaryandsucientconditiontodeterminewhetherthe anglebetweentwovectorsisacute,right,orobtuse. X 2.32 Generalizeto R n theconverseofthePythagoreanTheorem,thatif ~u and ~v areperpendicularthen k ~u + ~v k 2 = k ~u k 2 + k ~v k 2 2.33 Showthat k ~u k = k ~v k ifandonlyif ~u + ~v and ~u )]TJ/F32 8.9664 Tf 8.909 0 Td [(~v areperpendicular.Give anexamplein R 2 2.34 Showthatifavectorisperpendiculartoeachoftwoothersthenitisperpendiculartoeachvectorintheplanetheygenerate. Remark. Theycouldgenerate adegenerateplane|alineorapoint|butthestatementremainstrue. 2.35 Provethat,where ~u;~v 2 R n arenonzerovectors,thevector ~u k ~u k + ~v k ~v k bisectstheanglebetweenthem.Illustratein R 2 2.36 Verifythatthedenitionofangleisdimensionallycorrect:if k> 0then thecosineoftheanglebetween k~u and ~v equalsthecosineoftheanglebetween ~u and ~v ,andif k< 0thenthecosineoftheanglebetween k~u and ~v isthe negativeofthecosineoftheanglebetween ~u and ~v X 2.37 Showthattheinnerproductoperationis linear :for ~u;~v;~w 2 R n and k;m 2 R ~u k~v + m~w = k ~u ~v + m ~u ~w X 2.38 The geometricmean oftwopositivereals x;y is p xy .Itisanalogoustothe arithmeticmean x + y = 2.UsetheCauchy-Schwartzinequalitytoshowthatthe geometricmeanofany x;y 2 R islessthanorequaltothearithmeticmean. ? 2.39 Ashipissailingwithspeedanddirection ~v 1 ;thewindblowsapparently judgingbythevaneonthemastinthedirectionofavector ~a ;onchangingthe directionandspeedoftheshipfrom ~v 1 to ~v 2 theapparentwindisinthedirection ofavector ~ b Findthevectorvelocityofthewind.[Am.Math.Mon.,Feb.1933] 2.40 VerifytheCauchy-SchwartzinequalitybyrstprovingLagrange'sidentity: X 1 j n a j b j 2 = X 1 j n a 2 j X 1 j n b 2 j )]TJ/F1 9.9626 Tf 18.521 9.215 Td [(X 1 k
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SectionII.LinearGeometryof n -Space 45 ofthenotation.ThisresultisanimprovementoverCauchy-Schwartzbecause itgivesaformulaforthedierencebetweenthetwosides.Interpretthatdierence in R 2 .

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46 ChapterOne.LinearSystems IIIReducedEchelonForm AfterdevelopingthemechanicsofGauss'method,weobservedthatitcanbe doneinmorethanoneway.Oneexampleisthatwesometimeshavetoswap rowsandtherecanbemorethanonerowtochoosefrom.Anotherexampleis thatfromthismatrix 22 43 Gauss'methodcouldderiveanyoftheseechelonformmatrices. 22 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 11 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 20 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Therstresultsfrom )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 1 + 2 .Thesecondcomesfromfollowing = 2 1 with )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 1 + 2 .Thethirdcomesfrom )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 1 + 2 followedby2 2 + 1 aftertherst pivotthematrixisalreadyinechelonformsothesecondoneisextraworkbut itisnonethelessalegalrowoperation. ThefactthattheechelonformoutcomeofGauss'methodisnotunique leavesuswithsomequestions.Willanytwoechelonformversionsofasystem havethesamenumberoffreevariables?Willtheyinfacthaveexactlythesame variablesfree?Inthissectionwewillanswerbothquestionsyes".Wewill domorethananswerthequestions.Wewillgiveawaytodecideifonelinear systemcanbederivedfromanotherbyrowoperations.Theanswerstothetwo questionswillfollowfromthislargerresult. III.1Gauss-JordanReduction Gaussianeliminationcoupledwithback-substitutionsolveslinearsystems,but it'snottheonlymethodpossible.HereisanextensionofGauss'methodthat hassomeadvantages. 1.1Example Tosolve x + y )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 z = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 y +3 z =7 x )]TJ/F11 9.9626 Tf 14.944 0 Td [(z = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 wecanstartbygoingtoechelonformasusual. )]TJ/F10 6.9738 Tf 6.227 0 Td [( 1 + 3 )167(! 0 @ 11 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 013 7 0 )]TJ/F8 9.9626 Tf 7.748 0 Td [(11 1 1 A 2 + 3 )167(! 0 @ 11 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 013 7 004 8 1 A Wecankeepgoingtoasecondstagebymakingtheleadingentriesintoones = 4 3 )167(! 0 @ 11 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 013 7 001 2 1 A

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SectionIII.ReducedEchelonForm 47 andthentoathirdstagethatusestheleadingentriestoeliminateallofthe otherentriesineachcolumnbypivotingupwards. )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 3 + 2 )167(! 2 3 + 1 0 @ 110 2 010 1 001 2 1 A )]TJ/F10 6.9738 Tf 6.227 0 Td [( 2 + 1 )167(! 0 @ 100 1 010 1 001 2 1 A Theansweris x =1, y =1,and z =2. Notethatthepivotoperationsintherststageproceedfromcolumnoneto columnthreewhilethepivotoperationsinthethirdstageproceedfromcolumn threetocolumnone. 1.2Example Weoftencombinetheoperationsofthemiddlestageintoa singlestep,eventhoughtheyareoperationsondierentrows. 21 7 4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 6 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 1 + 2 )167(! 21 7 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 = 2 1 )167(! )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 4 2 11 = 2 7 = 2 01 2 )]TJ/F7 6.9738 Tf 6.226 0 Td [( = 2 2 + 1 )167(! 10 5 = 2 01 2 Theansweris x =5 = 2and y =2. ThisextensionofGauss'methodis Gauss-Jordanreduction .Itgoespast echelonformtoamorerened,morespecialized,matrixform. 1.3Denition Amatrixisin reducedechelonform if,inadditiontobeing inechelonform,eachleadingentryisaoneandistheonlynonzeroentryin itscolumn. ThedisadvantageofusingGauss-Jordanreductiontosolveasystemisthatthe additionalrowoperationsmeanadditionalarithmetic.Theadvantageisthat thesolutionsetcanjustbereado. Inanyechelonform,plainorreduced,wecanreadowhenasystemhas anemptysolutionsetbecausethereisacontradictoryequation,wecanreado whenasystemhasaone-elementsolutionsetbecausethereisnocontradiction andeveryvariableistheleadingvariableinsomerow,andwecanreadowhen asystemhasaninnitesolutionsetbecausethereisnocontradictionandat leastonevariableisfree. Inreducedechelonformwecanreadonotjustwhatkindofsolutionset thesystemhas,butalsoitsdescription.Whetherornottheechelonform isreduced,wehavenotroubledescribingthesolutionsetwhenitisempty, ofcourse.Thetwoexamplesaboveshowthatwhenthesystemhasasingle solutionthenthesolutioncanbereadofromtheright-handcolumn.Inthe casewhenthesolutionsetisinnite,itsparametrizationcanalsobereado

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48 ChapterOne.LinearSystems ofthereducedechelonform.Consider,forexample,thissystemthatisshown broughttoechelonformandthentoreducedechelonform. 0 @ 2612 5 0314 1 0312 5 1 A )]TJ/F10 6.9738 Tf 6.227 0 Td [( 2 + 3 )167(! 0 @ 2612 5 0314 1 000 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 4 1 A = 2 1 )167(! = 3 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [( = 2 3 = 3 3 + 2 )167(! )]TJ/F10 6.9738 Tf 6.227 0 Td [( 3 + 1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 2 + 1 )167(! 0 @ 10 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 20 )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 = 2 011 = 30 3 0001 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 1 A Startingwiththemiddlematrix,theechelonformversion,backsubstitution produces )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 x 4 =4sothat x 4 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2,thenanotherbacksubstitutiongives 3 x 2 + x 3 +4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2=1implyingthat x 2 =3 )]TJ/F8 9.9626 Tf 10.952 0 Td [( = 3 x 3 ,andthenthenal backsubstitutiongives2 x 1 +6 )]TJ/F8 9.9626 Tf 10.425 0 Td [( = 3 x 3 + x 3 +2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2=5implyingthat x 1 = )]TJ/F8 9.9626 Tf 7.749 0 Td [( = 2+ = 2 x 3 .Thusthesolutionsetisthis. S = f 0 B B @ x 1 x 2 x 3 x 4 1 C C A = 0 B B @ )]TJ/F8 9.9626 Tf 7.748 0 Td [(9 = 2 3 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 1 C C A + 0 B B @ 1 = 2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 = 3 1 0 1 C C A x 3 x 3 2 R g Now,consideringthenalmatrix,thereducedechelonformversion,notethat adjustingtheparametrizationbymovingthe x 3 termstotheothersidedoes indeedgivethedescriptionofthisinnitesolutionset. Partofthereasonthatthisworksisstraightforward.Whileasetcanhave manyparametrizationsthatdescribeit,e.g.,bothofthesealsodescribethe aboveset S take t tobe x 3 = 6and s tobe x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 f 0 B B @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 = 2 3 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 1 C C A + 0 B B @ 3 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 6 0 1 C C A t t 2 R gf 0 B B @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 8 = 3 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 1 C C A + 0 B B @ 1 = 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 3 1 0 1 C C A s s 2 R g nonethelesswehaveinthisbookstucktoaconventionofparametrizingusing theunmodiedfreevariablesthatis, x 3 = x 3 insteadof x 3 =6 t .Wecan easilyseethatareducedechelonformversionofasystemisequivalenttoa parametrizationintermsofunmodiedfreevariables.Forinstance, x 1 =4 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x 3 x 2 =3 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 3 0 @ 102 4 011 3 000 0 1 A tomovefromlefttorightwealsoneedtoknowhowmanyequationsareinthe system.So,theconventionofparametrizingwiththefreevariablesbysolving eachequationforitsleadingvariableandtheneliminatingthatleadingvariable fromeveryotherequationisexactlyequivalenttothereducedechelonform conditionsthateachleadingentrymustbeaoneandmustbetheonlynonzero entryinitscolumn.

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SectionIII.ReducedEchelonForm 49 Notasstraightforwardistheotherpartofthereasonthatthereduced echelonformversionallowsustoreadotheparametrizationthatwewould havegottenhadwestoppedatechelonformandthendonebacksubstitution. Thepriorparagraphshowsthatreducedechelonformcorrespondstosome parametrization,butwhythesameparametrization?Asolutionsetcanbe parametrizedinmanyways,andGauss'methodortheGauss-Jordanmethod canbedoneinmanyways,soarstguessmightbethatwecouldderivemany dierentreducedechelonformversionsofthesamestartingsystemandmany dierentparametrizations.Butweneverdo.Experienceshowsthatstarting withthesamesystemandproceedingwithrowoperationsinmanydierent waysalwaysyieldsthesamereducedechelonformandthesameparametrization usingtheunmodiedfreevariables. Intherestofthissectionwewillshowthatthereducedechelonformversion ofamatrixisunique.Itfollowsthattheparametrizationofalinearsystemin termsofitsunmodiedfreevariablesisuniquebecausetwodierentoneswould givetwodierentreducedechelonforms. Weshallusethisresult,andtheonesthatleaduptoit,intherestofthe bookbutperhapsarestatementinawaythatmakesitseemmoreimmediately usefulmaybeencouraging.Imaginethatwesolvealinearsystem,parametrize, andcheckinthebackofthebookfortheanswer.Buttheparametrizationthere appearsdierent.Havewemadeamistake,orcouldthesebedierent-looking descriptionsofthesameset,aswiththethreedescriptionsaboveof S ?Theprior paragraphnotesthatwewillshowherethatdierent-lookingparametrizations usingtheunmodiedfreevariablesdescribegenuinelydierentsets. Hereisaninformalargumentthatthereducedechelonformversionofa matrixisunique.Consideragaintheexamplethatstartedthissectionofa matrixthatreducestothreedierentechelonformmatrices.Therstmatrix ofthethreeisthenaturalechelonformversion.Thesecondmatrixisthesame astherstexceptthatarowhasbeenhalved.Thethirdmatrix,too,isjusta cosmeticvariantoftherst.Thedenitionofreducedechelonformoutlawsthis kindoffoolingaround.Inreducedechelonform,halvingarowisnotpossible becausethatwouldchangetherow'sleadingentryawayfromone,andneither iscombiningrowspossible,becausethenaleadingentrywouldnolongerbe aloneinitscolumn. Thisinformaljusticationisnotaproof;wehavearguedthatnotwodierent reducedechelonformmatricesarerelatedbyasinglerowoperationstep,but wehavenotruledoutthepossibilitythatmultiplestepsmightdo.Beforewego tothatproof,wenishthissubsectionbyrephrasingourworkinaterminology thatwillbeenlightening. Manydierentmatricesyieldthesamereducedechelonformmatrix.The threeechelonformmatricesfromthestartofthissection,andthematrixthey werederivedfrom,allgivethisreducedechelonformmatrix. 10 01 Wethinkofthesematricesasrelatedtoeachother.Thenextresultspeaksto

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50 ChapterOne.LinearSystems thisrelationship. 1.4Lemma Elementaryrowoperationsarereversible. Proof Foranymatrix A ,theeectofswappingrowsisreversedbyswapping themback,multiplyingarowbyanonzero k isundonebymultiplyingby1 =k andaddingamultipleofrow i torow j with i 6 = j isundonebysubtracting thesamemultipleofrow i fromrow j A i $ j )167(! j $ i )167(! AA k i )167(! =k i )167(! AA k i + j )167(! )]TJ/F10 6.9738 Tf 6.227 0 Td [(k i + j )167(! A The i 6 = j conditionsisneeded.SeeExercise13. QED Thislemmasuggeststhat`reducesto'ismisleading|where A )167(! B ,we shouldn'tthinkof B asafter" A orsimplerthan" A .Insteadweshouldthink ofthemasinterreducibleorinterrelated.Belowisapictureoftheidea.The matricesfromthestartofthissectionandtheirreducedechelonformversion areshowninacluster.Theyareallinterreducible;theserelationshipsareshown also. 10 01 22 43 20 0 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 11 0 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 22 0 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Wesaythatmatricesthatreducetoeachotherare`equivalentwithrespect totherelationshipofrowreducibility'.Thenextresultveriesthisstatement usingthedenitionofanequivalence. 1.5Lemma Betweenmatrices,`reducesto'isanequivalencerelation. Proof Wemustchecktheconditionsireexivity,thatanymatrixreducesto itself,iisymmetry,thatif A reducesto B then B reducesto A ,andiiitransitivity,thatif A reducesto B and B reducesto C then A reducesto C Reexivityiseasy;anymatrixreducestoitselfinzerorowoperations. ThattherelationshipissymmetricisLemma1.4|if A reducesto B by somerowoperationsthenalso B reducesto A byreversingthoseoperations. Fortransitivity,supposethat A reducesto B andthat B reducesto C Linkingthereductionstepsfrom A !! B withthosefrom B !! C givesareductionfrom A to C QED 1.6Denition Twomatricesthatareinterreduciblebytheelementaryrow operationsare rowequivalent Moreinformationonequivalencerelationsisintheappendix.

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SectionIII.ReducedEchelonForm 51 Thediagrambelowshowsthecollectionofallmatricesasabox.Insidethat box,eachmatrixliesinsomeclass.Matricesareinthesameclassifandonlyif theyareinterreducible.Theclassesaredisjoint|nomatrixisintwodistinct classes.Thecollectionofmatriceshasbeenpartitionedinto rowequivalence classes ... A B Oneoftheclassesinthispartitionistheclusterofmatricesshownabove, expandedtoincludeallofthenonsingular2 2matrices. Thenextsubsectionprovesthatthereducedechelonformofamatrixis unique;thateverymatrixreducestooneandonlyonereducedechelonform matrix.Rephrasedintermsoftherow-equivalencerelationship,weshallprove thateverymatrixisrowequivalenttooneandonlyonereducedechelonform matrix.Intermsofthepartitionwhatweshallproveis:everyequivalence classcontainsoneandonlyonereducedechelonformmatrix.Soeachreduced echelonformmatrixservesasarepresentativeofitsclass. Afterthatproofweshall,asmentionedintheintroductiontothissection, haveawaytodecideifonematrixcanbederivedfromanotherbyrowreduction. WejustapplytheGauss-Jordanproceduretobothandseewhetherornotthey cometothesamereducedechelonform. Exercises X 1.7 UseGauss-Jordanreductiontosolveeachsystem. a x + y =2 x )]TJ/F32 8.9664 Tf 9.215 0 Td [(y =0 b x )]TJ/F32 8.9664 Tf 9.215 0 Td [(z =4 2 x +2 y =1 c 3 x )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 y =1 6 x + y =1 = 2 d 2 x )]TJ/F32 8.9664 Tf 13.823 0 Td [(y = )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 x +3 y )]TJ/F32 8.9664 Tf 13.823 0 Td [(z =5 y +2 z =5 X 1.8 Findthereducedechelonformofeachmatrix. a 21 13 b 131 204 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(3 )]TJ/F29 8.9664 Tf 7.168 0 Td [(3 c 10312 14215 34812 d 0132 0056 1515 X 1.9 FindeachsolutionsetbyusingGauss-Jordanreduction,thenreadingothe parametrization. a 2 x + y )]TJ/F32 8.9664 Tf 9.215 0 Td [(z =1 4 x )]TJ/F32 8.9664 Tf 9.215 0 Td [(y =3 b x )]TJ/F32 8.9664 Tf 13.823 0 Td [(z =1 y +2 z )]TJ/F32 8.9664 Tf 9.215 0 Td [(w =3 x +2 y +3 z )]TJ/F32 8.9664 Tf 9.215 0 Td [(w =7 c x )]TJ/F32 8.9664 Tf 16.383 0 Td [(y + z =0 y + w =0 3 x )]TJ/F29 8.9664 Tf 11.775 0 Td [(2 y +3 z + w =0 )]TJ/F32 8.9664 Tf 7.168 0 Td [(y )]TJ/F32 8.9664 Tf 9.216 0 Td [(w =0 d a +2 b +3 c + d )]TJ/F32 8.9664 Tf 9.215 0 Td [(e =1 3 a )]TJ/F32 8.9664 Tf 13.823 0 Td [(b + c + d + e =3 Moreinformationonpartitionsandclassrepresentativesisintheappendix.

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52 ChapterOne.LinearSystems 1.10 Givetwodistinctechelonformversionsofthismatrix. 2113 6412 1515 X 1.11 Listthereducedechelonformspossibleforeachsize. a 2 2 b 2 3 c 3 2 d 3 3 X 1.12 WhatresultsfromapplyingGauss-Jordanreductiontoanonsingularmatrix? 1.13 TheproofofLemma1.4containsareferencetothe i 6 = j conditiononthe rowpivotingoperation. a Thedenitionofrowoperationshasan i 6 = j conditionontheswapoperation i $ j .Showthatin A i $ j )171(! i $ j )171(! A thisconditionisnotneeded. b Writedowna2 2matrixwithnonzeroentries,andshowthatthe )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 1 + 1 operationisnotreversedby1 1 + 1 c Expandtheproofofthatlemmatomakeexplicitexactlywherethe i 6 = j conditiononpivotingisused. III.2RowEquivalence Wewillclosethissectionandthischapterbyprovingthateverymatrixisrow equivalenttooneandonlyonereducedechelonformmatrix.Theideasthat appearherewillreappear,andbefurtherdeveloped,inthenextchapter. Theunderlyingthemehereisthatonewaytounderstandamathematical situationisbybeingabletoclassifythecasesthatcanhappen.Wehavemetthis themeseveraltimesalready.Wehaveclassiedsolutionsetsoflinearsystems intotheno-elements,one-element,andinnitely-manyelementscases.Wehave alsoclassiedlinearsystemswiththesamenumberofequationsasunknowns intothenonsingularandsingularcases.Weadoptedtheseclassicationsbecause theygiveusawaytounderstandthesituationsthatwewereinvestigating.Here, whereweareinvestigatingrowequivalence,weknowthatthesetofallmatrices breaksintotherowequivalenceclasses.Whenwenishtheproofhere,wewill haveawaytounderstandeachofthoseclasses|itsmatricescanbethoughtof asderivedbyrowoperationsfromtheuniquereducedechelonformmatrixin thatclass. Tounderstandhowrowoperationsacttotransformonematrixintoanother, weconsidertheeectthattheyhaveonthepartsofamatrix.Thecrucial observationisthatrowoperationscombinetherowslinearly. 2.1Denition A linearcombination of x 1 ;:::;x m isanexpressionofthe form c 1 x 1 + c 2 x 2 + + c m x m wherethe c 'sarescalars. Wehavealreadyusedthephrase`linearcombination'inthisbook.Themeaningisunchanged,butthenextresult'sstatementmakesamoreformaldenition inorder.

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SectionIII.ReducedEchelonForm 53 2.2LemmaLinearCombinationLemma Alinearcombinationoflinear combinationsisalinearcombination. Proof Giventhelinearcombinations c 1 ; 1 x 1 + + c 1 ;n x n through c m; 1 x 1 + + c m;n x n ,consideracombinationofthose d 1 c 1 ; 1 x 1 + + c 1 ;n x n + + d m c m; 1 x 1 + + c m;n x n wherethe d 'sarescalarsalongwiththe c 's.Distributingthose d 'sandregroupinggives = d 1 c 1 ; 1 + + d m c m; 1 x 1 + + d 1 c 1 ;n + + d m c m;n x n whichisalinearcombinationofthe x 's. QED Inthissubsectionwewillusetheconventionthat,whereamatrixisnamed withanuppercaseromanletter,thematchinglower-casegreekletternames therows. A = 0 B B B B @ 1 2 m 1 C C C C A B = 0 B B B B @ 1 2 m 1 C C C C A 2.3Corollary Whereonematrixreducestoanother,eachrowofthesecond isalinearcombinationoftherowsoftherst. Theproofbelowusesinductiononthenumberofrowoperationsusedto reduceonematrixtotheother.Beforeweproceed,hereisanoutlineoftheargumentreadersunfamiliarwithinductionmaywanttocomparethisargument withtheoneusedinthe`General=Particular+Homogeneous'proof. First, forthebasestepoftheargument,wewillverifythatthepropositionistrue whenreductioncanbedoneinzerorowoperations.Second,fortheinductive step,wewillarguethatifbeingabletoreducetherstmatrixtothesecond insomenumber t 0ofoperationsimpliesthateachrowofthesecondisa linearcombinationoftherowsoftherst,thenbeingabletoreducetherstto thesecondin t +1operationsimpliesthesamething.Together,thisbasestep andinductionstepprovethisresultbecausebythebasesteptheproposition istrueinthezerooperationscase,andbytheinductivestepthefactthatitis trueinthezerooperationscaseimpliesthatitistrueintheoneoperationcase, andtheinductivestepappliedagaingivesthatitisthereforetrueinthetwo operationscase,etc. Proof Weproceedbyinductionontheminimumnumberofrowoperations thattakearstmatrix A toasecondone B Moreinformationonmathematicalinductionisintheappendix.

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54 ChapterOne.LinearSystems Inthebasestep,thatzeroreductionoperationssuce,thetwomatrices areequalandeachrowof B isobviouslyacombinationof A 'srows: ~ i = 0 ~ 1 + +1 ~ i + +0 ~ m Fortheinductivestep,assumetheinductivehypothesis:with t 0,ifa matrixcanbederivedfrom A in t orfeweroperationsthenitsrowsarelinear combinationsofthe A 'srows.Considera B thattakes t +1operations.Because therearemorethanzerooperations,theremustbeanext-to-lastmatrix G so that A )167(!)167(! G )167(! B .This G isonly t operationsawayfrom A andsothe inductivehypothesisappliestoit,thatis,eachrowof G isalinearcombination oftherowsof A Ifthelastoperation,theonefrom G to B ,isarowswapthentherows of B arejusttherowsof G reorderedandthuseachrowof B isalsoalinear combinationoftherowsof A .Theothertwopossibilitiesforthislastoperation, thatitmultipliesarowbyascalarandthatitaddsamultipleofonerowto another,bothresultintherowsof B beinglinearcombinationsoftherowsof G .Buttherefore,bytheLinearCombinationLemma,eachrowof B isalinear combinationoftherowsof A Withthat,wehaveboththebasestepandtheinductivestep,andsothe propositionfollows. QED 2.4Example Inthereduction 02 11 1 $ 2 )167(! 11 02 = 2 2 )167(! 11 01 )]TJ/F10 6.9738 Tf 6.227 0 Td [( 2 + 1 )167(! 10 01 callthematrices A D G ,and B .Themethodsoftheproofshowthatthere arethreesetsoflinearrelationships. 1 =0 1 +1 2 2 =1 1 +0 2 1 =0 1 +1 2 2 = = 2 1 +0 2 1 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 = 2 1 +1 2 2 = = 2 1 +0 2 ThepriorresultgivesustheinsightthatGauss'methodworksbytaking linearcombinationsoftherows.Buttowhatend;whydowegotoechelon formasaparticularlysimple,orbasic,versionofalinearsystem?Theanswer, ofcourse,isthatechelonformissuitableforbacksubstitution,becausewehave isolatedthevariables.Forinstance,inthismatrix R = 0 B B @ 237800 001511 000330 000021 1 C C A x 1 hasbeenremovedfrom x 5 'sequation.Thatis,Gauss'methodhasmade x 5 's rowindependentof x 1 'srow. Independenceofacollectionofrowvectors,orofanykindofvectors,will bepreciselydenedandexploredinthenextchapter.Butarsttakeonitis thatwecanshowthat,say,thethirdrowaboveisnotcomprisedoftheother

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SectionIII.ReducedEchelonForm 55 rows,that 3 6 = c 1 1 + c 2 2 + c 4 4 .For,supposethattherearescalars c 1 c 2 and c 4 suchthatthisrelationshipholds. )]TJ/F8 9.9626 Tf 4.566 -7.971 Td [(000330 = c 1 )]TJ/F8 9.9626 Tf 4.566 -7.971 Td [(237800 + c 2 )]TJ/F8 9.9626 Tf 4.566 -7.97 Td [(001511 + c 4 )]TJ/F8 9.9626 Tf 4.566 -7.97 Td [(000021 Therstrow'sleadingentryisintherstcolumnandnarrowingourconsiderationoftheaboverelationshiptoconsiderationonlyoftheentriesfromtherst column0=2 c 1 +0 c 2 +0 c 4 givesthat c 1 =0.Thesecondrow'sleadingentryisin thethirdcolumnandtheequationofentriesinthatcolumn0=7 c 1 +1 c 2 +0 c 4 alongwiththeknowledgethat c 1 =0,givesthat c 2 =0.Now,tonish,the thirdrow'sleadingentryisinthefourthcolumnandtheequationofentries inthatcolumn3=8 c 1 +5 c 2 +0 c 4 ,alongwith c 1 =0and c 2 =0,givesan impossibility. Thefollowingresultshowsthatthiseectalwaysholds.Itshowsthatwhat Gauss'lineareliminationmethodeliminatesislinearrelationshipsamongthe rows. 2.5Lemma Inanechelonformmatrix,nononzerorowisalinearcombination oftheotherrows. Proof Let R beinechelonform.Suppose,toobtainacontradiction,that somenonzerorowisalinearcombinationoftheothers. i = c 1 1 + ::: + c i )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 i )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 + c i +1 i +1 + ::: + c m m Wewillrstuseinductiontoshowthatthecoecients c 1 ,..., c i )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 associated withrowsabove i areallzero.Thecontradictionwillcomefromconsideration of i andtherowsbelowit. Thebasestepoftheinductionargumentistoshowthattherstcoecient c 1 iszero.Lettherstrow'sleadingentrybeincolumnnumber ` 1 andconsider theequationofentriesinthatcolumn. i;` 1 = c 1 1 ;` 1 + ::: + c i )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 i )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 ;` 1 + c i +1 i +1 ;` 1 + ::: + c m m;` 1 Thematrixisinechelonformsotheentries 2 ;` 1 ,..., m;` 1 ,including i;` 1 ,are allzero. 0= c 1 1 ;` 1 + + c i )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 0+ c i +1 0+ + c m 0 Becausetheentry 1 ;` 1 isnonzeroasitleadsitsrow,thecoecient c 1 mustbe zero. Theinductivestepistoshowthatforeachrowindex k between1and i )]TJ/F8 9.9626 Tf 9.291 0 Td [(2, ifthecoecient c 1 andthecoecients c 2 ,..., c k areallzerothen c k +1 isalso zero.Thatargument,andthecontradictionthatnishesthisproof,issavedfor Exercise21. QED

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56 ChapterOne.LinearSystems Wecannowprovethateachmatrixisrowequivalenttooneandonlyone reducedechelonformmatrix.Wewillnditconvenienttobreakthersthalf oftheargumentoasapreliminarylemma.Foronething,itholdsforany echelonformwhatever,notjustreducedechelonform. 2.6Lemma Iftwoechelonformmatricesarerowequivalentthentheleading entriesintheirrstrowslieinthesamecolumn.Thesameistrueofallthe nonzerorows|theleadingentriesintheirsecondrowslieinthesamecolumn, etc. Fortheproofwerephrasetheresultinmoretechnicalterms.Denethe form ofan m n matrixtobethesequence h ` 1 ;` 2 ;:::;` m i where ` i isthecolumn numberoftheleadingentryinrow i and ` i = 1 ifthereisnoleadingentryin thatrow.Thelemmasaysthatiftwoechelonformmatricesarerowequivalent thentheirformsareequalsequences. Proof Let B and D beechelonformmatricesthatarerowequivalent.Because theyarerowequivalenttheymustbethesamesize,say m n .Letthecolumn numberoftheleadingentryinrow i of B be ` i andletthecolumnnumberof theleadingentryinrow j of D be k j .Wewillshowthat ` 1 = k 1 ,that ` 2 = k 2 etc.,byinduction. Thisinductionargumentreliesonthefactthatthematricesarerowequivalent,becausetheLinearCombinationLemmaanditscorollarythereforegive thateachrowof B isalinearcombinationoftherowsof D andviceversa: i = s i; 1 1 + s i; 2 2 + + s i;m m and j = t j; 1 1 + t j; 2 2 + + t j;m m wherethe s 'sand t 'sarescalars. Thebasestepoftheinductionistoverifythelemmafortherstrowsof thematrices,thatis,toverifythat ` 1 = k 1 .Ifeitherrowisazerorowthen theentirematrixisazeromatrixsinceitisinechelonform,andthereforeboth matricesarezeromatricesbyCorollary2.3,andsoboth ` 1 and k 1 are 1 .For thecasewhereneither 1 nor 1 isazerorow,considerthe i =1instanceof thelinearrelationshipabove. 1 = s 1 ; 1 1 + s 1 ; 2 2 + + s 1 ;m m )]TJ/F8 9.9626 Tf 4.566 -7.97 Td [(0 b 1 ;` 1 = s 1 ; 1 )]TJ/F8 9.9626 Tf 4.566 -7.97 Td [(0 d 1 ;k 1 + s 1 ; 2 )]TJ/F8 9.9626 Tf 4.567 -7.97 Td [(0 0 + s 1 ;m )]TJ/F8 9.9626 Tf 4.566 -7.971 Td [(0 0 First,notethat ` 1
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SectionIII.ReducedEchelonForm 57 Theinductivestepistoshowthatif ` 1 = k 1 ,and ` 2 = k 2 ,...,and ` r = k r thenalso ` r +1 = k r +1 for r intheinterval1 ::m )]TJ/F8 9.9626 Tf 10.047 0 Td [(1.Thisargumentissaved forExercise22. QED Thatlemmaanswerstwoofthequestionsthatwehaveposed:ianytwo echelonformversionsofamatrixhavethesamefreevariables,andconsequently, andiianytwoechelonformversionshavethesamenumberoffreevariables. Thereisnolinearsystemandnocombinationofrowoperationssuchthat,say, wecouldsolvethesystemonewayandget y and z freebutsolveitanother wayandget y and w free,orsolveitonewayandgettwofreevariableswhile solvingitanotherwayyieldsthree. Wenishnowbyspecializingtothecaseofreducedechelonformmatrices. 2.7Theorem Eachmatrixisrowequivalenttoauniquereducedechelon formmatrix. Proof Clearlyanymatrixisrowequivalenttoatleastonereducedechelon formmatrix,viaGauss-Jordanreduction.Fortheotherhalf,thatanymatrix isequivalenttoatmostonereducedechelonformmatrix,wewillshowthatif amatrixGauss-Jordanreducestoeachoftwoothersthenthosetwoareequal. Supposethatamatrixisrowequivalenttotworeducedechelonformmatrices B and D ,whicharethereforerowequivalenttoeachother.TheLinear CombinationLemmaanditscorollaryallowustowritetherowsofone,say B ,asalinearcombinationoftherowsoftheother i = c i; 1 1 + + c i;m m Thepreliminaryresult,Lemma2.6,saysthatinthetwomatrices,thesame collectionofrowsarenonzero.Thus,if 1 through r arethenonzerorowsof B thenthenonzerorowsof D are 1 through r .Zerorowsdon'tcontributeto thesumsowecanrewritetherelationshiptoincludejustthenonzerorows. i = c i; 1 1 + + c i;r r Thepreliminaryresultalsosaysthatforeachrow j between1and r ,the leadingentriesofthe j -throwof B and D appearinthesamecolumn,denoted ` j .Rewritingtheaboverelationshiptofocusontheentriesinthe ` j -thcolumn )]TJ/F14 9.9626 Tf 14.528 -7.97 Td [( b i;` j = c i; 1 )]TJ/F14 9.9626 Tf 14.529 -7.97 Td [( d 1 ;` j + c i; 2 )]TJ/F14 9.9626 Tf 14.529 -7.97 Td [( d 2 ;` j + c i;r )]TJ/F14 9.9626 Tf 14.529 -7.97 Td [( d r;` j givesthissetofequationsfor i =1upto i = r b 1 ;` j = c 1 ; 1 d 1 ;` j + + c 1 ;j d j;` j + + c 1 ;r d r;` j b j;` j = c j; 1 d 1 ;` j + + c j;j d j;` j + + c j;r d r;` j b r;` j = c r; 1 d 1 ;` j + + c r;j d j;` j + + c r;r d r;` j

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58 ChapterOne.LinearSystems Since D isinreducedechelonform,allofthe d 'sincolumn ` j arezeroexceptfor d j;` j ,whichis1.Thuseachequationabovesimpliesto b i;` j = c i;j d j;` j = c i;j 1. But B isalsoinreducedechelonformandsoallofthe b 'sincolumn ` j arezero exceptfor b j;` j ,whichis1.Therefore,each c i;j iszero,exceptthat c 1 ; 1 =1, and c 2 ; 2 =1,...,and c r;r =1. Wehaveshownthattheonlynonzerocoecientinthelinearcombination labelled is c j;j ,whichis1.Therefore j = j .Becausethisholdsforall nonzerorows, B = D QED Weendwitharecap.InGauss'methodwestartwithamatrixandthen deriveasequenceofothermatrices.Wedenedtwomatricestoberelatedifone canbederivedfromtheother.Thatrelationisanequivalencerelation,called rowequivalence,andsopartitionsthesetofallmatricesintorowequivalence classes. ... )]TJ/F6 4.9813 Tf 5.789 -4.98 Td [(13 27 )]TJ/F6 4.9813 Tf 5.788 -4.981 Td [(13 01 Thereareinnitelymanymatricesinthepicturedclass,butwe'veonlygot roomtoshowtwo.Wehaveprovedthereisoneandonlyonereducedechelon formmatrixineachrowequivalenceclass.Sothereducedechelonformisa canonicalform forrowequivalence:thereducedechelonformmatricesare representativesoftheclasses. ... ? ? ? ? )]TJ/F6 4.9813 Tf 5.788 -4.98 Td [(10 01 Wecananswerquestionsabouttheclassesbytranslatingthemintoquestions abouttherepresentatives. 2.8Example WecandecideifmatricesareinterreduciblebyseeingifGaussJordanreductionproducesthesamereducedechelonformresult.Thus,these arenotrowequivalent 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 )]TJ/F8 9.9626 Tf 7.748 0 Td [(26 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(25 Moreinformationoncanonicalrepresentativesisintheappendix.

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SectionIII.ReducedEchelonForm 59 becausetheirreducedechelonformsarenotequal. 1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 00 10 01 2.9Example Anynonsingular3 3matrixGauss-Jordanreducestothis. 0 @ 100 010 001 1 A 2.10Example Wecandescribetheclassesbylistingallpossiblereduced echelonformmatrices.Any2 2matrixliesinoneofthese:theclassofmatrices rowequivalenttothis, 00 00 theinnitelymanyclassesofmatricesrowequivalenttooneofthistype 1 a 00 where a 2 R including a =0,theclassofmatricesrowequivalenttothis, 01 00 andtheclassofmatricesrowequivalenttothis 10 01 thisistheclassofnonsingular2 2matrices. Exercises X 2.11 Decideifthematricesarerowequivalent. a 12 48 ; 01 12 b 102 3 )]TJ/F29 8.9664 Tf 7.167 0 Td [(11 5 )]TJ/F29 8.9664 Tf 7.167 0 Td [(15 ; 102 0210 204 c 21 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 110 43 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 ; 102 0210 d 111 )]TJ/F29 8.9664 Tf 7.167 0 Td [(122 ; 03 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 225 e 111 003 ; 012 1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(11 2.12 DescribethematricesineachoftheclassesrepresentedinExample2.10. 2.13 Describeallmatricesintherowequivalenceclassofthese.

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60 ChapterOne.LinearSystems a 10 00 b 12 24 c 11 13 2.14 Howmanyrowequivalenceclassesarethere? 2.15 Canrowequivalenceclassescontaindierent-sizedmatrices? 2.16 Howbigaretherowequivalenceclasses? a Showthattheclassofanyzeromatrixisnite. b Doanyotherclassescontainonlynitelymanymembers? X 2.17 Givetworeducedechelonformmatricesthathavetheirleadingentriesinthe samecolumns,butthatarenotrowequivalent. X 2.18 Showthatanytwo n n nonsingularmatricesarerowequivalent.Areany twosingularmatricesrowequivalent? X 2.19 Describealloftherowequivalenceclassescontainingthese. a 2 2matrices b 2 3matrices c 3 2matrices d 3 3matrices 2.20a Showthatavector ~ 0 isalinearcombinationofmembersoftheset f ~ 1 ;:::; ~ n g ifandonlyifthereisalinearrelationship ~ 0= c 0 ~ 0 + + c n ~ n where c 0 isnotzero. Hint. Watchoutforthe ~ 0 = ~ 0case. b UsethattosimplifytheproofofLemma2.5. X 2.21 FinishtheproofofLemma2.5. a Firstillustratetheinductivestepbyshowingthat c 2 =0. b Dothefullinductivestep:where1 n
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SectionIII.ReducedEchelonForm 61 Cananyequationbederivedfromaninconsistentsystem? 2.27 Extendthedenitionofrowequivalencetolinearsystems.Underyourdenition,doequivalentsystemshavethesamesolutionset?[Homan&Kunze] X 2.28 Inthismatrix 123 303 145 therstandsecondcolumnsaddtothethird. a Showthatremainstrueunderanyrowoperation. b Makeaconjecture. c Provethatitholds.

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62 ChapterOne.LinearSystems Topic:ComputerAlgebraSystems Thelinearsystemsinthischapteraresmallenoughthattheirsolutionbyhand iseasy.Butlargesystemsareeasiest,andsafest,todoonacomputer.There arespecialpurposeprogramssuchasLINPACKforthisjob.Anotherpopular toolisageneralpurposecomputeralgebrasystem,includingbothcommercial packagessuchasMaple,Mathematica,orMATLAB,orfreepackagessuchas SciLab,,MuPAD,orOctave. Forexample,intheTopiconNetworks,weneedtosolvethis. i 0 )]TJ/F11 9.9626 Tf 14.944 0 Td [(i 1 )]TJ/F11 9.9626 Tf 14.944 0 Td [(i 2 =0 i 1 )]TJ/F11 9.9626 Tf 19.925 0 Td [(i 3 )]TJ/F11 9.9626 Tf 19.925 0 Td [(i 5 =0 i 2 )]TJ/F11 9.9626 Tf 14.944 0 Td [(i 4 + i 5 =0 i 3 + i 4 )]TJ/F11 9.9626 Tf 9.963 0 Td [(i 6 =0 5 i 1 +10 i 3 =10 2 i 2 +4 i 4 =10 5 i 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 i 2 +50 i 5 =0 Itcanbedonebyhand,butitwouldtakeawhileandbeerror-prone.Usinga computerisbetter. WeillustratebysolvingthatsystemunderMapleforanothersystem,a user'smanualwouldobviouslydetailtheexactsyntaxneeded.Thearrayof coecientscanbeenteredinthisway >A:=array[[1,-1,-1,0,0,0,0], [0,1,0,-1,0,-1,0], [0,0,1,0,-1,1,0], [0,0,0,1,1,0,-1], [0,5,0,10,0,0,0], [0,0,2,0,4,0,0], [0,5,-1,0,0,10,0]]; puttingtherowsonseparatelinesisnotnecessary,butisdoneforclarity. Thevectorofconstantsisenteredsimilarly. >u:=array[0,0,0,0,10,10,0]; Thenthesystemissolved,likemagic. >linsolveA,u; 725257 [-,-,-,-,-,0,-] 333333 Systemswithinnitelymanysolutionsaresolvedinthesameway|thecomputersimplyreturnsaparametrization. Exercises AnswersforthisTopicuseMapleasthecomputeralgebrasystem.Inparticular, alloftheseweretestedonMaple V runningunderMS-DOSNTversion 4 : 0 .On allofthem,thepreliminarycommandtoloadthelinearalgebrapackagealongwith Maple'sresponsestotheEnterkey,havebeenomitted.Othersystemshavesimilar commands.

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Topic:ComputerAlgebraSystems 63 1 Usethecomputertosolvethetwoproblemsthatopenedthischapter. a ThisistheStaticsproblem. 40 h +15 c =100 25 c =50+50 h b ThisistheChemistryproblem. 7 h =7 j 8 h +1 i =5 j +2 k 1 i =3 j 3 i =6 j +1 k 2 Usethecomputertosolvethesesystemsfromtherstsubsection,orconclude `manysolutions'or`nosolutions'. a 2 x +2 y =5 x )]TJ/F29 8.9664 Tf 9.215 0 Td [(4 y =0 b )]TJ/F32 8.9664 Tf 7.168 0 Td [(x + y =1 x + y =2 c x )]TJ/F29 8.9664 Tf 9.215 0 Td [(3 y + z =1 x + y +2 z =14 d )]TJ/F32 8.9664 Tf 7.167 0 Td [(x )]TJ/F32 8.9664 Tf 13.823 0 Td [(y =1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(3 x )]TJ/F29 8.9664 Tf 9.215 0 Td [(3 y =2 e 4 y + z =20 2 x )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 y + z =0 x + z =5 x + y )]TJ/F32 8.9664 Tf 9.215 0 Td [(z =10 f 2 x + z + w =5 y )]TJ/F32 8.9664 Tf 9.215 0 Td [(w = )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 3 x )]TJ/F32 8.9664 Tf 13.823 0 Td [(z )]TJ/F32 8.9664 Tf 9.215 0 Td [(w =0 4 x + y +2 z + w =9 3 Usethecomputertosolvethesesystemsfromthesecondsubsection. a 3 x +6 y =18 x +2 y =6 b x + y =1 x )]TJ/F32 8.9664 Tf 9.215 0 Td [(y = )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 c x 1 + x 3 =4 x 1 )]TJ/F32 8.9664 Tf 9.216 0 Td [(x 2 +2 x 3 =5 4 x 1 )]TJ/F32 8.9664 Tf 9.216 0 Td [(x 2 +5 x 3 =17 d 2 a + b )]TJ/F32 8.9664 Tf 9.215 0 Td [(c =2 2 a + c =3 a )]TJ/F32 8.9664 Tf 9.215 0 Td [(b =0 e x +2 y )]TJ/F32 8.9664 Tf 9.215 0 Td [(z =3 2 x + y + w =4 x )]TJ/F32 8.9664 Tf 13.823 0 Td [(y + z + w =1 f x + z + w =4 2 x + y )]TJ/F32 8.9664 Tf 9.215 0 Td [(w =2 3 x + y + z =7 4 Whatdoesthecomputergiveforthesolutionofthegeneral2 2system? ax + cy = p bx + dy = q

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64 ChapterOne.LinearSystems Topic:Input-OutputAnalysis Aneconomyisanimmenselycomplicatednetworkofinterdependences.Changes inonepartcanrippleouttoaectotherparts.Economistshavestruggledto beabletodescribe,andtomakepredictionsabout,suchacomplicatedobject. Mathematicalmodelsusingsystemsoflinearequationshaveemergedasakey tool.OneisInput-OutputAnalysis,pioneeredbyW.Leontief,whowonthe 1973NobelPrizeinEconomics. Consideraneconomywithmanyparts,twoofwhicharethesteelindustry andtheautoindustry.Astheyworktomeetthedemandfortheirproductfrom otherpartsoftheeconomy,thatis,fromusersexternaltothesteelandauto sectors,thesetwointeracttightly.Forinstance,shouldtheexternaldemand forautosgoup,thatwouldleadtoanincreaseintheautoindustry'susageof steel.Or,shouldtheexternaldemandforsteelfall,thenitwouldleadtoafall insteel'spurchaseofautos.ThetypeofInput-Outputmodelwewillconsider takesintheexternaldemandsandthenpredictshowthetwointeracttomeet thosedemands. Westartwithalistingofproductionandconsumptionstatistics.These numbers,givingdollarvaluesinmillions,areexcerptedfrom[Leontief1965], describingthe1958U.S.economy.Today'sstatisticswouldbequitedierent, bothbecauseofinationandbecauseoftechnicalchangesintheindustries. usedby steel usedby auto usedby otherstotal valueof steel 5395266425448 valueof auto 48903030346 Forinstance,thedollarvalueofsteelusedbytheautoindustryinthisyearis 2 ; 664million.Notethatindustriesmayconsumesomeoftheirownoutput. Wecanllintheblanksfortheexternaldemand.Thisyear'svalueofthe steelusedbyothersthisyearis17 ; 389andthisyear'svalueoftheautoused byothersis21 ; 268.Withthat,wehaveacompletedescriptionoftheexternal demandsandofhowautoandsteelinteract,thisyear,tomeetthem. Now,imaginethattheexternaldemandforsteelhasrecentlybeengoingup by200peryearandsoweestimatethatnextyearitwillbe17 ; 589.Imagine alsothatforsimilarreasonsweestimatethatnextyear'sexternaldemandfor autoswillbedown25to21 ; 243.Wewishtopredictnextyear'stotaloutputs. Thatpredictionisn'tassimpleasadding200tothisyear'ssteeltotaland subtracting25fromthisyear'sautototal.Foronething,ariseinsteelwill causethatindustrytohaveanincreaseddemandforautos,whichwillmitigate, tosomeextent,thelossinexternaldemandforautos.Ontheotherhand,the dropinexternaldemandforautoswillcausetheautoindustrytouselesssteel, andsolessensomewhattheupswinginsteel'sbusiness.Inshort,thesetwo industriesformasystem,andweneedtopredictthetotalsatwhichthesystem asawholewillsettle.

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Topic:Input-OutputAnalysis 65 Forthatprediction,let s benextyearstotalproductionofsteelandlet a be nextyear'stotaloutputofautos.Weformtheseequations. nextyear'sproductionofsteel=nextyear'suseofsteelbysteel +nextyear'suseofsteelbyauto +nextyear'suseofsteelbyothers nextyear'sproductionofautos=nextyear'suseofautosbysteel +nextyear'suseofautosbyauto +nextyear'suseofautosbyothers Ontheleftsideofthoseequationsgotheunknowns s and a .Attheendsofthe rightsidesgoourexternaldemandestimatesfornextyear17 ; 589and21 ; 243. Fortheremainingfourterms,welooktothetableofthisyear'sinformation abouthowtheindustriesinteract. Forinstance,fornextyear'suseofsteelbysteel,wenotethatthisyearthe steelindustryused5395unitsofsteelinputtoproduce25 ; 448unitsofsteel output.Sonextyear,whenthesteelindustrywillproduce s unitsout,we expectthatdoingsowilltake s = 448unitsofsteelinput|thisis simplytheassumptionthatinputisproportionaltooutput.Weareassuming thattheratioofinputtooutputremainsconstantovertime;inpractice,models maytrytotakeaccountoftrendsofchangeintheratios. Nextyear'suseofsteelbytheautoindustryissimilar.Thisyear,theauto industryuses2664unitsofsteelinputtoproduce30346unitsofautooutput.So nextyear,whentheautoindustry'stotaloutputis a ,weexpectittoconsume a = unitsofsteel. Fillingintheotherequationinthesameway,wegetthissystemoflinear equation. 5395 25448 s + 2664 30346 a +17589= s 48 25448 s + 9030 30346 a +21243= a Gauss'methodonthissystem. 053 = 25448 s )]TJ/F8 9.9626 Tf 14.944 0 Td [(664 = 30346 a =17589 )]TJ/F8 9.9626 Tf 7.748 0 Td [( = 25448 s +316 = 30346 a =21243 gives s =25698and a =30311. Lookingback,recallthatabovewedescribedwhythepredictionofnext year'stotalsisn'tassimpleasadding200tolastyear'ssteeltotalandsubtracting25fromlastyear'sautototal.Infact,comparingthesetotalsfornextyear totheonesgivenatthestartforthecurrentyearshowsthat,despitethedrop inexternaldemand,thetotalproductionoftheautoindustryispredictedto rise.Theincreaseininternaldemandforautoscausedbysteel'ssharprisein businessmorethanmakesupforthelossinexternaldemandforautos. Oneoftheadvantagesofhavingamathematicalmodelisthatwecanask Whatif...?"questions.Forinstance,wecanaskWhatiftheestimatesfor

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66 ChapterOne.LinearSystems nextyear'sexternaldemandsaresomewhato?"Totrytounderstandhow muchthemodel'spredictionschangeinreactiontochangesinourestimates,we cantryrevisingourestimateofnextyear'sexternalsteeldemandfrom17 ; 589 downto17 ; 489,whilekeepingtheassumptionofnextyear'sexternaldemand forautosxedat21 ; 243.Theresultingsystem 053 = 25448 s )]TJ/F8 9.9626 Tf 14.944 0 Td [(664 = 30346 a =17489 )]TJ/F8 9.9626 Tf 7.749 0 Td [( = 25448 s +316 = 30346 a =21243 whensolvedgives s =25571and a =30311.Thiskindofexplorationofthe modelis sensitivityanalysis .Weareseeinghowsensitivethepredictionsofour modelaretotheaccuracyoftheassumptions. Obviously,wecanconsiderlargermodelsthatdetailtheinteractionsamong moresectorsofaneconomy.Thesemodelsaretypicallysolvedonacomputer, usingthetechniquesofmatrixalgebrathatwewilldevelopinChapterThree. Someexamplesaregivenintheexercises.Obviouslyalso,asinglemodeldoes notsuiteverycase;expertjudgmentisneededtoseeiftheassumptionsunderlyingthemodelarereasonableforaparticularcase.Withthosecaveats, however,thismodelhasproveninpracticetobeausefulandaccuratetoolfor economicanalysis.Forfurtherreading,try[Leontief1951]and[Leontief1965]. Exercises Hint:thesesystemsareeasiesttosolveonacomputer. 1 Withthesteel-autosystemgivenabove,estimatenextyear'stotalproductions inthesecases. a Nextyear'sexternaldemandsare:up200fromthisyearforsteel,andunchangedforautos. b Nextyear'sexternaldemandsare:up100forsteel,andup200forautos. c Nextyear'sexternaldemandsare:up200forsteel,andup200forautos. 2 Inthesteel-autosystem,theratiofortheuseofsteelbytheautoindustryis 2664 = 30346,about0 : 0878.Imaginethatanewprocessformakingautosreduces thisratioto : 0500. a Howwillthepredictionsfornextyear'stotalproductionschangecompared totherstexamplediscussedabovei.e.,takingnextyear'sexternaldemands tobe17 ; 589forsteeland21 ; 243forautos? b Predictnextyear'stotalsif,inaddition,theexternaldemandforautosrises tobe21 ; 500becausethenewcarsarecheaper. 3 Thistablegivesthenumbersfortheauto-steelsystemfromadierentyear,1947 see[Leontief1951].Theunitsherearebillionsof1947dollars. usedby steel usedby auto usedby otherstotal valueof steel 6 : 901 : 2818 : 69 valueof autos 04 : 4014 : 27 a Solvefortotaloutputifnextyear'sexternaldemandsare:steel'sdemandup 10%andauto'sdemandup15%. b Howdotheratioscomparetothosegivenaboveinthediscussionforthe 1958economy?

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Topic:Input-OutputAnalysis 67 c Solvethe1947equationswiththe1958externaldemandsnotethedierence inunits;a1947dollarbuysaboutwhat$1 : 30in1958dollarsbuys.Howfaro arethepredictionsfortotaloutput? 4 Predictnextyear'stotalproductionsofeachofthethreesectorsofthehypotheticaleconomyshownbelow usedby farm usedby rail usedby shipping usedby otherstotal valueof farm 2550100800 valueof rail 255050300 valueof shipping 15100500 ifnextyear'sexternaldemandsareasstated. a 625forfarm,200forrail,475forshipping b 650forfarm,150forrail,450forshipping 5 Thistablegivestheinterrelationshipsamongthreesegmentsofaneconomysee [Clark&Coupe]. usedby food usedby wholesale usedby retail usedby others total valueof food 02318467911869 valueof wholesale 393108922459122242 valueof retail 35375116041 WewilldoanInput-Outputanalysisonthissystem. a Fillinthenumbersforthisyear'sexternaldemands. b Setupthelinearsystem,leavingnextyear'sexternaldemandsblank. c Solvethesystemwherenextyear'sexternaldemandsarecalculatedbytakingthisyear'sexternaldemandsandinatingthem10%.Doallthreesectors increasetheirtotalbusinessby10%?Dotheyallevenincreaseatthesamerate? d Solvethesystemwherenextyear'sexternaldemandsarecalculatedbytaking thisyear'sexternaldemandsandreducingthem7%.Thestudyfromwhich thesenumbersaretakenconcludedthatbecauseoftheclosingofalocalmilitary facility,overallpersonalincomeintheareawouldfall7%,sothismightbea rstguessatwhatwouldactuallyhappen.

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68 ChapterOne.LinearSystems Topic:AccuracyofComputations Gauss'methodlendsitselfnicelytocomputerization.Thecodebelowillustrates. Itoperatesonan n n matrix a ,pivotingwiththerstrow,thenwiththesecond row,etc. forpivot_row=1;pivot_row<=n-1;pivot_row++{ forrow_below=pivot_row+1;row_below<=n;row_below++{ multiplier=a[row_below,pivot_row]/a[pivot_row,pivot_row]; forcol=pivot_row;col<=n;col++{ a[row_below,col]-=multiplier*a[pivot_row,col]; } } } ThiscodeisintheClanguage.Hereisabrieftranslation.Theloopconstruct forpivot row=1;pivot row<=n-1;pivot row++{ } sets pivot row to1andtheniterateswhile pivot row islessthanorequalto n )]TJ/F8 9.9626 Tf 9.686 0 Td [(1,eachtime throughincrementing pivot row byonewiththe` ++ 'operation.Theother non-obviousconstructisthatthe` -= 'intheinnermostloopamountstothe a[row below,col] = )]TJ/F47 9.9626 Tf 7.748 0 Td [(multiplier a[pivot row,col] + a[row below,col] operation. WhilethiscodeprovidesaquicktakeonhowGauss'methodcanbemechanized,itisnotreadytouse.Itisnaiveinmanyways.Themostglaringwayisthatitassumesthatanonzeronumberisalwaysfoundinthe pivot row ; pivot row positionforuseasthepivotentry.Tomakeitpractical,onewayinwhichthiscodeneedstobereworkedistocoverthecasewhere ndingazerointhatlocationleadstoarowswap,ortotheconclusionthat thematrixissingular. Addingsome if statementstocoverthosecasesisnothard,butwe willinsteadconsidersomemoresubtlewaysinwhichthecodeisnaive.There arepitfallsarisingfromthecomputer'srelianceonnite-precisionoatingpoint arithmetic. Forexample,wehaveseenabovethatwemusthandleasaseparatecasea systemthatissingular.Butsystemsthatarenearlysingularalsorequirecare. Considerthisone. x +2 y =3 1 : 00000001 x +2 y =3 : 00000001 Byeyewegetthesolution x =1and y =1.Butacomputerhasmoretrouble.A computerthatrepresentsrealnumberstoeightsignicantplacesasiscommon, usuallycalled singleprecision willrepresentthesecondequationinternallyas 1 : 0000000 x +2 y =3 : 0000000,losingthedigitsintheninthplace.Insteadof reportingthecorrectsolution,thiscomputerwillreportsomethingthatisnot evenclose|thiscomputerthinksthatthesystemissingularbecausethetwo equationsarerepresentedinternallyasequal. Forsomeintuitionabouthowthecomputercouldcomeupwithsomething thatfaro,wecangraphthesystem.

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Topic:AccuracyofComputations 69 ; 1 Atthescaleofthisgraph,thetwolinescannotberesolvedapart.Thissystem isnearlysingularinthesensethatthetwolinesarenearlythesameline.Nearsingularitygivesthissystemthepropertythatasmallchangeinthesystem cancausealargechangeinitssolution;forinstance,changingthe3 : 00000001 to3 : 00000003changestheintersectionpointfrom ; 1to ; 0.Thissystem changesradicallydependingonaninthdigit,whichexplainswhytheeightplacecomputerhastrouble.Aproblemthatisverysensitivetoinaccuracyor uncertaintiesintheinputvaluesis ill-conditioned Theaboveexamplegivesonewayinwhichasystemcanbediculttosolve onacomputer.Ithastheadvantagethatthepictureofnearly-equallines givesamemorableinsightintoonewaythatnumericaldicultiescanarise. Unfortunatelythisinsightisn'tveryusefulwhenwewishtosolvesomelarge system.Wecannot,typically,hopetounderstandthegeometryofanarbitrary largesystem.Inaddition,therearewaysthatacomputer'sresultsmaybe unreliableotherthanthattheanglebetweensomeofthelinearsurfacesisquite small. Foranexample,considerthesystembelow,from[Hamming]. 0 : 001 x + y =1 x )]TJ/F11 9.9626 Tf 9.962 0 Td [(y =0 Thesecondequationgives x = y ,so x = y =1 = 1 : 001andthusbothvariables havevaluesthatarejustlessthan1.Acomputerusingtwodigitsrepresents thesysteminternallyinthiswaywewilldothisexampleintwo-digitoating pointarithmetic,butasimilaronewitheightdigitsiseasytoinvent. : 0 10 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 x + : 0 10 0 y =1 : 0 10 0 : 0 10 0 x )]TJ/F8 9.9626 Tf 9.963 0 Td [( : 0 10 0 y =0 : 0 10 0 Thecomputer'srowreductionstep )]TJ/F8 9.9626 Tf 7.748 0 Td [(1000 1 + 2 producesasecondequation )]TJ/F8 9.9626 Tf 7.749 0 Td [(1001 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(999,whichthecomputerroundstotwoplacesas )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 : 0 10 3 y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 : 0 10 3 .Thenthecomputerdecidesfromthesecondequationthat y =1 andfromtherstequationthat x =0.This y valueisfairlygood,butthe x isquitebad.Thus,anothercauseofunreliableoutputisamixtureofoating pointarithmeticandarelianceonpivotsthataresmall. Anexperiencedprogrammermayrespondthatweshouldgoto doubleprecision wheresixteensignicantdigitsareretained.Thiswillindeedsolvemany problems.However,therearesomedicultieswithitasageneralapproach. Foronething,doubleprecisiontakeslongerthansingleprecisionona'486

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70 ChapterOne.LinearSystems chip,multiplicationtakeseleventicksinsingleprecisionbutfourteenindoubleprecision[Programmer'sRef.]andhastwicethememoryrequirements.So attemptingtodoallcalculationsindoubleprecisionisjustnotpractical.And besides,theabovesystemscanobviouslybetweakedtogivethesametroublein theseventeenthdigit,sodoubleprecisionwon'txallproblems.Whatweneed isastrategytominimizethenumericaltroublearisingfromsolvingsystems onacomputer,andsomeguidanceastohowfarthereportedsolutionscanbe trusted. Mathematicianshavemadeacarefulstudyofhowtogetthemostreliable results.Abasicimprovementonthenaivecodeaboveistonotsimplytake theentryinthe pivot row ; pivot row positionforthepivot,butrathertolook atalloftheentriesinthe pivot row columnbelowthe pivot row row,andtake theonethatismostlikelytogivereliableresultse.g.,takeonethatisnottoo small.Thisstrategyis partialpivoting .Forexample,tosolvethetroublesome system above,westartbylookingatbothequationsforabestrstpivot, andtakingthe1inthesecondequationasmorelikelytogivegoodresults. Then,thepivotstepof )]TJ/F11 9.9626 Tf 7.748 0 Td [(: 001 2 + 1 givesarstequationof1 : 001 y =1,which thecomputerwillrepresentas : 0 10 0 y =1 : 0 10 0 ,leadingtotheconclusion that y =1and,afterback-substitution, x =1,bothofwhichareclosetoright. Thecodefromabovecanbeadaptedtothispurpose. forpivot_row=1;pivot_row<=n-1;pivot_row++{ /*findthelargestpivotinthiscolumninrowmax*/ max=pivot_row; forrow_below=pivot_row+1;pivot_row<=n;row_below++{ ifabsa[row_below,pivot_row]>absa[max,row_below] max=row_below; } /*swaprowstomovethatpivotentryup*/ forcol=pivot_row;col<=n;col++{ temp=a[pivot_row,col]; a[pivot_row,col]=a[max,col]; a[max,col]=temp; } /*proceedasbefore*/ forrow_below=pivot_row+1;row_below<=n;row_below++{ multiplier=a[row_below,pivot_row]/a[pivot_row,pivot_row]; forcol=pivot_row;col<=n;col++{ a[row_below,col]-=multiplier*a[pivot_row,col]; } } } AfullanalysisofthebestwaytoimplementGauss'methodisoutsidethe scopeofthebooksee[Wilkinson1965],butthemethodrecommendedbymost expertsisavariationonthecodeabovethatrstndsthebestpivotamong thecandidates,andthenscalesittoanumberthatislesslikelytogivetrouble. Thisis scaledpartialpivoting .

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Topic:AccuracyofComputations 71 Inadditiontoreturningaresultthatislikelytobereliable,mostwell-done codewillreturnanumber,calledthe conditioningnumber thatdescribesthe factorbywhichuncertaintiesintheinputnumberscouldbemagniedtobecome inaccuraciesintheresultsreturnedsee[Rice]. ThelessonofthisdiscussionisthatjustbecauseGauss'methodalwaysworks intheory,andjustbecausecomputercodecorrectlyimplementsthatmethod, andjustbecausetheanswerappearsongreen-barpaper,doesn'tmeanthatthe answerisreliable.Inpractice,alwaysuseapackagewhereexpertshaveworked hardtocounterwhatcangowrong. Exercises 1 Usingtwodecimalplaces,add253and2 = 3. 2 Thisintersect-the-linesproblemcontrastswiththeexamplediscussedabove. ; 1 x +2 y =3 3 x )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 y =1 Illustratethatinthissystemsomesmallchangeinthenumberswillproduceonly asmallchangeinthesolutionbychangingtheconstantinthebottomequationto 1 : 008andsolving.Compareittothesolutionoftheunchangedsystem. 3 Solvethissystembyhand[Rice]. 0 : 0003 x +1 : 556 y =1 : 569 0 : 3454 x )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 : 346 y =1 : 018 a Solveitaccurately,byhand. b Solveitbyroundingateachstepto foursignicantdigits. 4 Roundinginsidethecomputeroftenhasaneectontheresult.Assumethat yourmachinehaseightsignicantdigits. a Showthatthemachinewillcompute = 3+ = 3 )]TJ/F29 8.9664 Tf 9.518 0 Td [( = 3asunequalto = 3+ = 3 )]TJ/F29 8.9664 Tf 9.216 0 Td [( = 3.Thus,computerarithmeticisnotassociative. b Comparethecomputer'sversionof = 3 x + y =0and = 3 x +2 y =0.Is twicetherstequationthesameasthesecond? 5 Ill-conditioningisnotonlydependentonthematrixofcoecients.Thisexample [Hamming]showsthatitcanarisefromaninteractionbetweentheleftandright sidesofthesystem.Let beasmallreal. 3 x +2 y + z =6 2 x +2 "y +2 "z =2+4 x +2 "y )]TJ/F32 8.9664 Tf 13.822 0 Td [("z =1+ a Solvethesystembyhand.Noticethatthe 'sdivideoutonlybecausethere isanexactcancelationoftheintegerpartsontherightsideaswellasonthe left. b Solvethesystembyhand,roundingtotwodecimalplaces,andwith = 0 : 001.

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72 ChapterOne.LinearSystems Topic:AnalyzingNetworks Thediagrambelowshowssomeofacar'selectricalnetwork.Thebatteryison theleft,drawnasstackedlinesegments.Thewiresaredrawnaslines,shown straightandwithsharprightanglesforneatness.Eachlightisacircleenclosing aloop. 12V Dome Light Door Actuated Switch Brake Lights L R Brake Actuated Switch Light Switch O Dimmer Hi Lo L R L R Headlights L R Rear Lights L R Parking Lights Thedesignerofsuchanetworkneedstoanswerquestionslike:Howmuch electricityowswhenboththehi-beamheadlightsandthebrakelightsare on?Below,wewilluselinearsystemstoanalyzesimplerversionsofelectrical networks. Fortheanalysisweneedtwofactsaboutelectricityandtwofactsabout electricalnetworks. Therstfactaboutelectricityisthatabatteryislikeapump:itprovides aforceimpellingtheelectricitytoowthroughthecircuitsconnectingthebattery'sends,ifthereareanysuchcircuits.Wesaythatthebatteryprovidesa potential toow.Ofcourse,thisnetworkaccomplishesitsfunctionwhen,as theelectricityowsthroughacircuit,itgoesthroughalight.Forinstance, whenthedriverstepsonthebrakethentheswitchmakescontactandacircuitisformedontheleftsideofthediagram,andtheelectricalcurrentowing throughthatcircuitwillmakethebrakelightsgoon,warningdriversbehind. Thesecondelectricalfactisthatinsomekindsofnetworkcomponentsthe amountofowisproportionaltotheforceprovidedbythebattery.Thatis,for eachsuchcomponentthereisanumber,it's resistance ,suchthatthepotentialis equaltotheowtimestheresistance.Theunitsofmeasurementare:potential isdescribedin volts ,therateofowisin amperes ,andresistancetotheowis in ohms .Theseunitsaredenedsothatvolts=amperes ohms. Componentswiththisproperty,thatthevoltage-amperageresponsecurve isalinethroughtheorigin,arecalled resistors .Lightbulbssuchastheones shownabovearenotthiskindofcomponent,becausetheirohmagechangesas theyheatup.Forexample,ifaresistormeasures2ohmsthenwiringittoa 12voltbatteryresultsinaowof6amperes.Conversely,ifwehaveowof electricalcurrentof2amperesthroughitthentheremustbea4voltpotential

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Topic:AnalyzingNetworks 73 dierencebetweenit'sends.Thisisthe voltagedrop acrosstheresistor.One waytothinkofaelectricalcircuitsliketheoneaboveisthatthebatteryprovides avoltagerisewhiletheothercomponentsarevoltagedrops. ThetwofactsthatweneedaboutnetworksareKirchho'sLaws. CurrentLaw. Foranypointinanetwork,theowinequalstheowout. VoltageLaw. Aroundanycircuitthetotaldropequalsthetotalrise. Intheabovenetworkthereisonlyonevoltagerise,atthebattery,butsome networkshavemorethanone. Forastartwecanconsiderthenetworkbelow.Ithasabatterythatprovides thepotentialtoowandthreeresistorsresistorsaredrawnaszig-zags.When componentsarewiredoneafteranother,ashere,theyaresaidtobein series 20 volt potential 2 ohm resistance 5 ohm resistance 3 ohm resistance ByKirchho'sVoltageLaw,becausethevoltageriseis20volts,thetotalvoltage dropmustalsobe20volts.Sincetheresistancefromstarttonishis10ohms theresistanceofthewiresisnegligible,wegetthatthecurrentis = 10= 2amperes.Now,byKirchho'sCurrentLaw,thereare2amperesthrougheach resistor.Andthereforethevoltagedropsare:4voltsacrossthe2ohmresistor, 10voltsacrossthe5ohmresistor,and6voltsacrossthe3ohmresistor. Thepriornetworkissosimplethatwedidn'tusealinearsystem,butthe nextnetworkismorecomplicated.Inthisone,theresistorsarein parallel .This networkismorelikethecarlightingdiagramshownearlier. 20 volt 12 ohm 8 ohm Webeginbylabelingthebranches,shownbelow.Letthecurrentthroughthe leftbranchoftheparallelportionbe i 1 andthatthroughtherightbranchbe i 2 andalsoletthecurrentthroughthebatterybe i 0 .WearefollowingKircho's CurrentLaw;forinstance,allpointsintherightbranchhavethesamecurrent, whichwecall i 2 .Notethatwedon'tneedtoknowtheactualdirectionofow| ifcurrentowsinthedirectionoppositetoourarrowthenwewillsimplygeta negativenumberinthesolution.

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74 ChapterOne.LinearSystems i 0 i 1 ## i 2 TheCurrentLaw,appliedtothepointintheupperrightwheretheow i 0 meets i 1 and i 2 ,givesthat i 0 = i 1 + i 2 .Appliedtothelowerrightitgives i 1 + i 2 = i 0 .Inthecircuitthatloopsoutofthetopofthebattery,downthe leftbranchoftheparallelportion,andbackintothebottomofthebattery, thevoltageriseis20whilethevoltagedropis i 1 12,sotheVoltageLawgives that12 i 1 =20.Similarly,thecircuitfromthebatterytotherightbranchand backtothebatterygivesthat8 i 2 =20.And,inthecircuitthatsimplyloops aroundintheleftandrightbranchesoftheparallelportionarbitrarilytaken clockwise,thereisavoltageriseof0andavoltagedropof8 i 2 )]TJ/F8 9.9626 Tf 10.306 0 Td [(12 i 1 sothe VoltageLawgivesthat8 i 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(12 i 1 =0. i 0 )]TJ/F11 9.9626 Tf 27.674 0 Td [(i 1 )]TJ/F11 9.9626 Tf 14.944 0 Td [(i 2 =0 )]TJ/F11 9.9626 Tf 7.749 0 Td [(i 0 + i 1 + i 2 =0 12 i 1 =20 8 i 2 =20 )]TJ/F8 9.9626 Tf 7.748 0 Td [(12 i 1 +8 i 2 =0 Thesolutionis i 0 =25 = 6, i 1 =5 = 3,and i 2 =5 = 2,allinamperes.Incidentally, thisillustratesthatredundantequationsdoindeedariseinpractice. Kirchho'slawscanbeusedtoestablishtheelectricalpropertiesofnetworks ofgreatcomplexity.Thenextdiagramshowsveresistors,wiredina seriesparallel way. 10 volt 5 ohm 2 ohm 50 ohm 10 ohm 4 ohm Thisnetworkisa Wheatstonebridge seeExercise4.Toanalyzeit,wecan placethearrowsinthisway. i 0 i 1 .& i 2 i 5 i 3 &. i 4

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Topic:AnalyzingNetworks 75 Kircho'sCurrentLaw,appliedtothetopnode,theleftnode,therightnode, andthebottomnodegivesthese. i 0 = i 1 + i 2 i 1 = i 3 + i 5 i 2 + i 5 = i 4 i 3 + i 4 = i 0 Kirchho'sVoltageLaw,appliedtotheinsideloopthe i 0 to i 1 to i 3 to i 0 loop, theoutsideloop,andtheupperloopnotinvolvingthebattery,givesthese. 5 i 1 +10 i 3 =10 2 i 2 +4 i 4 =10 5 i 1 +50 i 5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 i 2 =0 Thosesucetodeterminethesolution i 0 =7 = 3, i 1 =2 = 3, i 2 =5 = 3, i 3 =2 = 3, i 4 =5 = 3,and i 5 =0. Networksofotherkinds,notjustelectricalones,canalsobeanalyzedinthis way.Forinstance,networksofstreetsaregivenintheexercises. Exercises Manyofthesystemsfortheseproblemsaremostlyeasilysolvedonacomputer. 1 Calculatetheamperagesineachpartofeachnetwork. a Thisisasimplenetwork. 9 volt 3 ohm 2 ohm 2 ohm b Comparethisonewiththeparallelcasediscussedabove. 9 volt 3 ohm 2 ohm 2 ohm 2 ohm c Thisisareasonablycomplicatednetwork. 9 volt 3 ohm 3 ohm 2 ohm 2 ohm 3 ohm 4 ohm 2 ohm

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76 ChapterOne.LinearSystems 2 Intherstnetworkthatweanalyzed,withthethreeresistorsinseries,wejust addedtogetthattheyactedtogetherlikeasingleresistorof10ohms.Wecando asimilarthingforparallelcircuits.Inthesecondcircuitanalyzed, 20 volt 12 ohm 8 ohm theelectriccurrentthroughthebatteryis25 = 6amperes.Thus,theparallelportion is equivalent toasingleresistorof20 = = 6=4 : 8ohms. a Whatistheequivalentresistanceifwechangethe12ohmresistorto5ohms? b Whatistheequivalentresistanceifthetwoareeach8ohms? c Findtheformulafortheequivalentresistanceifthetworesistorsinparallel are r 1 ohmsand r 2 ohms. 3 ForthecardashboardexamplethatopensthisTopic,solvefortheseamperages assumethatallresistancesare2ohms. a Ifthedriverissteppingonthebrakes,sothebrakelightsareon,andno othercircuitisclosed. b Ifthehi-beamheadlightsandthebrakelightsareon. 4 Showthat,inthisWheatstoneBridge, r 1 r 3 r g r 2 r 4 r 2 =r 1 equals r 4 =r 3 ifandonlyifthecurrentowingthrough r g iszero.The waythatthisdeviceisusedinpracticeisthatanunknownresistanceat r 4 is comparedtotheotherthree r 1 r 2 ,and r 3 .At r g isplacedameterthatshowsthe current.Thethreeresistances r 1 r 2 ,and r 3 arevaried|typicallytheyeachhave acalibratedknob|untilthecurrentinthemiddlereads0,andthentheabove equationgivesthevalueof r 4 Therearenetworksotherthanelectricalones,andwecanaskhowwellKircho's lawsapplytothem.Theremainingquestionsconsideranextensiontonetworksof streets. 5 Considerthistraccircle. MainStreet NorthAvenue PierBoulevard

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Topic:AnalyzingNetworks 77 Thisisthetracvolume,inunitsofcarsperveminutes. NorthPierMain into 10015025 outof 7515050 Wecansetupequationstomodelhowthetracows. a AdaptKircho'sCurrentLawtothiscircumstance.Isitareasonablemodellingassumption? b Labelthethreebetween-roadarcsinthecirclewithavariable.Usingthe adaptedCurrentLaw,foreachofthethreein-outintersectionsstateanequationdescribingthetracowatthatnode. c Solvethatsystem. d Interpretyoursolution. e RestatetheVoltageLawforthiscircumstance.Howreasonableisit? 6 Thisisanetworkofstreets. ShelburneSt Willow WinooskiAve west east JayLn Thehourlyowofcarsintothisnetwork'sentrances,andoutofitsexitscanbe observed. eastWinooskiwestWinooskiWillowJayShelburne into 805065{40 outof 305705575 NotethattoreachJayacarmustenterthenetworkviasomeotherroadrst, whichiswhythereisno`intoJay'entryinthetable.Notealsothatoveralong periodoftime,thetotalinmustapproximatelyequalthetotalout,whichiswhy bothrowsaddto235cars.Onceinsidethenetwork,thetracmayowindierentways,perhapsllingWillowandleavingJaymostlyempty,orperhapsowing insomeotherway.Kirchho'sLawsgivethelimitsonthatfreedom. a Determinetherestrictionsontheowinsidethisnetworkofstreetsbysetting upavariableforeachblock,establishingtheequations,andsolvingthem.Notice thatsomestreetsareone-wayonly. Hint: thiswillnotyieldauniquesolution, sincetraccanowthroughthisnetworkinvariousways;youshouldgetat leastonefreevariable. b SupposethatsomeconstructionisproposedforWinooskiAvenueEastbetweenWillowandJay,sotraconthatblockwillbereduced.Whatistheleast amountoftracowthatcanbeallowedonthatblockwithoutdisruptingthe hourlyowintoandoutofthenetwork?

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ChapterTwo VectorSpaces TherstchapterbeganbyintroducingGauss'methodandnishedwithafair understanding,keyedontheLinearCombinationLemma,ofhowitndsthe solutionsetofalinearsystem.Gauss'methodsystematicallytakeslinearcombinationsoftherows.Withthatinsight,wenowmovetoageneralstudyof linearcombinations. Weneedasettingforthisstudy.Attimesintherstchapter,we'vecombinedvectorsfrom R 2 ,atothertimesvectorsfrom R 3 ,andatothertimesvectors fromevenhigher-dimensionalspaces.Thus,ourrstimpulsemightbetowork in R n ,leaving n unspecied.Thiswouldhavetheadvantagethatanyofthe resultswouldholdfor R 2 andfor R 3 andformanyotherspaces,simultaneously. But,ifhavingtheresultsapplytomanyspacesatonceisadvantageousthen stickingonlyto R n 'sisoverlyrestrictive.We'dliketheresultstoalsoapplyto combinationsofrowvectors,asinthenalsectionoftherstchapter.We've evenseensomespacesthatarenotjustacollectionofallofthesame-sized columnvectorsorrowvectors.Forinstance,we'veseenasolutionsetofa homogeneoussystemthatisaplane,insideof R 3 .Thissolutionsetisaclosed systeminthesensethatalinearcombinationofthesesolutionsisalsoasolution. Butitisnotjustacollectionofallofthethree-tallcolumnvectors;onlysome ofthemareinthissolutionset. Wewanttheresultsaboutlinearcombinationstoapplyanywherethatlinear combinationsaresensible.Weshallcallanysuchseta vectorspace .Ourresults, insteadofbeingphrasedasWheneverwehaveacollectioninwhichwecan sensiblytakelinearcombinations...",willbestatedasInanyvectorspace ...". Suchastatementdescribesatoncewhathappensinmanyspaces.Thestep upinabstractionfromstudyingasinglespaceatatimetostudyingaclass ofspacescanbehardtomake.Tounderstanditsadvantages,considerthis analogy.Imaginethatthegovernmentmadelawsonepersonatatime:Leslie Jonescan'tjaywalk."Thatwouldbeabadidea;statementshavethevirtueof economywhentheyapplytomanycasesatonce.Or,supposethattheyruled, KimKemuststopwhenpassingthesceneofanaccident."Contrastthatwith, Anydoctormuststopwhenpassingthesceneofanaccident."Moregeneral statements,insomeways,areclearer. 79

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80 ChapterTwo.VectorSpaces IDenitionofVectorSpace Weshallstudystructureswithtwooperations,anadditionandascalarmultiplication,thataresubjecttosomesimpleconditions.Wewillreectmoreon theconditionslater,butonrstreadingnoticehowreasonabletheyare.For instance,surelyanyoperationthatcanbecalledanadditione.g.,columnvectoraddition,rowvectoraddition,orrealnumberadditionwillsatisfyallthe conditionsinbelow. I.1DenitionandExamples 1.1Denition A vectorspace over R consistsofaset V alongwithtwo operations`+'and` 'subjecttotheseconditions. Where ~v;~w 2 V ,their vectorsum ~v + ~w isanelementof V .If ~u;~v;~w 2 V then ~v + ~w = ~w + ~v and ~v + ~w + ~u = ~v + ~w + ~u .Thereisa zero vector ~ 0 2 V suchthat ~v + ~ 0= ~v forall ~v 2 V .5Each ~v 2 V hasan additive inverse ~w 2 V suchthat ~w + ~v = ~ 0. If r;s are scalars ,membersof R ,and ~v;~w 2 V theneach scalarmultiple r ~v isin V .If r;s 2 R and ~v;~w 2 V then r + s ~v = r ~v + s ~v ,and r ~v + ~w = r ~v + r ~w ,and rs ~v = r s ~v ,and1 ~v = ~v 1.2Remark Becauseitinvolvestwokindsofadditionandtwokindsofmultiplication,thatdenitionmayseemconfused.Forinstance,incondition ` r + s ~v = r ~v + s ~v ',therst`+'istherealnumberadditionoperatorwhile the`+'totherightoftheequalssignrepresentsvectoradditioninthestructure V .Theseexpressionsaren'tambiguousbecause,e.g., r and s arerealnumbers so` r + s 'canonlymeanrealnumberaddition. Thebestwaytogothroughtheexamplesbelowistocheckalltenconditions inthedenition.Thatcheckiswrittenoutatlengthintherstexample.Use itasamodelfortheothers.Especiallyimportantaretherstcondition` ~v + ~w isin V 'andthesixthcondition` r ~v isin V '.Thesearethe closure conditions. Theyspecifythattheadditionandscalarmultiplicationoperationsarealways sensible|theyaredenedforeverypairofvectors,andeveryscalarandvector, andtheresultoftheoperationisamemberofthesetseeExample1.4. 1.3Example Theset R 2 isavectorspaceiftheoperations`+'and` 'have theirusualmeaning. x 1 x 2 + y 1 y 2 = x 1 + y 1 x 2 + y 2 r x 1 x 2 = rx 1 rx 2 Weshallcheckalloftheconditions.

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SectionI.DenitionofVectorSpace 81 Thereareveconditionsinitem.For,closureofaddition,notethat forany v 1 ;v 2 ;w 1 ;w 2 2 R theresultofthesum v 1 v 2 + w 1 w 2 = v 1 + w 1 v 2 + w 2 isacolumnarraywithtworealentries,andsoisin R 2 .For,thataddition ofvectorscommutes,takeallentriestoberealnumbersandcompute v 1 v 2 + w 1 w 2 = v 1 + w 1 v 2 + w 2 = w 1 + v 1 w 2 + v 2 = w 1 w 2 + v 1 v 2 thesecondequalityfollowsfromthefactthatthecomponentsofthevectorsare realnumbers,andtheadditionofrealnumbersiscommutative.Condition, associativityofvectoraddition,issimilar. v 1 v 2 + w 1 w 2 + u 1 u 2 = v 1 + w 1 + u 1 v 2 + w 2 + u 2 = v 1 + w 1 + u 1 v 2 + w 2 + u 2 = v 1 v 2 + w 1 w 2 + u 1 u 2 Forthefourthconditionwemustproduceazeroelement|thevectorofzeroes isit. v 1 v 2 + 0 0 = v 1 v 2 For,toproduceanadditiveinverse,notethatforany v 1 ;v 2 2 R wehave )]TJ/F11 9.9626 Tf 7.749 0 Td [(v 1 )]TJ/F11 9.9626 Tf 7.749 0 Td [(v 2 + v 1 v 2 = 0 0 sotherstvectoristhedesiredadditiveinverseofthesecond. Thechecksfortheveconditionshavingtodowithscalarmultiplicationare justasroutine.For,closureunderscalarmultiplication,where r;v 1 ;v 2 2 R r v 1 v 2 = rv 1 rv 2 isacolumnarraywithtworealentries,andsoisin R 2 .Next,thischecks. r + s v 1 v 2 = r + s v 1 r + s v 2 = rv 1 + sv 1 rv 2 + sv 2 = r v 1 v 2 + s v 1 v 2 For,thatscalarmultiplicationdistributesfromtheleftovervectoraddition, wehavethis. r v 1 v 2 + w 1 w 2 = r v 1 + w 1 r v 2 + w 2 = rv 1 + rw 1 rv 2 + rw 2 = r v 1 v 2 + r w 1 w 2

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82 ChapterTwo.VectorSpaces Theninth rs v 1 v 2 = rs v 1 rs v 2 = r sv 1 r sv 2 = r s v 1 v 2 andtenthconditionsarealsostraightforward. 1 v 1 v 2 = 1 v 1 1 v 2 = v 1 v 2 Inasimilarway,each R n isavectorspacewiththeusualoperationsof vectoradditionandscalarmultiplication.In R 1 ,weusuallydonotwritethe membersascolumnvectors,i.e.,weusuallydonotwrite` '.Insteadwejust write` '. 1.4Example Thissubsetof R 3 thatisaplanethroughtheorigin P = f 0 @ x y z 1 A x + y + z =0 g isavectorspaceif`+'and` 'areinterpretedinthisway. 0 @ x 1 y 1 z 1 1 A + 0 @ x 2 y 2 z 2 1 A = 0 @ x 1 + x 2 y 1 + y 2 z 1 + z 2 1 A r 0 @ x y z 1 A = 0 @ rx ry rz 1 A Theadditionandscalarmultiplicationoperationsherearejusttheonesof R 3 reusedonitssubset P .Wesaythat P inherits theseoperationsfrom R 3 .This exampleofanadditionin P 0 @ 1 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 1 A + 0 @ )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 0 1 1 A = 0 @ 0 1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 1 A illustratesthat P isclosedunderaddition.We'veaddedtwovectorsfrom P | thatis,withthepropertythatthesumoftheirthreeentriesiszero|andthe resultisavectoralsoin P .Ofcourse,thisexampleofclosureisnotaproofof closure.Toprovethat P isclosedunderaddition,taketwoelementsof P 0 @ x 1 y 1 z 1 1 A 0 @ x 2 y 2 z 2 1 A membershipin P meansthat x 1 + y 1 + z 1 =0and x 2 + y 2 + z 2 =0,and observethattheirsum 0 @ x 1 + x 2 y 1 + y 2 z 1 + z 2 1 A

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SectionI.DenitionofVectorSpace 83 isalsoin P sinceitsentriesadd x 1 + x 2 + y 1 + y 2 + z 1 + z 2 = x 1 + y 1 + z 1 + x 2 + y 2 + z 2 to0.Toshowthat P isclosedunderscalarmultiplication, startwithavectorfrom P 0 @ x y z 1 A sothat x + y + z =0andthenfor r 2 R observethatthescalarmultiple r 0 @ x y z 1 A = 0 @ rx ry rz 1 A satisesthat rx + ry + rz = r x + y + z =0.Thusthetwoclosureconditions aresatised.Vericationoftheotherconditionsinthedenitionofavector spacearejustasstraightforward. 1.5Example Example1.3showsthatthesetofalltwo-tallvectorswithreal entriesisavectorspace.Example1.4givesasubsetofan R n thatisalsoa vectorspace.Incontrastwiththosetwo,considerthesetoftwo-tallcolumns withentriesthatareintegersundertheobviousoperations.Thisisasubset ofavectorspace,butitisnotitselfavectorspace.Thereasonisthatthissetis notclosedunderscalarmultiplication,thatis,itdoesnotsatisfycondition. Hereisacolumnwithintegerentries,andascalar,suchthattheoutcomeof theoperation 0 : 5 4 3 = 2 1 : 5 isnotamemberoftheset,sinceitsentriesarenotallintegers. 1.6Example Thesingletonset f 0 B B @ 0 0 0 0 1 C C A g isavectorspaceundertheoperations 0 B B @ 0 0 0 0 1 C C A + 0 B B @ 0 0 0 0 1 C C A = 0 B B @ 0 0 0 0 1 C C A r 0 B B @ 0 0 0 0 1 C C A = 0 B B @ 0 0 0 0 1 C C A thatitinheritsfrom R 4 Avectorspacemusthaveatleastoneelement,itszerovector.Thusa one-elementvectorspaceisthesmallestonepossible. 1.7Denition Aone-elementvectorspaceisa trivial space.

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84 ChapterTwo.VectorSpaces Warning!Theexamplessofarinvolvesetsofcolumnvectorswiththeusual operations.Butvectorspacesneednotbecollectionsofcolumnvectors,oreven ofrowvectors.Belowaresomeothertypesofvectorspaces.Theterm`vector space'doesnotmean`collectionofcolumnsofreals'.Itmeanssomethingmore like`collectioninwhichanylinearcombinationissensible'. 1.8Example Consider P 3 = f a 0 + a 1 x + a 2 x 2 + a 3 x 3 a 0 ;:::;a 3 2 R g ,the setofpolynomialsofdegreethreeorlessinthisbook,we'lltakeconstant polynomials,includingthezeropolynomial,tobeofdegreezero.Itisavector spaceundertheoperations a 0 + a 1 x + a 2 x 2 + a 3 x 3 + b 0 + b 1 x + b 2 x 2 + b 3 x 3 = a 0 + b 0 + a 1 + b 1 x + a 2 + b 2 x 2 + a 3 + b 3 x 3 and r a 0 + a 1 x + a 2 x 2 + a 3 x 3 = ra 0 + ra 1 x + ra 2 x 2 + ra 3 x 3 thevericationiseasy.Thisvectorspaceisworthyofattentionbecausethese arethepolynomialoperationsfamiliarfromhighschoolalgebra.Forinstance, 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 x +3 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x + x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [( = 2 x 3 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1+7 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(11 x 3 Althoughthisspaceisnotasubsetofany R n ,thereisasenseinwhichwe canthinkof P 3 asthesame"as R 4 .Ifweidentifythesetwospaces'selements inthisway a 0 + a 1 x + a 2 x 2 + a 3 x 3 correspondsto 0 B B @ a 0 a 1 a 2 a 3 1 C C A thentheoperationsalsocorrespond.Hereisanexampleofcorrespondingadditions. 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 x +0 x 2 +1 x 3 +2+3 x +7 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x 3 3+1 x +7 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x 3 correspondsto 0 B B @ 1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 0 1 1 C C A + 0 B B @ 2 3 7 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 1 C C A = 0 B B @ 3 1 7 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 1 C C A Thingswearethinkingofasthesame"addtothesame"sum.ChapterThree makesprecisethisideaofvectorspacecorrespondence.Fornowweshalljust leaveitasanintuition. 1.9Example Theset M 2 2 of2 2matriceswithrealnumberentriesisa vectorspaceunderthenaturalentry-by-entryoperations. ab cd + wx yz = a + wb + x c + yd + z r ab cd = rarb rcrd Asinthepriorexample,wecanthinkofthisspaceasthesame"as R 4 .

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SectionI.DenitionofVectorSpace 85 1.10Example Theset f f f : N R g ofallreal-valuedfunctionsofone naturalnumbervariableisavectorspaceundertheoperations f 1 + f 2 n = f 1 n + f 2 n r f n = rf n sothatif,forexample, f 1 n = n 2 +2sin n and f 2 n = )]TJ/F8 9.9626 Tf 9.409 0 Td [(sin n +0 : 5then f 1 +2 f 2 n = n 2 +1. WecanviewthisspaceasageneralizationofExample1.3|insteadof2-tall vectors,thesefunctionsarelikeinnitely-tallvectors. n f n = n 2 +1 0 1 1 2 2 5 3 10 correspondsto 0 B B B B B @ 1 2 5 10 1 C C C C C A Additionandscalarmultiplicationarecomponent-wise,asinExample1.3.We canformalizeinnitely-tall"bysayingthatitmeansaninnitesequence,or thatitmeansafunctionfrom N to R 1.11Example Thesetofpolynomialswithrealcoecients f a 0 + a 1 x + + a n x n n 2 N and a 0 ;:::;a n 2 R g makesavectorspacewhengiventhenatural`+' a 0 + a 1 x + + a n x n + b 0 + b 1 x + + b n x n = a 0 + b 0 + a 1 + b 1 x + + a n + b n x n and` '. r a 0 + a 1 x + :::a n x n = ra 0 + ra 1 x + ::: ra n x n Thisspacediersfromthespace P 3 ofExample1.8.Thisspacecontainsnotjust degreethreepolynomials,butdegreethirtypolynomialsanddegreethreehundredpolynomials,too.Eachindividualpolynomialofcourseisofanitedegree, butthesethasnosingleboundonthedegreeofallofitsmembers. Thisexample,likethepriorone,canbethoughtofintermsofinnite-tuples. Forinstance,wecanthinkof1+3 x +5 x 2 ascorrespondingto ; 3 ; 5 ; 0 ; 0 ;::: However,don'tconfusethisspacewiththeonefromExample1.10.Eachmemberofthissethasaboundeddegree,sounderourcorrespondencethereareno elementsfromthisspacematching ; 2 ; 5 ; 10 ;::: .Thevectorsinthisspace correspondtoinnite-tuplesthatendinzeroes. 1.12Example Theset f f f : R R g ofallreal-valuedfunctionsofonereal variableisavectorspaceunderthese. f 1 + f 2 x = f 1 x + f 2 x r f x = rf x ThedierencebetweenthisandExample1.10isthedomainofthefunctions.

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86 ChapterTwo.VectorSpaces 1.13Example Theset F = f a cos + b sin a;b 2 R g ofreal-valuedfunctions oftherealvariable isavectorspaceundertheoperations a 1 cos + b 1 sin + a 2 cos + b 2 sin = a 1 + a 2 cos + b 1 + b 2 sin and r a cos + b sin = ra cos + rb sin inheritedfromthespaceinthepriorexample.Wecanthinkof F asthesame" as R 2 inthat a cos + b sin correspondstothevectorwithcomponents a and b 1.14Example Theset f f : R R d 2 f dx 2 + f =0 g isavectorspaceunderthe,bynownatural,interpretation. f + g x = f x + g x r f x = rf x Inparticular,noticethatclosureisaconsequence: d 2 f + g dx 2 + f + g = d 2 f dx 2 + f + d 2 g dx 2 + g and d 2 rf dx 2 + rf = r d 2 f dx 2 + f ofbasicCalculus.Thisturnsouttoequalthespacefromthepriorexample| functionssatisfyingthisdierentialequationhavetheform a cos + b sin | butthisdescriptionsuggestsanextensiontosolutionssetsofotherdierential equations. 1.15Example Thesetofsolutionsofahomogeneouslinearsystemin n variablesisavectorspaceundertheoperationsinheritedfrom R n .Forclosure underaddition,if ~v = 0 B @ v 1 v n 1 C A ~w = 0 B @ w 1 w n 1 C A bothsatisfytheconditionthattheirentriesaddto0then ~v + ~w alsosatises thatcondition: c 1 v 1 + w 1 + + c n v n + w n = c 1 v 1 + + c n v n + c 1 w 1 + + c n w n =0.Thechecksoftheotherconditionsarejustasroutine. Aswe'vedoneinthoseequations,weoftenomitthemultiplicationsymbol` '. Wecandistinguishthemultiplicationin` c 1 v 1 'fromthatin` r~v 'sinceifboth multiplicandsarerealnumbersthenreal-realmultiplicationmustbemeant, whileifoneisavectorthenscalar-vectormultiplicationmustbemeant. Thepriorexamplehasbroughtusfullcirclesinceitisoneofourmotivating examples.

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SectionI.DenitionofVectorSpace 87 1.16Remark Now,withsomefeelforthekindsofstructuresthatsatisfythe denitionofavectorspace,wecanreectonthatdenition.Forexample,why specifyinthedenitiontheconditionthat1 ~v = ~v butnotaconditionthat 0 ~v = ~ 0? Oneansweristhatthisisjustadenition|itgivestherulesofthegame fromhereon,andifyoudon'tlikeit,putthebookdownandwalkaway. Anotheranswerisperhapsmoresatisfying.Peopleinthisareahaveworked hardtodeveloptherightbalanceofpowerandgenerality.Thisdenitionhas beenshapedsothatitcontainstheconditionsneededtoprovealloftheinterestingandimportantpropertiesofspacesoflinearcombinations.Asweproceed, weshallderiveallofthepropertiesnaturaltocollectionsoflinearcombinations fromtheconditionsgiveninthedenition. Thenextresultisanexample.Wedonotneedtoincludetheseproperties inthedenitionofvectorspacebecausetheyfollowfromthepropertiesalready listedthere. 1.17Lemma Inanyvectorspace V ,forany ~v 2 V and r 2 R ,wehave 0 ~v = ~ 0,and )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ~v + ~v = ~ 0,and r ~ 0= ~ 0. Proof For,notethat ~v =+0 ~v = ~v + ~v .Addtobothsidesthe additiveinverseof ~v ,thevector ~w suchthat ~w + ~v = ~ 0. ~w + ~v = ~w + ~v +0 ~v ~ 0= ~ 0+0 ~v ~ 0=0 ~v Theseconditemiseasy: )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ~v + ~v = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1+1 ~v =0 ~v = ~ 0showsthat wecanwrite` )]TJ/F11 9.9626 Tf 7.362 0 Td [(~v 'fortheadditiveinverseof ~v withoutworryingaboutpossible confusionwith )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ~v For,this r ~ 0= r ~ 0= r 0 ~ 0= ~ 0willdo. QED Wenishwitharecap. OurstudyinChapterOneofGaussianreductionledustoconsidercollectionsoflinearcombinations.Sointhischapterwehavedenedavectorspace tobeastructureinwhichwecanformsuchcombinations,expressionsofthe form c 1 ~v 1 + + c n ~v n subjecttosimpleconditionsontheadditionandscalar multiplicationoperations.Inaphrase:vectorspacesaretherightcontextin whichtostudylinearity. Finally,acomment.Fromthefactthatitformsawholechapter,andespeciallybecausethatchapteristherstone,areadercouldcometothinkthat thestudyoflinearsystemsisourpurpose.Thetruthis,wewillnotsomuch usevectorspacesinthestudyoflinearsystemsaswewillinsteadhavelinear systemsstartusonthestudyofvectorspaces.Thewidevarietyofexamples fromthissubsectionshowsthatthestudyofvectorspacesisinterestingandimportantinitsownright,asidefromhowithelpsusunderstandlinearsystems. Linearsystemswon'tgoaway.Butfromnowonourprimaryobjectsofstudy willbevectorspaces.

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88 ChapterTwo.VectorSpaces Exercises 1.18 Namethezerovectorforeachofthesevectorspaces. a Thespaceofdegreethreepolynomialsunderthenaturaloperations b Thespaceof2 4matrices c Thespace f f :[0 :: 1] R f iscontinuous g d Thespaceofreal-valuedfunctionsofonenaturalnumbervariable X 1.19 Findtheadditiveinverse,inthevectorspace,ofthevector. a In P 3 ,thevector )]TJ/F29 8.9664 Tf 7.167 0 Td [(3 )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 x + x 2 b Inthespace2 2, 1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 03 : c In f ae x + be )]TJ/F33 5.9776 Tf 5.757 0 Td [(x a;b 2 R g ,thespaceoffunctionsoftherealvariable x under thenaturaloperations,thevector3 e x )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 e )]TJ/F33 5.9776 Tf 5.756 0 Td [(x X 1.20 Showthateachoftheseisavectorspace. a Thesetoflinearpolynomials P 1 = f a 0 + a 1 x a 0 ;a 1 2 R g undertheusual polynomialadditionandscalarmultiplicationoperations. b Thesetof2 2matriceswithrealentriesundertheusualmatrixoperations. c Thesetofthree-componentrowvectorswiththeirusualoperations. d Theset L = f 0 B @ x y z w 1 C A 2 R 4 x + y )]TJ/F32 8.9664 Tf 9.216 0 Td [(z + w =0 g undertheoperationsinheritedfrom R 4 X 1.21 Showthateachoftheseisnotavectorspace. Hint. Startbylistingtwo membersofeachset. a Undertheoperationsinheritedfrom R 3 ,thisset f x y z 2 R 3 x + y + z =1 g b Undertheoperationsinheritedfrom R 3 ,thisset f x y z 2 R 3 x 2 + y 2 + z 2 =1 g c Undertheusualmatrixoperations, f a 1 bc a;b;c 2 R g d Undertheusualpolynomialoperations, f a 0 + a 1 x + a 2 x 2 a 0 ;a 1 ;a 2 2 R + g where R + isthesetofrealsgreaterthanzero e Undertheinheritedoperations, f x y 2 R 2 x +3 y =4and2 x )]TJ/F32 8.9664 Tf 9.216 0 Td [(y =3and6 x +4 y =10 g 1.22 Deneadditionandscalarmultiplicationoperationstomakethecomplex numbersavectorspaceover R .

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SectionI.DenitionofVectorSpace 89 X 1.23 Isthesetofrationalnumbersavectorspaceover R undertheusualaddition andscalarmultiplicationoperations? 1.24 Showthatthesetoflinearcombinationsofthevariables x;y;z isavector spaceunderthenaturaladditionandscalarmultiplicationoperations. 1.25 Provethatthisisnotavectorspace:thesetoftwo-tallcolumnvectorswith realentriessubjecttotheseoperations. x 1 y 1 + x 2 y 2 = x 1 )]TJ/F32 8.9664 Tf 9.215 0 Td [(x 2 y 1 )]TJ/F32 8.9664 Tf 9.215 0 Td [(y 2 r x y = rx ry 1.26 Proveordisprovethat R 3 isavectorspaceundertheseoperations. a x 1 y 1 z 1 + x 2 y 2 z 2 = 0 0 0 and r x y z = rx ry rz b x 1 y 1 z 1 + x 2 y 2 z 2 = 0 0 0 and r x y z = 0 0 0 X 1.27 Foreach,decideifitisavectorspace;theintendedoperationsarethenatural ones. a The diagonal 2 2matrices f a 0 0 b a;b 2 R g b Thissetof2 2matrices f xx + y x + yy x;y 2 R g c Thisset f 0 B @ x y z w 1 C A 2 R 4 x + y + w =1 g d Thesetoffunctions f f : R R df=dx +2 f =0 g e Thesetoffunctions f f : R R df=dx +2 f =1 g X 1.28 Proveordisprovethatthisisavectorspace:thereal-valuedfunctions f of onerealvariablesuchthat f =0. X 1.29 Showthattheset R + ofpositiverealsisavectorspacewhen` x + y 'isinterpretedtomeantheproductof x and y sothat2+3is6,and` r x 'isinterpreted asthe r -thpowerof x 1.30 Is f x;y x;y 2 R g avectorspaceundertheseoperations? a x 1 ;y 1 + x 2 ;y 2 = x 1 + x 2 ;y 1 + y 2 and r x;y = rx;y b x 1 ;y 1 + x 2 ;y 2 = x 1 + x 2 ;y 1 + y 2 and r x;y = rx; 0 1.31 Proveordisprovethatthisisavectorspace:thesetofpolynomialsofdegree greaterthanorequaltotwo,alongwiththezeropolynomial. 1.32 Atthispointthesame"isonlyanintuition,butnonethelessforeachvector spaceidentifythe k forwhichthespaceisthesame"as R k a The2 3matricesundertheusualoperations b The n m matricesundertheirusualoperations c Thissetof2 2matrices f a 0 bc a;b;c 2 R g

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90 ChapterTwo.VectorSpaces d Thissetof2 2matrices f a 0 bc a + b + c =0 g X 1.33 Using ~ +torepresentvectoradditionand ~ forscalarmultiplication,restate thedenitionofvectorspace. X 1.34 Provethese. a Anyvectoristheadditiveinverseoftheadditiveinverseofitself. b Vectoradditionleft-cancels:if ~v;~s; ~ t 2 V then ~v + ~s = ~v + ~ t impliesthat ~s = ~ t 1.35 Thedenitionofvectorspacesdoesnotexplicitlysaythat ~ 0+ ~v = ~v itinstead saysthat ~v + ~ 0= ~v .Showthatitmustnonethelessholdinanyvectorspace. X 1.36 Proveordisprovethatthisisavectorspace:thesetofallmatrices,under theusualoperations. 1.37 Inavectorspaceeveryelementhasanadditiveinverse.Cansomeelements havetwoormore? 1.38a Provethateverypoint,line,orplanethrutheoriginin R 3 isavector spaceundertheinheritedoperations. b Whatifitdoesn'tcontaintheorigin? X 1.39 Usingtheideaofavectorspacewecaneasilyreprovethatthesolutionsetof ahomogeneouslinearsystemhaseitheroneelementorinnitelymanyelements. Assumethat ~v 2 V isnot ~ 0. a Provethat r ~v = ~ 0ifandonlyif r =0. b Provethat r 1 ~v = r 2 ~v ifandonlyif r 1 = r 2 c Provethatanynontrivialvectorspaceisinnite. d Usethefactthatanonemptysolutionsetofahomogeneouslinearsystemis avectorspacetodrawtheconclusion. 1.40 Isthisavectorspaceunderthenaturaloperations:thereal-valuedfunctions ofonerealvariablethataredierentiable? 1.41 A vectorspaceoverthecomplexnumbers C hasthesamedenitionasavector spaceovertherealsexceptthatscalarsaredrawnfrom C insteadoffrom R .Show thateachoftheseisavectorspaceoverthecomplexnumbers.Recallhowcomplex numbersaddandmultiply: a 0 + a 1 i + b 0 + b 1 i = a 0 + b 0 + a 1 + b 1 i and a 0 + a 1 i b 0 + b 1 i = a 0 b 0 )]TJ/F32 8.9664 Tf 9.215 0 Td [(a 1 b 1 + a 0 b 1 + a 1 b 0 i a Thesetofdegreetwopolynomialswithcomplexcoecients b Thisset f 0 a b 0 a;b 2 C and a + b =0+0 i g 1.42 Nameapropertysharedbyallofthe R n 'sbutnotlistedasarequirementfor avectorspace. X 1.43a Provethatasumoffourvectors ~v 1 ;:::;~v 4 2 V canbeassociatedinany waywithoutchangingtheresult. ~v 1 + ~v 2 + ~v 3 + ~v 4 = ~v 1 + ~v 2 + ~v 3 + ~v 4 = ~v 1 + ~v 2 + ~v 3 + ~v 4 = ~v 1 + ~v 2 + ~v 3 + ~v 4 = ~v 1 + ~v 2 + ~v 3 + ~v 4 Thisallowsustosimplywrite` ~v 1 + ~v 2 + ~v 3 + ~v 4 'withoutambiguity.

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SectionI.DenitionofVectorSpace 91 b Provethatanytwowaysofassociatingasumofanynumberofvectorsgive thesamesum. Hint. Useinductiononthenumberofvectors. 1.44 Foranyvectorspace,asubsetthatisitselfavectorspaceundertheinherited operationse.g.,aplanethroughtheorigininsideof R 3 isa subspace a Showthat f a 0 + a 1 x + a 2 x 2 a 0 + a 1 + a 2 =0 g isasubspaceofthevector spaceofdegreetwopolynomials. b Showthatthisisasubspaceofthe2 2matrices. f ab c 0 a + b =0 g c Showthatanonemptysubset S ofarealvectorspaceisasubspaceifandonly ifitisclosedunderlinearcombinationsofpairsofvectors:whenever c 1 ;c 2 2 R and ~s 1 ;~s 2 2 S thenthecombination c 1 ~v 1 + c 2 ~v 2 isin S I.2SubspacesandSpanningSets Oneoftheexamplesthatledustointroducetheideaofavectorspacewasthe solutionsetofahomogeneoussystem.Forinstance,we'veseeninExample1.4 suchaspacethatisaplanarsubsetof R 3 .There,thevectorspace R 3 contains insideitanothervectorspace,theplane. 2.1Denition Foranyvectorspace,a subspace isasubsetthatisitselfa vectorspace,undertheinheritedoperations. 2.2Example Theplanefromthepriorsubsection, P = f 0 @ x y z 1 A x + y + z =0 g isasubspaceof R 3 .Asspeciedinthedenition,theoperationsaretheones thatareinheritedfromthelargerspace,thatis,vectorsaddin P astheyadd in R 3 0 @ x 1 y 1 z 1 1 A + 0 @ x 2 y 2 z 2 1 A = 0 @ x 1 + x 2 y 1 + y 2 z 1 + z 2 1 A andscalarmultiplicationisalsothesameasitisin R 3 .Toshowthat P isa subspace,weneedonlynotethatitisasubsetandthenverifythatitisaspace. Checkingthat P satisestheconditionsinthedenitionofavectorspaceis routine.Forinstance,forclosureunderaddition,justnotethatifthesummands satisfythat x 1 + y 1 + z 1 =0and x 2 + y 2 + z 2 =0thenthesumsatisesthat x 1 + x 2 + y 1 + y 2 + z 1 + z 2 = x 1 + y 1 + z 1 + x 2 + y 2 + z 2 =0.

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92 ChapterTwo.VectorSpaces 2.3Example The x -axisin R 2 isasubspacewheretheadditionandscalar multiplicationoperationsaretheinheritedones. x 1 0 + x 2 0 = x 1 + x 2 0 r x 0 = rx 0 Asabove,toverifythatthisisasubspace,wesimplynotethatitisasubset andthencheckthatitsatisestheconditionsindenitionofavectorspace. Forinstance,thetwoclosureconditionsaresatised:addingtwovectors withasecondcomponentofzeroresultsinavectorwithasecondcomponent ofzero,andmultiplyingascalartimesavectorwithasecondcomponentof zeroresultsinavectorwithasecondcomponentofzero. 2.4Example Anothersubspaceof R 2 is f 0 0 g itstrivialsubspace. Anyvectorspacehasatrivialsubspace f ~ 0 g .Attheoppositeextreme,any vectorspacehasitselfforasubspace.Thesetwoarethe improper subspaces. Othersubspacesare proper 2.5Example Theconditioninthedenitionrequiringthattheadditionand scalarmultiplicationoperationsmustbetheonesinheritedfromthelargerspace isimportant.Considerthesubset f 1 g ofthevectorspace R 1 .Undertheoperations1+1=1and r 1=1thatsetisavectorspace,specically,atrivialspace. Butitisnotasubspaceof R 1 becausethosearen'ttheinheritedoperations,since ofcourse R 1 has1+1=2. 2.6Example Allkindsofvectorspaces,notjust R n 's,havesubspaces.The vectorspaceofcubicpolynomials f a + bx + cx 2 + dx 3 a;b;c;d 2 R g hasasubspacecomprisedofalllinearpolynomials f m + nx m;n 2 R g 2.7Example Anotherexampleofasubspacenottakenfroman R n isone fromtheexamplesfollowingthedenitionofavectorspace.Thespaceofall real-valuedfunctionsofonerealvariable f : R R hasasubspaceoffunctions satisfyingtherestriction d 2 f=dx 2 + f =0. 2.8Example Beingvectorspacesthemselves,subspacesmustsatisfytheclosureconditions.Theset R + isnotasubspaceofthevectorspace R 1 because withtheinheritedoperationsitisnotclosedunderscalarmultiplication:if ~v =1then )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ~v 62 R + ThenextresultsaysthatExample2.8isprototypical.Theonlywaythata subsetcanfailtobeasubspaceifitisnonemptyandtheinheritedoperations areusedisifitisn'tclosed.

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SectionI.DenitionofVectorSpace 93 2.9Lemma Foranonemptysubset S ofavectorspace,undertheinherited operations,thefollowingareequivalentstatements. S isasubspaceofthatvectorspace S isclosedunderlinearcombinationsofpairsofvectors:foranyvectors ~s 1 ;~s 2 2 S andscalars r 1 ;r 2 thevector r 1 ~s 1 + r 2 ~s 2 isin S S isclosedunderlinearcombinationsofanynumberofvectors:forany vectors ~s 1 ;:::;~s n 2 S andscalars r 1 ;:::;r n thevector r 1 ~s 1 + + r n ~s n is in S Briey,thewaythatasubsetgetstobeasubspaceisbybeingclosedunder linearcombinations. Proof `Thefollowingareequivalent'meansthateachpairofstatementsare equivalent. Wewillshowthisequivalencebyestablishingthat= = = .Thisstrategyissuggestedbynoticingthat= and= areeasyandsoweneedonlyarguethesingleimplication= Forthatargument,assumethat S isanonemptysubsetofavectorspace V andthat S isclosedundercombinationsofpairsofvectors.Wewillshowthat S isavectorspacebycheckingtheconditions. Therstiteminthevectorspacedenitionhasveconditions.First,for closureunderaddition,if ~s 1 ;~s 2 2 S then ~s 1 + ~s 2 2 S ,as ~s 1 + ~s 2 =1 ~s 1 +1 ~s 2 Second,forany ~s 1 ;~s 2 2 S ,becauseadditionisinheritedfrom V ,thesum ~s 1 + ~s 2 in S equalsthesum ~s 1 + ~s 2 in V ,andthatequalsthesum ~s 2 + ~s 1 in V because V isavectorspace,itsadditioniscommutative,andthatinturnequalsthe sum ~s 2 + ~s 1 in S .Theargumentforthethirdconditionissimilartothatforthe second.Forthefourth,considerthezerovectorof V andnotethatclosureof S underlinearcombinationsofpairsofvectorsgivesthatwhere ~s isanymember ofthenonemptyset S 0 ~s +0 ~s = ~ 0isin S ;showingthat ~ 0actsunderthe inheritedoperationsastheadditiveidentityof S iseasy.Thefthconditionis satisedbecauseforany ~s 2 S ,closureunderlinearcombinationsshowsthat thevector0 ~ 0+ )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ~s isin S ;showingthatitistheadditiveinverseof ~s undertheinheritedoperationsisroutine. ThechecksforitemaresimilarandaresavedforExercise32. QED Weusuallyshowthatasubsetisasubspacewith= 2.10Remark Atthestartofthischapterweintroducedvectorspacesas collectionsinwhichlinearcombinationsaresensible".Theaboveresultspeaks tothis. Thevectorspacedenitionhastenconditionsbuteightofthem|theconditionsnotaboutclosure|simplyensurethatreferringtotheoperationsasan `addition'anda`scalarmultiplication'issensible.Theproofabovechecksthat theseeightareinheritedfromthesurroundingvectorspaceprovidedthatthe Moreinformationonequivalenceofstatementsisintheappendix.

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94 ChapterTwo.VectorSpaces nonemptyset S satisesTheorem2.9'sstatemente.g.,commutativityof additionin S followsrightfromcommutativityofadditionin V .So,inthis context,thismeaningofsensible"isautomaticallysatised. Inassuringusthatthisrstmeaningofthewordismet,theresultdraws ourattentiontothesecondmeaningofsensible".Ithastodowiththetwo remainingconditions,theclosureconditions.Above,thetwoseparateclosure conditionsinherentinstatementarecombinedinstatementintothe singleconditionofclosureunderalllinearcombinationsoftwovectors,which isthenextendedinstatementtoclosureundercombinationsofanynumber ofvectors.Thelattertwostatementssaythatwecanalwaysmakesenseofan expressionlike r 1 ~s 1 + r 2 ~s 2 ,withoutrestrictionsonthe r 's|suchexpressions aresensible"inthatthevectordescribedisdenedandisintheset S Thissecondmeaningsuggeststhatagoodwaytothinkofavectorspace isasacollectionofunrestrictedlinearcombinations.Thenexttwoexamples takesomespacesanddescribetheminthisway.Thatis,intheseexamples weparametrize,justaswedidinChapterOnetodescribethesolutionsetofa homogeneouslinearsystem. 2.11Example Thissubsetof R 3 S = f 0 @ x y z 1 A x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y + z =0 g isasubspaceundertheusualadditionandscalarmultiplicationoperationsof columnvectorsthecheckthatitisnonemptyandclosedunderlinearcombinationsoftwovectorsisjustliketheoneinExample2.2.Toparametrize,we cantake x )]TJ/F8 9.9626 Tf 10.041 0 Td [(2 y + z =0tobeaone-equationlinearsystemandexpressingthe leadingvariableintermsofthefreevariables x =2 y )]TJ/F11 9.9626 Tf 9.963 0 Td [(z S = f 0 @ 2 y )]TJ/F11 9.9626 Tf 9.963 0 Td [(z y z 1 A y;z 2 R g = f y 0 @ 2 1 0 1 A + z 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 0 1 1 A y;z 2 R g Nowthesubspaceisdescribedasthecollectionofunrestrictedlinearcombinationsofthosetwovectors.Ofcourse,ineitherdescription,thisisaplane throughtheorigin. 2.12Example Thisisasubspaceofthe2 2matrices L = f a 0 bc a + b + c =0 g checkingthatitisnonemptyandclosedunderlinearcombinationsiseasy.To parametrize,expresstheconditionas a = )]TJ/F11 9.9626 Tf 7.749 0 Td [(b )]TJ/F11 9.9626 Tf 9.963 0 Td [(c L = f )]TJ/F11 9.9626 Tf 7.749 0 Td [(b )]TJ/F11 9.9626 Tf 9.963 0 Td [(c 0 bc b;c 2 R g = f b )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 10 + c )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 01 b;c 2 R g Asabove,we'vedescribedthesubspaceasacollectionofunrestrictedlinear combinationsbycoincidence,alsooftwoelements.

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SectionI.DenitionofVectorSpace 95 Parametrizationisaneasytechnique,butitisimportant.Weshalluseit often. 2.13Denition The span or linearclosure ofanonemptysubset S ofa vectorspaceisthesetofalllinearcombinationsofvectorsfrom S [ S ]= f c 1 ~s 1 + + c n ~s n c 1 ;:::;c n 2 R and ~s 1 ;:::;~s n 2 S g Thespanoftheemptysubsetofavectorspaceisthetrivialsubspace. Nonotationforthespaniscompletelystandard.Thesquarebracketsusedhere arecommon,butsoare`span S 'and`sp S '. 2.14Remark InChapterOne,afterweshowedthatthesolutionsetofahomogeneouslinearsystemcanbewrittenas f c 1 ~ 1 + + c k ~ k c 1 ;:::;c k 2 R g wedescribedthatastheset`generated'bythe ~ 's.Wenowhavethetechnical term;wecallthatthe`span'oftheset f ~ 1 ;:::; ~ k g Recallalsothediscussionofthetrickypoint"inthatproof.Thespanof theemptysetisdenedtobetheset f ~ 0 g becausewefollowtheconventionthat alinearcombinationofnovectorssumsto ~ 0.Besides,deningtheemptyset's spantobethetrivialsubspaceisaconvienenceinthatitkeepsresultslikethe nextonefromhavingannoyingexceptionalcases. 2.15Lemma Inavectorspace,thespanofanysubsetisasubspace. Proof Callthesubset S .If S isemptythenbydenitionitsspanisthetrivial subspace.If S isnotemptythenbyLemma2.9weneedonlycheckthatthe span[ S ]isclosedunderlinearcombinations.Forapairofvectorsfromthat span, ~v = c 1 ~s 1 + + c n ~s n and ~w = c n +1 ~s n +1 + + c m ~s m ,alinearcombination p c 1 ~s 1 + + c n ~s n + r c n +1 ~s n +1 + + c m ~s m = pc 1 ~s 1 + + pc n ~s n + rc n +1 ~s n +1 + + rc m ~s m p r scalarsisalinearcombinationofelementsof S andsoisin[ S ]possibly someofthe ~s i 'sforming ~v equalsomeofthe ~s j 'sfrom ~w ,butitdoesnot matter. QED Theconverseofthelemmaholds:anysubspaceisthespanofsomeset, becauseasubspaceisobviouslythespanofthesetofitsmembers.Thusa subsetofavectorspaceisasubspaceifandonlyifitisaspan.Thiststhe intuitionthatagoodwaytothinkofavectorspaceisasacollectioninwhich linearcombinationsaresensible. Takentogether,Lemma2.9andLemma2.15showthatthespanofasubset S ofavectorspaceisthesmallestsubspacecontainingallthemembersof S 2.16Example Inanyvectorspace V ,foranyvector ~v ,theset f r ~v r 2 R g isasubspaceof V .Forinstance,foranyvector ~v 2 R 3 ,thelinethroughthe origincontainingthatvector, f k~v k 2 R g isasubspaceof R 3 .Thisistrueeven when ~v isthezerovector,inwhichcasethesubspaceisthedegenerateline,the trivialsubspace.

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96 ChapterTwo.VectorSpaces 2.17Example Thespanofthissetisallof R 2 f 1 1 ; 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 g Tocheckthiswemustshowthatanymemberof R 2 isalinearcombinationof thesetwovectors.Soweask:forwhichvectorswithrealcomponents x and y aretherescalars c 1 and c 2 suchthatthisholds? c 1 1 1 + c 2 1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 = x y Gauss'method c 1 + c 2 = x c 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(c 2 = y )]TJ/F10 6.9738 Tf 6.227 0 Td [( 1 + 2 )167(! c 1 + c 2 = x )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 c 2 = )]TJ/F11 9.9626 Tf 7.749 0 Td [(x + y withbacksubstitutiongives c 2 = x )]TJ/F11 9.9626 Tf 10.58 0 Td [(y = 2and c 1 = x + y = 2.Thesetwo equationsshowthatforany x and y thatwestartwith,thereareappropriate coecients c 1 and c 2 makingtheabovevectorequationtrue.Forinstance,for x =1and y =2thecoecients c 2 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 = 2and c 1 =3 = 2willdo.Thatis,any vectorin R 2 canbewrittenasalinearcombinationofthetwogivenvectors. Sincespansaresubspaces,andweknowthatagoodwaytounderstanda subspaceistoparametrizeitsdescription,wecantrytounderstandaset'sspan inthatway. 2.18Example Consider,in P 2 ,thespanoftheset f 3 x )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 2 ; 2 x g .Bythe denitionofspan,itisthesetofunrestrictedlinearcombinationsofthetwo f c 1 x )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 2 + c 2 x c 1 ;c 2 2 R g .Clearlypolynomialsinthisspanmusthave aconstanttermofzero.Isthatnecessaryconditionalsosucient? Weareasking:forwhichmembers a 2 x 2 + a 1 x + a 0 of P 2 arethere c 1 and c 2 suchthat a 2 x 2 + a 1 x + a 0 = c 1 x )]TJ/F11 9.9626 Tf 9.764 0 Td [(x 2 + c 2 x ?Sincepolynomialsareequal ifandonlyiftheircoecientsareequal,wearelookingforconditionson a 2 a 1 ,and a 0 satisfyingthese. )]TJ/F11 9.9626 Tf 7.749 0 Td [(c 1 = a 2 3 c 1 +2 c 2 = a 1 0= a 0 Gauss'methodgivesthat c 1 = )]TJ/F11 9.9626 Tf 7.749 0 Td [(a 2 c 2 = = 2 a 2 + = 2 a 1 ,and0= a 0 .Thus theonlyconditiononpolynomialsinthespanistheconditionthatweknew of|aslongas a 0 =0,wecangiveappropriatecoecients c 1 and c 2 todescribe thepolynomial a 0 + a 1 x + a 2 x 2 asinthespan.Forinstance,forthepolynomial 0 )]TJ/F8 9.9626 Tf 9.993 0 Td [(4 x +3 x 2 ,thecoecients c 1 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3and c 2 =5 = 2willdo.Sothespanofthe givensetis f a 1 x + a 2 x 2 a 1 ;a 2 2 R g Thisshows,incidentally,thattheset f x;x 2 g alsospansthissubspace.A spacecanhavemorethanonespanningset.Twoothersetsspanningthissubspaceare f x;x 2 ; )]TJ/F11 9.9626 Tf 7.749 0 Td [(x +2 x 2 g and f x;x + x 2 ;x +2 x 2 ;::: g .Naturally,weusually prefertoworkwithspanningsetsthathaveonlyafewmembers.

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SectionI.DenitionofVectorSpace 97 2.19Example Thesearethesubspacesof R 3 thatwenowknowof,thetrivial subspace,thelinesthroughtheorigin,theplanesthroughtheorigin,andthe wholespaceofcourse,thepictureshowsonlyafewoftheinnitelymany subspaces.Inthenextsectionwewillprovethat R 3 hasnoothertypeof subspaces,soinfactthispictureshowsthemall. f x 1 0 0 + y 0 1 0 + z 0 0 1 g f x 1 0 0 + y 0 1 0 g f x 1 0 0 + z 0 0 1 g )]TJ -5.977 -5.978 Td [()]TJ/F12 4.9813 Tf -35.866 -17.185 Td [(f x 1 1 0 + z 0 0 1 g ... f x 1 0 0 g A A f y 0 1 0 g H H H H f y 2 1 0 g )]TJ -2.989 -2.989 Td [()]TJ/F12 4.9813 Tf -30.884 -16.189 Td [(f y 1 1 1 g ... X X X X X X X X X X X X P P P P P P P P H H H H H @ @ f 0 0 0 g Thesubsetsaredescribedasspansofsets,usingaminimalnumberofmembers, andareshownconnectedtotheirsupersets.Notethatthesesubspacesfall naturallyintolevels|planesononelevel,linesonanother,etc.|accordingto howmanyvectorsareinaminimal-sizedspanningset. Sofarinthischapterwehaveseenthattostudythepropertiesoflinear combinations,therightsettingisacollectionthatisclosedunderthesecombinations.Intherstsubsectionweintroducedsuchcollections,vectorspaces, andwesawagreatvarietyofexamples.Inthissubsectionwesawstillmore spaces,onesthathappentobesubspacesofothers.Inallofthevarietywe've seenacommonality.Example2.19abovebringsitout:vectorspacesandsubspacesarebestunderstoodasaspan,andespeciallyasaspanofasmallnumber ofvectors.Thenextsectionstudiesspanningsetsthatareminimal. Exercises X 2.20 Whichofthesesubsetsofthevectorspaceof2 2matricesaresubspaces undertheinheritedoperations?Foreachonethatisasubspace,parametrizeits description.Foreachthatisnot,giveaconditionthatfails. a f a 0 0 b a;b 2 R g b f a 0 0 b a + b =0 g c f a 0 0 b a + b =5 g d f ac 0 b a + b =0 ;c 2 R g

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98 ChapterTwo.VectorSpaces X 2.21 Isthisasubspaceof P 2 : f a 0 + a 1 x + a 2 x 2 a 0 +2 a 1 + a 2 =4 g ?Ifitisthen parametrizeitsdescription. X 2.22 Decideifthevectorliesinthespanoftheset,insideofthespace. a 2 0 1 f 1 0 0 ; 0 0 1 g ,in R 3 b x )]TJ/F32 8.9664 Tf 9.215 0 Td [(x 3 f x 2 ; 2 x + x 2 ;x + x 3 g ,in P 3 c 01 42 f 10 11 ; 20 23 g ,in M 2 2 2.23 Whichofthesearemembersofthespan[ f cos 2 x; sin 2 x g ]inthevectorspace ofreal-valuedfunctionsofonerealvariable? a f x =1 b f x =3+ x 2 c f x =sin x d f x =cos x X 2.24 Whichofthesesetsspans R 3 ?Thatis,whichofthesesetshastheproperty thatanythree-tallvectorcanbeexpressedasasuitablelinearcombinationofthe set'selements? a f 1 0 0 ; 0 2 0 ; 0 0 3 g b f 2 0 1 ; 1 1 0 ; 0 0 1 g c f 1 1 0 ; 3 0 0 g d f 1 0 1 ; 3 1 0 ; )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 0 0 ; 2 1 5 g e f 2 1 1 ; 3 0 1 ; 5 1 2 ; 6 0 2 g X 2.25 Parametrizeeachsubspace'sdescription.Thenexpresseachsubspaceasa span. a Thesubset f )]TJ/F32 8.9664 Tf 4.566 -7.771 Td [(abc a )]TJ/F32 8.9664 Tf 9.216 0 Td [(c =0 g ofthethree-widerowvectors b Thissubsetof M 2 2 f ab cd a + d =0 g c Thissubsetof M 2 2 f ab cd 2 a )]TJ/F32 8.9664 Tf 9.215 0 Td [(c )]TJ/F32 8.9664 Tf 9.216 0 Td [(d =0and a +3 b =0 g d Thesubset f a + bx + cx 3 a )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 b + c =0 g of P 3 e Thesubsetof P 2 ofquadraticpolynomials p suchthat p =0 X 2.26 Findasettospanthegivensubspaceofthegivenspace. Hint. Parametrize each. a the xz -planein R 3 b f x y z 3 x +2 y + z =0 g in R 3 c f 0 B @ x y z w 1 C A 2 x + y + w =0and y +2 z =0 g in R 4 d f a 0 + a 1 x + a 2 x 2 + a 3 x 3 a 0 + a 1 =0and a 2 )]TJ/F32 8.9664 Tf 9.216 0 Td [(a 3 =0 g in P 3 e Theset P 4 inthespace P 4 f M 2 2 in M 2 2 2.27 Is R 2 asubspaceof R 3 ?

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SectionI.DenitionofVectorSpace 99 X 2.28 Decideifeachisasubspaceofthevectorspaceofreal-valuedfunctionsofone realvariable. a The even functions f f : R R f )]TJ/F32 8.9664 Tf 7.167 0 Td [(x = f x forall x g .Forexample,two membersofthissetare f 1 x = x 2 and f 2 x =cos x b The odd functions f f : R R f )]TJ/F32 8.9664 Tf 7.167 0 Td [(x = )]TJ/F32 8.9664 Tf 7.167 0 Td [(f x forall x g .Twomembersare f 3 x = x 3 and f 4 x =sin x 2.29 Example2.16saysthatforanyvector ~v thatisanelementofavectorspace V ,theset f r ~v r 2 R g isasubspaceof V .Thisisofcourse,simplythespanof thesingletonset f ~v g .Mustanysuchsubspacebeapropersubspace,orcanitbe improper? 2.30 Anexamplefollowingthedenitionofavectorspaceshowsthatthesolution setofahomogeneouslinearsystemisavectorspace.Intheterminologyofthis subsection,itisasubspaceof R n wherethesystemhas n variables.Whatabout anon-homogeneouslinearsystem;doitssolutionsformasubspaceunderthe inheritedoperations? 2.31 Example2.19showsthat R 3 hasinnitelymanysubspaces.Doeseverynontrivialspacehaveinnitelymanysubspaces? 2.32 FinishtheproofofLemma2.9. 2.33 Showthateachvectorspacehasonlyonetrivialsubspace. X 2.34 Showthatforanysubset S ofavectorspace,thespanofthespanequalsthe span[[ S ]]=[ S ]. Hint. Membersof[ S ]arelinearcombinationsofmembersof S Membersof[[ S ]]arelinearcombinationsoflinearcombinationsofmembersof S 2.35 Allofthesubspacesthatwe'veseenusezerointheirdescriptioninsome way.Forexample,thesubspaceinExample2.3consistsofallthevectorsfrom R 2 withasecondcomponentofzero.Incontrast,thecollectionofvectorsfrom R 2 withasecondcomponentofonedoesnotformasubspaceitisnotclosedunder scalarmultiplication.AnotherexampleisExample2.2,wheretheconditionon thevectorsisthatthethreecomponentsaddtozero.Iftheconditionwerethatthe threecomponentsaddtoonethenitwouldnotbeasubspaceagain,itwouldfail tobeclosed.Thisexerciseshowsthatarelianceonzeroisnotstrictlynecessary. Considertheset f x y z x + y + z =1 g undertheseoperations. x 1 y 1 z 1 + x 2 y 2 z 2 = x 1 + x 2 )]TJ/F29 8.9664 Tf 9.215 0 Td [(1 y 1 + y 2 z 1 + z 2 r x y z = rx )]TJ/F32 8.9664 Tf 9.215 0 Td [(r +1 ry rz a Showthatitisnotasubspaceof R 3 Hint. SeeExample2.5. b Showthatitisavectorspace.Notethatbytheprioritem,Lemma2.9can notapply. c Showthatanysubspaceof R 3 mustpassthroughtheorigin,andsoany subspaceof R 3 mustinvolvezeroinitsdescription.Doestheconversehold? Doesanysubsetof R 3 thatcontainstheoriginbecomeasubspacewhengiven theinheritedoperations? 2.36 Wecangiveajusticationfortheconventionthatthesumofzero-many vectorsequalsthezerovector.Considerthissumofthreevectors ~v 1 + ~v 2 +

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100 ChapterTwo.VectorSpaces ~v 3 a Whatisthedierencebetweenthissumofthreevectorsandthesumofthe rsttwoofthesethree? b Whatisthedierencebetweenthepriorsumandthesumofjusttherst onevector? c Whatshouldbethedierencebetweenthepriorsumofonevectorandthe sumofnovectors? d Sowhatshouldbethedenitionofthesumofnovectors? 2.37 Isaspacedeterminedbyitssubspaces?Thatis,iftwovectorspaceshavethe samesubspaces,mustthetwobeequal? 2.38a Giveasetthatisclosedunderscalarmultiplicationbutnotaddition. b Giveasetclosedunderadditionbutnotscalarmultiplication. c Giveasetclosedunderneither. 2.39 Showthatthespanofasetofvectorsdoesnotdependontheorderinwhich thevectorsarelistedinthatset. 2.40 Whichtrivialsubspaceisthespanoftheemptyset?Isit f 0 0 0 g R 3 ; or f 0+0 x gP 1 ; orsomeothersubspace? 2.41 Showthatifavectorisinthespanofasetthenaddingthatvectortotheset won'tmakethespananybigger.Isthatalso`onlyif'? X 2.42 Subspacesaresubsetsandsowenaturallyconsiderhow`isasubspaceof' interactswiththeusualsetoperations. a If A;B aresubspacesofavectorspace,must A B beasubspace?Always? Sometimes?Never? b Must A [ B beasubspace? c If A isasubspace,mustitscomplementbeasubspace? Hint. TrysometestsubspacesfromExample2.19. X 2.43 Doesthespanofasetdependontheenclosingspace?Thatis,if W isa subspaceof V and S isasubsetof W andsoalsoasubsetof V ,mightthespan of S in W dierfromthespanof S in V ? 2.44 Istherelation`isasubspaceof'transitive?Thatis,if V isasubspaceof W and W isasubspaceof X ,must V beasubspaceof X ? X 2.45 Because`spanof'isanoperationonsetswenaturallyconsiderhowitinteracts withtheusualsetoperations. a If S T aresubsetsofavectorspace,is[ S ] [ T ]?Always?Sometimes? Never? b If S;T aresubsetsofavectorspace,is[ S [ T ]=[ S ] [ [ T ]? c If S;T aresubsetsofavectorspace,is[ S T ]=[ S ] [ T ]? d Isthespanofthecomplementequaltothecomplementofthespan? 2.46 ReproveLemma2.15withoutdoingtheemptysetseparately. 2.47 Findastructurethatisclosedunderlinearcombinations,andyetisnota vectorspace. Remark. Thisisabitofatrickquestion.

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SectionII.LinearIndependence 101 IILinearIndependence Thepriorsectionshowsthatavectorspacecanbeunderstoodasanunrestricted linearcombinationofsomeofitselements|thatis,asaspan.Forexample, thespaceoflinearpolynomials f a + bx a;b 2 R g isspannedbytheset f 1 ;x g Thepriorsectionalsoshowedthataspacecanhavemanysetsthatspanit. Thespaceoflinearpolynomialsisalsospannedby f 1 ; 2 x g and f 1 ;x; 2 x g Attheendofthatsectionwedescribedsomespanningsetsas`minimal', butweneverpreciselydenedthatword.Wecouldtake`minimal'tomeanone oftwothings.Wecouldmeanthataspanningsetisminimalifitcontainsthe smallestnumberofmembersofanysetwiththesamespan.Withthismeaning f 1 ;x; 2 x g isnotminimalbecauseithasonemembermorethantheothertwo. Orwecouldmeanthataspanningsetisminimalwhenithasnoelementsthat canberemovedwithoutchangingthespan.Underthismeaning f 1 ;x; 2 x g isnot minimalbecauseremovingthe2 x andgetting f 1 ;x g leavesthespanunchanged. Therstsenseofminimalityappearstobeaglobalrequirement,inthatto checkifaspanningsetisminimalweseeminglymustlookatallthespanningsets ofasubspaceandndonewiththeleastnumberofelements.Thesecondsense ofminimalityislocalinthatweneedtolookonlyatthesetunderdiscussion andconsiderthespanwithandwithoutvariouselements.Forinstance,using thesecondsense,wecouldcomparethespanof f 1 ;x; 2 x g withthespanof f 1 ;x g andnotethatthe2 x isarepeat"inthatitsremovaldoesn'tshrinkthespan. Inthissectionwewillusethesecondsenseof`minimalspanningset'because ofthistechnicalconvenience.However,themostimportantresultofthisbook isthatthetwosensescoincide;wewillprovethatinthesectionafterthisone. II.1DenitionandExamples Werstcharacterizewhenavectorcanberemovedfromasetwithoutchanging thespanofthatset. 1.1Lemma Where S isasubsetofavectorspace V [ S ]=[ S [f ~v g ]ifandonlyif ~v 2 [ S ] forany ~v 2 V Proof Thelefttorightimplicationiseasy.If[ S ]=[ S [f ~v g ]then,since ~v 2 [ S [f ~v g ],theequalityofthetwosetsgivesthat ~v 2 [ S ]. Fortherighttoleftimplicationassumethat ~v 2 [ S ]toshowthat[ S ]=[ S [ f ~v g ]bymutualinclusion.Theinclusion[ S ] [ S [f ~v g ]isobvious.Fortheother inclusion[ S ] [ S [f ~v g ],writeanelementof[ S [f ~v g ]as d 0 ~v + d 1 ~s 1 + + d m ~s m andsubstitute ~v 'sexpansionasalinearcombinationofmembersofthesameset d 0 c 0 ~ t 0 + + c k ~ t k + d 1 ~s 1 + + d m ~s m .Thisisalinearcombinationoflinear combinationsandsodistributing d 0 resultsinalinearcombinationofvectors from S .Henceeachmemberof[ S [f ~v g ]isalsoamemberof[ S ]. QED

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102 ChapterTwo.VectorSpaces 1.2Example In R 3 ,where ~v 1 = 0 @ 1 0 0 1 A ~v 2 = 0 @ 0 1 0 1 A ~v 3 = 0 @ 2 1 0 1 A thespans[ f ~v 1 ;~v 2 g ]and[ f ~v 1 ;~v 2 ;~v 3 g ]areequalsince ~v 3 isinthespan[ f ~v 1 ;~v 2 g ]. Thelemmasaysthatifwehaveaspanningsetthenwecanremovea ~v to getanewset S withthesamespanifandonlyif ~v isalinearcombinationof vectorsfrom S .Thus,underthesecondsensedescribedabove,aspanningset isminimalifandonlyifitcontainsnovectorsthatarelinearcombinationsof theothersinthatset.Wehaveatermforthisimportantproperty. 1.3Denition Asubsetofavectorspaceis linearlyindependent ifnone ofitselementsisalinearcombinationoftheothers.Otherwiseitis linearly dependent Hereisanimportantobservation:althoughthiswayofwritingonevector asacombinationoftheothers ~s 0 = c 1 ~s 1 + c 2 ~s 2 + + c n ~s n visuallysets ~s 0 ofromtheothervectors,algebraicallythereisnothingspecial inthatequationabout ~s 0 .Forany ~s i withacoecient c i thatisnonzero,we canrewritetherelationshiptoseto ~s i ~s i = =c i ~s 0 + )]TJ/F11 9.9626 Tf 7.749 0 Td [(c 1 =c i ~s 1 + + )]TJ/F11 9.9626 Tf 7.748 0 Td [(c n =c i ~s n Whenwedon'twanttosingleoutanyvectorbywritingitaloneononesideofthe equationwewillinsteadsaythat ~s 0 ;~s 1 ;:::;~s n areina linearrelationship and writetherelationshipwithallofthevectorsonthesameside.Thenextresult rephrasesthelinearindependencedenitioninthisstyle.Itgiveswhatisusually theeasiestwaytocomputewhetheranitesetisdependentorindependent. 1.4Lemma Asubset S ofavectorspaceislinearlyindependentifandonlyif foranydistinct ~s 1 ;:::;~s n 2 S theonlylinearrelationshipamongthosevectors c 1 ~s 1 + + c n ~s n = ~ 0 c 1 ;:::;c n 2 R isthetrivialone: c 1 =0 ;:::;c n =0. Proof Thisisadirectconsequenceoftheobservationabove. Iftheset S islinearlyindependentthennovector ~s i canbewrittenasalinear combinationoftheothervectorsfrom S sothereisnolinearrelationshipwhere someofthe ~s 'shavenonzerocoecients.If S isnotlinearlyindependentthen some ~s i isalinearcombination ~s i = c 1 ~s 1 + + c i )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ~s i )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 + c i +1 ~s i +1 + + c n ~s n of othervectorsfrom S ,andsubtracting ~s i frombothsidesofthatequationgives alinearrelationshipinvolvinganonzerocoecient,namelythe )]TJ/F8 9.9626 Tf 7.749 0 Td [(1infrontof ~s i QED

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SectionII.LinearIndependence 103 1.5Example Inthevectorspaceoftwo-widerowvectors,thetwo-elementset f )]TJ/F8 9.9626 Tf 4.566 -7.97 Td [(4015 ; )]TJ/F14 9.9626 Tf 4.566 -7.97 Td [()]TJ/F8 9.9626 Tf 7.749 0 Td [(5025 g islinearlyindependent.Tocheckthis,set c 1 )]TJ/F8 9.9626 Tf 4.566 -7.97 Td [(4015 + c 2 )]TJ/F14 9.9626 Tf 4.566 -7.97 Td [()]TJ/F8 9.9626 Tf 7.749 0 Td [(5025 = )]TJ/F8 9.9626 Tf 4.566 -7.97 Td [(00 andsolvingtheresultingsystem 40 c 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(50 c 2 =0 15 c 1 +25 c 2 =0 )]TJ/F7 6.9738 Tf 6.227 0 Td [( = 40 1 + 2 )167(! 40 c 1 )]TJ/F8 9.9626 Tf 32.655 0 Td [(50 c 2 =0 = 4 c 2 =0 showsthatboth c 1 and c 2 arezero.Sotheonlylinearrelationshipbetweenthe twogivenrowvectorsisthetrivialrelationship. Inthesamevectorspace, f )]TJ/F8 9.9626 Tf 4.566 -7.97 Td [(4015 ; )]TJ/F8 9.9626 Tf 4.567 -7.97 Td [(207 : 5 g islinearlydependentsince wecansatisfy c 1 )]TJ/F8 9.9626 Tf 4.566 -7.97 Td [(4015 + c 2 )]TJ/F8 9.9626 Tf 4.566 -7.97 Td [(207 : 5 = )]TJ/F8 9.9626 Tf 4.567 -7.97 Td [(00 with c 1 =1and c 2 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2. 1.6Remark RecalltheStaticsexamplethatbeganthisbook.Werstsetthe unknown-massobjectsat40cmand15cmandgotabalance,andthenweset theobjectsat )]TJ/F8 9.9626 Tf 7.749 0 Td [(50cmand25cmandgotabalance.Withthosetwopiecesof informationwecouldcomputevaluesoftheunknownmasses.Hadweinstead rstsettheunknown-massobjectsat40cmand15cm,andthenat20cmand 7 : 5cm,wewouldnothavebeenabletocomputethevaluesoftheunknown massestryit.Intuitively,theproblemisthatthe )]TJ/F8 9.9626 Tf 4.566 -7.97 Td [(207 : 5 informationisa repeat"ofthe )]TJ/F8 9.9626 Tf 4.566 -7.97 Td [(4015 information|thatis, )]TJ/F8 9.9626 Tf 4.566 -7.97 Td [(207 : 5 isinthespanofthe set f )]TJ/F8 9.9626 Tf 4.566 -7.971 Td [(4015 g |andsowewouldbetryingtosolveatwo-unknownsproblem withwhatisessentiallyonepieceofinformation. 1.7Example Theset f 1+ x; 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x g islinearlyindependentin P 2 ,thespace ofquadraticpolynomialswithrealcoecients,because 0+0 x +0 x 2 = c 1 + x + c 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x = c 1 + c 2 + c 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(c 2 x +0 x 2 gives c 1 + c 2 =0 c 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(c 2 =0 )]TJ/F10 6.9738 Tf 6.226 0 Td [( 1 + 2 )167(! c 1 + c 2 =0 2 c 2 =0 sincepolynomialsareequalonlyiftheircoecientsareequal.Thus,theonly linearrelationshipbetweenthesetwomembersof P 2 isthetrivialone. 1.8Example In R 3 ,where ~v 1 = 0 @ 3 4 5 1 A ~v 2 = 0 @ 2 9 2 1 A ~v 3 = 0 @ 4 18 4 1 A theset S = f ~v 1 ;~v 2 ;~v 3 g islinearlydependentbecausethisisarelationship 0 ~v 1 +2 ~v 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ~v 3 = ~ 0 wherenotallofthescalarsarezerothefactthatsomeofthescalarsarezero doesn'tmatter.

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104 ChapterTwo.VectorSpaces 1.9Remark Thatexampleillustrateswhy,althoughDenition1.3isaclearer statementofwhatindependenceis,Lemma1.4ismoreusefulforcomputations. Workingstraightfromthedenition,someonetryingtocomputewhether S is linearlyindependentwouldstartbysetting ~v 1 = c 2 ~v 2 + c 3 ~v 3 andconcluding thattherearenosuch c 2 and c 3 .Butknowingthattherstvectorisnot dependentontheothertwoisnotenough.Thispersonwouldhavetogoonto try ~v 2 = c 1 ~v 1 + c 3 ~v 3 tondthedependence c 1 =0, c 3 =1 = 2.Lemma1.4gets thesameconclusionwithonlyonecomputation. 1.10Example Theemptysubsetofavectorspaceislinearlyindependent. Thereisnonontriviallinearrelationshipamongitsmembersasithasnomembers. 1.11Example Inanyvectorspace,anysubsetcontainingthezerovectoris linearlydependent.Forexample,inthespace P 2 ofquadraticpolynomials, considerthesubset f 1+ x;x + x 2 ; 0 g OnewaytoseethatthissubsetislinearlydependentistouseLemma1.4:we have0 ~v 1 +0 ~v 2 +1 ~ 0= ~ 0,andthisisanontrivialrelationshipasnotallofthe coecientsarezero.Anotherwaytoseethatthissubsetislinearlydependent istogostraighttoDenition1.3:wecanexpressthethirdmemberofthesubset asalinearcombinationofthersttwo,namely, c 1 ~v 1 + c 2 ~v 2 = ~ 0issatisedby taking c 1 =0and c 2 =0incontrasttothelemma,thedenitionallowsallof thecoecientstobezero. Thereisstillanotherwaytoseethatthissubsetisdependentthatissubtler. Thezerovectorisequaltothetrivialsum,thatis,itisthesumofnovectors. Soinasetcontainingthezerovector,thereisanelementthatcanbewritten asacombinationofacollectionofothervectorsfromtheset,specically,the zerovectorcanbewrittenasacombinationoftheemptycollection. Theaboveexamples,especiallyExample1.5,underlinethediscussionthat beginsthissection.Thenextresultsaysthatgivenaniteset,wecanproduce alinearlyindependentsubsetbydiscardingwhatRemark1.6callsrepeats". 1.12Theorem Inavectorspace,anynitesubsethasalinearlyindependent subsetwiththesamespan. Proof Iftheset S = f ~s 1 ;:::;~s n g islinearlyindependentthen S itselfsatises thestatement,soassumethatitislinearlydependent. Bythedenitionofdependence,thereisavector ~s i thatisalinearcombinationoftheothers.Callthatvector ~v 1 .Discardit|denetheset S 1 = S )-75(f ~v 1 g ByLemma1.1,thespandoesnotshrink[ S 1 ]=[ S ]. Now,if S 1 islinearlyindependentthenwearenished.Otherwiseiteratethe priorparagraph:takeavector ~v 2 thatisalinearcombinationofothermembers of S 1 anddiscardittoderive S 2 = S 1 )-182(f ~v 2 g suchthat[ S 2 ]=[ S 1 ].Repeatthis untilalinearlyindependentset S j appears;onemustappeareventuallybecause S isniteandtheemptysetislinearlyindependent.Formally,thisargument usesinductionon n ,thenumberofelementsinthestartingset.Exercise37 asksforthedetails. QED

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SectionII.LinearIndependence 105 1.13Example Thissetspans R 3 S = f 0 @ 1 0 0 1 A ; 0 @ 0 2 0 1 A ; 0 @ 1 2 0 1 A ; 0 @ 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 1 A ; 0 @ 3 3 0 1 A g Lookingforalinearrelationship c 1 0 @ 1 0 0 1 A + c 2 0 @ 0 2 0 1 A + c 3 0 @ 1 2 0 1 A + c 4 0 @ 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 1 A + c 5 0 @ 3 3 0 1 A = 0 @ 0 0 0 1 A givesathreeequations/veunknownslinearsystemwhosesolutionsetcanbe parametrizedinthisway. f 0 B B B B @ c 1 c 2 c 3 c 4 c 5 1 C C C C A = c 3 0 B B B B @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 0 0 1 C C C C A + c 5 0 B B B B @ )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 = 2 0 0 1 1 C C C C A c 3 ;c 5 2 R g So S islinearlydependent.Setting c 3 =0and c 5 =1showsthatthefthvector isalinearcombinationofthersttwo.Thus,Lemma1.1saysthatdiscarding thefthvector S 1 = f 0 @ 1 0 0 1 A ; 0 @ 0 2 0 1 A ; 0 @ 1 2 0 1 A ; 0 @ 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 1 A g leavesthespanunchanged[ S 1 ]=[ S ].Now,thethirdvectorof S 1 isalinear combinationofthersttwoandweget S 2 = f 0 @ 1 0 0 1 A ; 0 @ 0 2 0 1 A ; 0 @ 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 1 A g withthesamespanas S 1 ,andthereforethesamespanas S ,butwithone dierence.Theset S 2 islinearlyindependentthisiseasilychecked,andso discardinganyofitselementswillshrinkthespan. Theorem1.12describesproducingalinearlyindependentsetbyshrinking, thatis,bytakingsubsets.Wenishthissubsectionbyconsideringhowlinear independenceanddependence,whicharepropertiesofsets,interactwiththe subsetrelationbetweensets. 1.14Lemma Anysubsetofalinearlyindependentsetisalsolinearlyindependent.Anysupersetofalinearlydependentsetisalsolinearlydependent. Proof Thisisclear. QED

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106 ChapterTwo.VectorSpaces Restated,independenceispreservedbysubsetanddependenceispreserved bysuperset. Thosearetwoofthefourpossiblecasesofinteractionthatwecanconsider. Thethirdcase,whetherlineardependenceispreservedbythesubsetoperation, iscoveredbyExample1.13,whichgivesalinearlydependentset S withasubset S 1 thatislinearlydependentandanothersubset S 2 thatislinearlyindependent. Thatleavesonecase,whetherlinearindependenceispreservedbysuperset. Thenextexampleshowswhatcanhappen. 1.15Example Ineachofthesethreeparagraphsthesubset S islinearly independent. Fortheset S = f 0 @ 1 0 0 1 A g thespan[ S ]isthe x axis.Herearetwosupersetsof S ,onelinearlydependent andtheotherlinearlyindependent. dependent: f 0 @ 1 0 0 1 A ; 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 0 0 1 A g independent: f 0 @ 1 0 0 1 A ; 0 @ 0 1 0 1 A g Checkingthedependenceorindependenceofthesesetsiseasy. For S = f 0 @ 1 0 0 1 A ; 0 @ 0 1 0 1 A g thespan[ S ]isthe xy plane.Thesearetwosupersets. dependent: f 0 @ 1 0 0 1 A ; 0 @ 0 1 0 1 A ; 0 @ 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 0 1 A g independent: f 0 @ 1 0 0 1 A ; 0 @ 0 1 0 1 A ; 0 @ 0 0 1 1 A g If S = f 0 @ 1 0 0 1 A ; 0 @ 0 1 0 1 A ; 0 @ 0 0 1 1 A g then[ S ]= R 3 .Alinearlydependentsupersetis dependent: f 0 @ 1 0 0 1 A ; 0 @ 0 1 0 1 A ; 0 @ 0 0 1 1 A ; 0 @ 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 3 1 A g buttherearenolinearlyindependentsupersetsof S .Thereasonisthatforany vectorthatwewouldaddtomakeasuperset,thelineardependenceequation 0 @ x y z 1 A = c 1 0 @ 1 0 0 1 A + c 2 0 @ 0 1 0 1 A + c 3 0 @ 0 0 1 1 A hasasolution c 1 = x c 2 = y ,and c 3 = z .

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SectionII.LinearIndependence 107 So,ingeneral,alinearlyindependentsetmayhaveasupersetthatisdependent.And,ingeneral,alinearlyindependentsetmayhaveasupersetthatis independent.Wecancharacterizewhenthesupersetisoneandwhenitisthe other. 1.16Lemma Where S isalinearlyindependentsubsetofavectorspace V S [f ~v g islinearlydependentifandonlyif ~v 2 [ S ] forany ~v 2 V with ~v 62 S Proof Oneimplicationisclear:if ~v 2 [ S ]then ~v = c 1 ~s 1 + c 2 ~s 2 + + c n ~s n whereeach ~s i 2 S and c i 2 R ,andso ~ 0= c 1 ~s 1 + c 2 ~s 2 + + c n ~s n + )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ~v isa nontriviallinearrelationshipamongelementsof S [f ~v g Theotherimplicationrequirestheassumptionthat S islinearlyindependent. With S [f ~v g linearlydependent,thereisanontriviallinearrelationship c 0 ~v + c 1 ~s 1 + c 2 ~s 2 + + c n ~s n = ~ 0andindependenceof S thenimpliesthat c 0 6 =0,or elsethatwouldbeanontrivialrelationshipamongmembersof S .Nowrewriting thisequationas ~v = )]TJ/F8 9.9626 Tf 7.749 0 Td [( c 1 =c 0 ~s 1 )-222()]TJ/F8 9.9626 Tf 33.762 0 Td [( c n =c 0 ~s n showsthat ~v 2 [ S ]. QED ComparethisresultwithLemma1.1.Bothsay,roughly,that ~v isarepeat" ifitisinthespanof S .However,notetheadditionalhypothesishereoflinear independence. 1.17Corollary Asubset S = f ~s 1 ;:::;~s n g ofavectorspaceislinearlydependentifandonlyifsome ~s i isalinearcombinationofthevectors ~s 1 ,..., ~s i )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 listedbeforeit. Proof Consider S 0 = fg S 1 = f ~s 1 g S 2 = f ~s 1 ;~s 2 g ,etc.Someindex i 1is therstonewith S i )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 [f ~s i g linearlydependent,andthere ~s i 2 [ S i )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 ]. QED Lemma1.16canberestatedintermsofindependenceinsteadofdependence: if S islinearlyindependentand ~v 62 S thentheset S [f ~v g isalsolinearly independentifandonlyif ~v 62 [ S ] : ApplyingLemma1.1,weconcludethatif S islinearlyindependentand ~v 62 S then S [f ~v g isalsolinearlyindependentif andonlyif[ S [f ~v g ] 6 =[ S ].Briey,whenpassingfrom S toasuperset S 1 ,to preservelinearindependencewemustexpandthespan[ S 1 ] [ S ]. Example1.15showsthatsomelinearlyindependentsetsaremaximal|have asmanyelementsaspossible|inthattheyhavenosupersetsthatarelinearly independent.Bythepriorparagraph,alinearlyindependentsetsismaximalif andonlyifitspanstheentirespace,becausethennovectorexiststhatisnot alreadyinthespan. Thistablesummarizestheinteractionbetweenthepropertiesofindependenceanddependenceandtherelationsofsubsetandsuperset. S 1 SS 1 S S independent S 1 mustbeindependent S 1 maybeeither S dependent S 1 maybeeither S 1 mustbedependent

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108 ChapterTwo.VectorSpaces Indevelopingthistablewe'veuncoveredanintimaterelationshipbetweenlinear independenceandspan.Complementingthefactthataspanningsetisminimal ifandonlyifitislinearlyindependent,alinearlyindependentsetismaximalif andonlyifitspansthespace. Insummary,wehaveintroducedthedenitionoflinearindependenceto formalizetheideaoftheminimalityofaspanningset.Wehavedevelopedsome propertiesofthisidea.ThemostimportantisLemma1.16,whichtellsusthat alinearlyindependentsetismaximalwhenitspansthespace. Exercises X 1.18 Decidewhethereachsubsetof R 3 islinearlydependentorlinearlyindependent. a f 1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(3 5 ; 2 2 4 ; 4 )]TJ/F29 8.9664 Tf 7.168 0 Td [(4 14 g b f 1 7 7 ; 2 7 7 ; 3 7 7 g c f 0 0 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 ; 1 0 4 g d f 9 9 0 ; 2 0 1 ; 3 5 )]TJ/F29 8.9664 Tf 7.168 0 Td [(4 ; 12 12 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 g X 1.19 Whichofthesesubsetsof P 3 arelinearlydependentandwhichareindependent? a f 3 )]TJ/F32 8.9664 Tf 9.215 0 Td [(x +9 x 2 ; 5 )]TJ/F29 8.9664 Tf 9.215 0 Td [(6 x +3 x 2 ; 1+1 x )]TJ/F29 8.9664 Tf 9.216 0 Td [(5 x 2 g b f)]TJ/F32 8.9664 Tf 11.775 0 Td [(x 2 ; 1+4 x 2 g c f 2+ x +7 x 2 ; 3 )]TJ/F32 8.9664 Tf 9.216 0 Td [(x +2 x 2 ; 4 )]TJ/F29 8.9664 Tf 9.215 0 Td [(3 x 2 g d f 8+3 x +3 x 2 ;x +2 x 2 ; 2+2 x +2 x 2 ; 8 )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 x +5 x 2 g X 1.20 Provethateachset f f;g g islinearlyindependentinthevectorspaceofall functionsfrom R + to R a f x = x and g x =1 =x b f x =cos x and g x =sin x c f x = e x and g x =ln x X 1.21 Whichofthesesubsetsofthespaceofreal-valuedfunctionsofonerealvariableislinearlydependentandwhichislinearlyindependent?Notethatwehave abbreviatedsomeconstantfunctions;e.g.,intherstitem,the`2'standsforthe constantfunction f x =2. a f 2 ; 4sin 2 x ; cos 2 x g b f 1 ; sin x ; sin x g c f x; cos x g d f + x 2 ;x 2 +2 x; 3 g e f cos x ; sin 2 x ; cos 2 x g f f 0 ;x;x 2 g 1.22 Doestheequationsin 2 x = cos 2 x =tan 2 x showthatthissetoffunctions f sin 2 x ; cos 2 x ; tan 2 x g isalinearlydependentsubsetofthesetofallreal-valued functionswithdomaintheinterval )]TJ/F32 8.9664 Tf 7.168 0 Td [(= 2 ::= 2ofrealnumbersbetween )]TJ/F32 8.9664 Tf 7.168 0 Td [(= 2and = 2? 1.23 WhydoesLemma1.4saydistinct"? X 1.24 Showthatthenonzerorowsofanechelonformmatrixformalinearlyindependentset.

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SectionII.LinearIndependence 109 X 1.25a Showthatiftheset f ~u;~v;~w g islinearlyindependentsetthensoisthe set f ~u;~u + ~v;~u + ~v + ~w g b Whatistherelationshipbetweenthelinearindependenceordependenceof theset f ~u;~v;~w g andtheindependenceordependenceof f ~u )]TJ/F32 8.9664 Tf 8.876 0 Td [(~v;~v )]TJ/F32 8.9664 Tf 10.413 0 Td [(~w;~w )]TJ/F32 8.9664 Tf 9.129 0 Td [(~u g ? 1.26 Example1.10showsthattheemptysetislinearlyindependent. a Whenisaone-elementsetlinearlyindependent? b Howaboutasetwithtwoelements? 1.27 Inanyvectorspace V ,theemptysetislinearlyindependent.Whataboutall of V ? 1.28 Showthatif f ~x;~y;~z g islinearlyindependentthensoareallofitsproper subsets: f ~x;~y g f ~x;~z g f ~y;~z g f ~x g f ~y g f ~z g ,and fg .Isthat`onlyif'also? 1.29a Showthatthis S = f 1 1 0 ; )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 2 0 g isalinearlyindependentsubsetof R 3 b Showthat 3 2 0 isinthespanof S bynding c 1 and c 2 givingalinearrelationship. c 1 1 1 0 + c 2 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 2 0 = 3 2 0 Showthatthepair c 1 ;c 2 isunique. c Assumethat S isasubsetofavectorspaceandthat ~v isin[ S ],sothat ~v is alinearcombinationofvectorsfrom S .Provethatif S islinearlyindependent thenalinearcombinationofvectorsfrom S addingto ~v isuniquethatis,unique uptoreorderingandaddingortakingawaytermsoftheform0 ~s .Thus S asaspanningsetisminimalinthisstrongsense:eachvectorin[ S ]ishit"a minimumnumberoftimes|onlyonce. d Provethatitcanhappenwhen S isnotlinearlyindependentthatdistinct linearcombinationssumtothesamevector. 1.30 Provethatapolynomialgivesrisetothezerofunctionifandonlyifitis thezeropolynomial. Comment. ThisquestionisnotaLinearAlgebramatter, butweoftenusetheresult.Apolynomialgivesrisetoafunctionintheobvious way: x 7! c n x n + + c 1 x + c 0 1.31 ReturntoSection1.2andredenepoint,line,plane,andotherlinearsurfaces toavoiddegeneratecases. 1.32a Showthatanysetoffourvectorsin R 2 islinearlydependent. b Isthistrueforanysetofve?Anysetofthree? c Whatisthemostnumberofelementsthatalinearlyindependentsubsetof R 2 canhave? X 1.33 Isthereasetoffourvectorsin R 3 ,anythreeofwhichformalinearlyindependentset? 1.34 Musteverylinearlydependentsethaveasubsetthatisdependentanda subsetthatisindependent?

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110 ChapterTwo.VectorSpaces 1.35 In R 4 ,whatisthebiggestlinearlyindependentsetyoucannd?Thesmallest? Thebiggestlinearlydependentset?Thesmallest?`Biggest'and`smallest'mean thattherearenosupersetsorsubsetswiththesameproperty. X 1.36 Linearindependenceandlineardependencearepropertiesofsets.Wecan thusnaturallyaskhowthosepropertiesactwithrespecttothefamiliarelementary setrelationsandoperations.Inthisbodyofthissubsectionwehavecoveredthe subsetandsupersetrelations.Wecanalsoconsidertheoperationsofintersection, complementation,andunion. a Howdoeslinearindependencerelatetointersection:cananintersectionof linearlyindependentsetsbeindependent?Mustitbe? b Howdoeslinearindependencerelatetocomplementation? c Showthattheunionoftwolinearlyindependentsetsneednotbelinearly independent. d Characterizewhentheunionoftwolinearlyindependentsetsislinearlyindependent,intermsoftheintersectionofthespanofeach. X 1.37 ForTheorem1.12, a llintheinductionfortheproof; b giveanalternateproofthatstartswiththeemptysetandbuildsasequence oflinearlyindependentsubsetsofthegivennitesetuntiloneappearswiththe samespanasthegivenset. 1.38 Withalittlecalculationwecangetformulastodeterminewhetherornota setofvectorsislinearlyindependent. a Showthatthissubsetof R 2 f a c ; b d g islinearlyindependentifandonlyif ad )]TJ/F32 8.9664 Tf 9.216 0 Td [(bc 6 =0. b Showthatthissubsetof R 3 f a d g ; b e h ; c f i g islinearlyindependenti aei + bfg + cdh )]TJ/F32 8.9664 Tf 9.216 0 Td [(hfa )]TJ/F32 8.9664 Tf 9.215 0 Td [(idb )]TJ/F32 8.9664 Tf 9.216 0 Td [(gec 6 =0. c Whenisthissubsetof R 3 f a d g ; b e h g linearlyindependent? d Thisisanopinionquestion:forasetoffourvectorsfrom R 4 ,musttherebe aformulainvolvingthesixteenentriesthatdeterminesindependenceoftheset? Youneedn'tproducesuchaformula,justdecideifoneexists. X 1.39a Provethatasetoftwoperpendicularnonzerovectorsfrom R n islinearly independentwhen n> 1. b Whatif n =1? n =0? c Generalizetomorethantwovectors. 1.40 Considerthesetoffunctionsfromtheopeninterval )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 :: 1to R a Showthatthissetisavectorspaceundertheusualoperations. b Recalltheformulaforthesumofaninnitegeometricseries:1+ x + x 2 + = 1 = )]TJ/F32 8.9664 Tf 7.28 0 Td [(x forall x 2 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 :: 1.Whydoesthisnotexpressadependenceinsideofthe set f g x =1 = )]TJ/F32 8.9664 Tf 9.216 0 Td [(x ;f 0 x =1 ;f 1 x = x;f 2 x = x 2 ;::: g inthevectorspace thatweareconsidering? Hint. Reviewthedenitionoflinearcombination.

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SectionII.LinearIndependence 111 c Showthatthesetintheprioritemislinearlyindependent. Thisshowsthatsomevectorspacesexistwithlinearlyindependentsubsetsthat areinnite. 1.41 Showthat,where S isasubspaceof V ,ifasubset T of S islinearlyindependentin S then T isalsolinearlyindependentin V .Isthat`onlyif'?

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112 ChapterTwo.VectorSpaces IIIBasisandDimension Thepriorsectionendswiththestatementthataspanningsetisminimalwhenit islinearlyindependentandalinearlyindependentsetismaximalwhenitspans thespace.Sothenotionsofminimalspanningsetandmaximalindependent setcoincide.Inthissectionwewillnamethisideaandstudyitsproperties. III.1Basis 1.1Denition A basis foravectorspaceisasequenceofvectorsthatform asetthatislinearlyindependentandthatspansthespace. Wedenoteabasiswithanglebrackets h ~ 1 ; ~ 2 ;::: i tosignifythatthiscollectionisasequence |theorderoftheelementsissignicant.Therequirement thatabasisbeorderedwillbeneeded,forinstance,inDenition1.13. 1.2Example Thisisabasisfor R 2 h 2 4 ; 1 1 i Itislinearlyindependent c 1 2 4 + c 2 1 1 = 0 0 = 2 c 1 +1 c 2 =0 4 c 1 +1 c 2 =0 = c 1 = c 2 =0 anditspans R 2 2 c 1 +1 c 2 = x 4 c 1 +1 c 2 = y = c 2 =2 x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y and c 1 = y )]TJ/F11 9.9626 Tf 9.962 0 Td [(x = 2 1.3Example Thisbasisfor R 2 h 1 1 ; 2 4 i diersfromtheprioronebecausethevectorsareinadierentorder.The vericationthatitisabasisisjustasinthepriorexample. 1.4Example Thespace R 2 hasmanybases.Anotheroneisthis. h 1 0 ; 0 1 i Thevericationiseasy. Moreinformationonsequencesisintheappendix.

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SectionIII.BasisandDimension 113 1.5Denition Forany R n E n = h 0 B B B @ 1 0 0 1 C C C A ; 0 B B B @ 0 1 0 1 C C C A ;:::; 0 B B B @ 0 0 1 1 C C C A i isthe standard or natural basis.Wedenotethesevectorsby ~e 1 ;:::;~e n Calculusbooksreferto R 2 'sstandardbasisvectors ~{ and ~| insteadof ~e 1 and ~e 2 ,andtheyreferto R 3 'sstandardbasisvectors ~{ ~| ,and ~ k insteadof ~e 1 ~e 2 and ~e 3 .Notethatthesymbol` ~e 1 'meanssomethingdierentinadiscussionof R 3 thanitmeansinadiscussionof R 2 1.6Example Considerthespace f a cos + b sin a;b 2 R g offunctionsof therealvariable .Thisisanaturalbasis. h 1 cos +0 sin ; 0 cos +1 sin i = h cos ; sin i Another,moregeneric,basisis h cos )]TJ/F8 9.9626 Tf 10.108 0 Td [(sin ; 2cos +3sin i .Vercationthat thesetwoarebasesisExercise22. 1.7Example Anaturalbasisforthevectorspaceofcubicpolynomials P 3 is h 1 ;x;x 2 ;x 3 i .Twootherbasesforthisspaceare h x 3 ; 3 x 2 ; 6 x; 6 i and h 1 ; 1+ x; 1+ x + x 2 ; 1+ x + x 2 + x 3 i .Checkingthatthesearelinearlyindependentandspan thespaceiseasy. 1.8Example Thetrivialspace f ~ 0 g hasonlyonebasis,theemptyone hi 1.9Example Thespaceofnite-degreepolynomialshasabasiswithinnitely manyelements h 1 ;x;x 2 ;::: i 1.10Example Wehaveseenbasesbefore.Intherstchapterwedescribed thesolutionsetofhomogeneoussystemssuchasthisone x + y )]TJ/F11 9.9626 Tf 9.962 0 Td [(w =0 z + w =0 byparametrizing. f 0 B B @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 0 0 1 C C A y + 0 B B @ 1 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 1 C C A w y;w 2 R g Thatis,wedescribedthevectorspaceofsolutionsasthespanofatwo-element set.Wecaneasilycheckthatthistwo-vectorsetisalsolinearlyindependent. Thusthesolutionsetisasubspaceof R 4 withatwo-elementbasis.

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114 ChapterTwo.VectorSpaces 1.11Example Parameterizationhelpsndbasesforothervectorspaces,not justforsolutionsetsofhomogeneoussystems.Tondabasisforthissubspace of M 2 2 f ab c 0 a + b )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 c =0 g werewritetheconditionas a = )]TJ/F11 9.9626 Tf 7.748 0 Td [(b +2 c f )]TJ/F11 9.9626 Tf 7.749 0 Td [(b +2 cb c 0 b;c 2 R g = f b )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 00 + c 20 10 b;c 2 R g Thus,thisisanaturalcandidateforabasis. h )]TJ/F8 9.9626 Tf 7.748 0 Td [(11 00 ; 20 10 i Theaboveworkshowsthatitspansthespace.Toshowthatitislinearly independentisroutine. ConsideragainExample1.2.Itinvolvestwoverications. Intherst,tocheckthatthesetislinearlyindependentwelookedatlinear combinationsoftheset'smembersthattotaltothezerovector c 1 ~ 1 + c 2 ~ 2 = )]TJ/F7 6.9738 Tf 4.567 -3.649 Td [(0 0 Theresultingcalculationshowsthatsuchacombinationisunique,that c 1 must be0and c 2 mustbe0. Thesecondverication,thatthesetspansthespace,looksatlinearcombinationsthattotaltoanymemberofthespace c 1 ~ 1 + c 2 ~ 2 = )]TJ/F10 6.9738 Tf 4.566 -3.649 Td [(x y .InExample1.2 wenotedonlythattheresultingcalculationshowsthatsuchacombinationexists,thatforeach x;y thereisa c 1 ;c 2 .However,infactthecalculationalso showsthatthecombinationisunique: c 1 mustbe y )]TJ/F11 9.9626 Tf 10.593 0 Td [(x = 2and c 2 mustbe 2 x )]TJ/F11 9.9626 Tf 9.962 0 Td [(y Thatis,therstcalculationisaspecialcaseofthesecond.Thenextresult saysthatthisholdsingeneralforaspanningset:thecombinationtotalingto thezerovectorisuniqueifandonlyifthecombinationtotalingtoanyvector isunique. 1.12Theorem Inanyvectorspace,asubsetisabasisifandonlyifeach vectorinthespacecanbeexpressedasalinearcombinationofelementsofthe subsetinauniqueway. Weconsidercombinationstobethesameiftheydieronlyintheorderof summandsorintheadditionordeletionoftermsoftheform`0 ~ '. Proof Bydenition,asequenceisabasisifandonlyifitsvectorsformboth aspanningsetandalinearlyindependentset.Asubsetisaspanningsetif andonlyifeachvectorinthespaceisalinearcombinationofelementsofthat subsetinatleastoneway. Thus,tonishweneedonlyshowthatasubsetislinearlyindependentif andonlyifeveryvectorinthespaceisalinearcombinationofelementsfrom thesubsetinatmostoneway.Considertwoexpressionsofavectorasalinear

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SectionIII.BasisandDimension 115 combinationofthemembersofthebasis.Wecanrearrangethetwosums,and ifnecessaryaddsome0 ~ i terms,sothatthetwosumscombinethesame ~ 'sin thesameorder: ~v = c 1 ~ 1 + c 2 ~ 2 + + c n ~ n and ~v = d 1 ~ 1 + d 2 ~ 2 + + d n ~ n Now c 1 ~ 1 + c 2 ~ 2 + + c n ~ n = d 1 ~ 1 + d 2 ~ 2 + + d n ~ n holdsifandonlyif c 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(d 1 ~ 1 + + c n )]TJ/F11 9.9626 Tf 9.962 0 Td [(d n ~ n = ~ 0 holds,andsoassertingthateachcoecientinthelowerequationiszeroisthe samethingasassertingthat c i = d i foreach i QED 1.13Denition Inavectorspacewithbasis B the representationof ~v with respectto B isthecolumnvectorofthecoecientsusedtoexpress ~v asalinear combinationofthebasisvectors: Rep B ~v = 0 B B B @ c 1 c 2 c n 1 C C C A where B = h ~ 1 ;:::; ~ n i and ~v = c 1 ~ 1 + c 2 ~ 2 + + c n ~ n .The c 'sarethe coordinatesof ~v withrespectto B Wewilllaterdorepresentationsincontextsthatinvolvemorethanonebasis. Tohelpwiththebookkeeping,weshalloftenattachasubscript B tothecolumn vector. 1.14Example In P 3 ,withrespecttothebasis B = h 1 ; 2 x; 2 x 2 ; 2 x 3 i ,the representationof x + x 2 is Rep B x + x 2 = 0 B B @ 0 1 = 2 1 = 2 0 1 C C A B notethatthecoordinatesarescalars,notvectors.Withrespecttoadierent basis D = h 1+ x; 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x;x + x 2 ;x + x 3 i ,therepresentation Rep D x + x 2 = 0 B B @ 0 0 1 0 1 C C A D isdierent.

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116 ChapterTwo.VectorSpaces 1.15Remark Thisuseofcolumnnotationandtheterm`coordinates'has bothadownsideandanupside. Thedownsideisthatrepresentationslooklikevectorsfrom R n ,whichcan beconfusingwhenthevectorspaceweareworkingwithis R n ,especiallysince wesometimesomitthesubscriptbase.Wemusttheninfertheintentfromthe context.Forexample,thephrase`in R 2 ,where ~v = )]TJ/F7 6.9738 Tf 4.566 -3.649 Td [(3 2 'referstotheplane vectorthat,whenincanonicalposition,endsat ; 2.Tondthecoordinates ofthatvectorwithrespecttothebasis B = h 1 1 ; 0 2 i wesolve c 1 1 1 + c 2 0 2 = 3 2 togetthat c 1 =3and c 2 =1 = 2.Thenwehavethis. Rep B ~v = 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 Here,althoughwe'veommitedthesubscript B fromthecolumn,thefactthat therightsideisarepresentationisclearfromthecontext. Theupsideofthenotationandtheterm`coordinates'isthattheygeneralize theusethatwearefamiliarwith:in R n andwithrespecttothestandard basis E n ,thevectorstartingattheoriginandendingat v 1 ;:::;v n hasthis representation. Rep E n 0 B @ v 1 v n 1 C A = 0 B @ v 1 v n 1 C A E n Ourmainuseofrepresentationswillcomeinthethirdchapter.Thedenitionappearsherebecausethefactthateveryvectorisalinearcombination ofbasisvectorsinauniquewayisacrucialpropertyofbases,andalsotohelp maketwopoints.First,wexanorderfortheelementsofabasissothat coordinatescanbestatedinthatorder.Second,forcalculationofcoordinates, amongotherthings,weshallrestrictourattentiontospaceswithbaseshaving onlynitelymanyelements.Wewillseethatinthenextsubsection. Exercises X 1.16 Decideifeachisabasisfor R 3 a h 1 2 3 ; 3 2 1 ; 0 0 1 i b h 1 2 3 ; 3 2 1 i c h 0 2 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 ; 1 1 1 ; 2 5 0 i d h 0 2 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 ; 1 1 1 ; 1 3 0 i

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SectionIII.BasisandDimension 117 X 1.17 Representthevectorwithrespecttothebasis. a 1 2 B = h 1 1 ; )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 1 i R 2 b x 2 + x 3 D = h 1 ; 1+ x; 1+ x + x 2 ; 1+ x + x 2 + x 3 iP 3 c 0 B @ 0 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 0 1 1 C A E 4 R 4 1.18 Findabasisfor P 2 ,thespaceofallquadraticpolynomials.Mustanysuch basiscontainapolynomialofeachdegree:degreezero,degreeone,anddegreetwo? 1.19 Findabasisforthesolutionsetofthissystem. x 1 )]TJ/F29 8.9664 Tf 9.215 0 Td [(4 x 2 +3 x 3 )]TJ/F32 8.9664 Tf 13.823 0 Td [(x 4 =0 2 x 1 )]TJ/F29 8.9664 Tf 9.215 0 Td [(8 x 2 +6 x 3 )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 x 4 =0 X 1.20 Findabasisfor M 2 2 ,thespaceof2 2matrices. X 1.21 Findabasisforeach. a Thesubspace f a 2 x 2 + a 1 x + a 0 a 2 )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 a 1 = a 0 g of P 2 b Thespaceofthree-widerowvectorswhoserstandsecondcomponentsadd tozero c Thissubspaceofthe2 2matrices f ab 0 c c )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 b =0 g 1.22 CheckExample1.6. X 1.23 Findthespanofeachsetandthenndabasisforthatspan. a f 1+ x; 1+2 x g in P 2 b f 2 )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 x; 3+4 x 2 g in P 2 X 1.24 Findabasisforeachofthesesubspacesofthespace P 3 ofcubicpolynomials. a Thesubspaceofcubicpolynomials p x suchthat p =0 b Thesubspaceofpolynomials p x suchthat p =0and p =0 c Thesubspaceofpolynomials p x suchthat p =0, p =0,and p =0 d Thespaceofpolynomials p x suchthat p =0, p =0, p =0, and p =0 1.25 We'veseenthatitispossibleforabasistoremainabasiswhenitisreordered. Mustitremainabasis? 1.26 Canabasiscontainazerovector? X 1.27 Let h ~ 1 ; ~ 2 ; ~ 3 i beabasisforavectorspace. a Showthat h c 1 ~ 1 ;c 2 ~ 2 ;c 3 ~ 3 i isabasiswhen c 1 ;c 2 ;c 3 6 =0.Whathappens whenatleastone c i is0? b Provethat h ~ 1 ;~ 2 ;~ 3 i isabasiswhere ~ i = ~ 1 + ~ i 1.28 Findonevector ~v thatwillmakeeachintoabasisforthespace. a h 1 1 ;~v i in R 2 b h 1 1 0 ; 0 1 0 ;~v i in R 3 c h x; 1+ x 2 ;~v i in P 2 X 1.29 Where h ~ 1 ;:::; ~ n i isabasis,showthatinthisequation c 1 ~ 1 + + c k ~ k = c k +1 ~ k +1 + + c n ~ n eachofthe c i 'siszero.Generalize. 1.30 Abasiscontainssomeofthevectorsfromavectorspace;canitcontainthem all?

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118 ChapterTwo.VectorSpaces 1.31 Theorem1.12showsthat,withrespecttoabasis,everylinearcombinationis unique.Ifasubsetisnotabasis,canlinearcombinationsbenotunique?Ifso, musttheybe? X 1.32 Asquarematrixis symmetric ifforallindices i and j ,entry i;j equalsentry j;i a Findabasisforthevectorspaceofsymmetric2 2matrices. b Findabasisforthespaceofsymmetric3 3matrices. c Findabasisforthespaceofsymmetric n n matrices. X 1.33 Wecanshowthateverybasisfor R 3 containsthesamenumberofvectors. a Showthatnolinearlyindependentsubsetof R 3 containsmorethanthree vectors. b Showthatnospanningsubsetof R 3 containsfewerthanthreevectors. Hint: recallhowtocalculatethespanofasetandshowthatthismethodcannotyield allof R 3 whenitisappliedtofewerthanthreevectors. 1.34 OneoftheexercisesintheSubspacessubsectionshowsthattheset f x y z x + y + z =1 g isavectorspaceundertheseoperations. x 1 y 1 z 1 + x 2 y 2 z 2 = x 1 + x 2 )]TJ/F29 8.9664 Tf 9.215 0 Td [(1 y 1 + y 2 z 1 + z 2 r x y z = rx )]TJ/F32 8.9664 Tf 9.215 0 Td [(r +1 ry rz Findabasis. III.2Dimension Inthepriorsubsectionwedenedthebasisofavectorspace,andwesawthat aspacecanhavemanydierentbases.Forexample,followingthedenitionof abasis,wesawthreedierentbasesfor R 2 .Sowecannottalkaboutthe"basis foravectorspace.True,somevectorspaceshavebasesthatstrikeusasmore naturalthanothers,forinstance, R 2 'sbasis E 2 or R 3 'sbasis E 3 or P 2 'sbasis h 1 ;x;x 2 i .But,forexampleinthespace f a 2 x 2 + a 1 x + a 0 2 a 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(a 0 = a 1 g ,no particularbasisleapsoutatusasthemostnaturalone.Wecannot,ingeneral, associatewithaspaceanysinglebasisthatbestdescribesthatspace. Wecan,however,ndsomethingaboutthebasesthatisuniquelyassociated withthespace.Thissubsectionshowsthatanytwobasesforaspacehavethe samenumberofelements.So,witheachspacewecanassociateanumber,the numberofvectorsinanyofitsbases. Thisbringsusbacktowhenweconsideredthetwothingsthatcouldbe meantbytheterm`minimalspanningset'.Atthatpointwedened`minimal' aslinearlyindependent,butwenotedthatanotherreasonableinterpretationof thetermisthataspanningsetis`minimal'whenithasthefewestnumberof elementsofanysetwiththesamespan.Attheendofthissubsection,afterwe

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SectionIII.BasisandDimension 119 haveshownthatallbaseshavethesamenumberofelements,thenwewillhave shownthatthetwosensesof`minimal'areequivalent. Beforewestart,werstlimitourattentiontospaceswhereatleastonebasis hasonlynitelymanymembers. 2.1Denition Avectorspaceis nite-dimensional ifithasabasiswithonly nitelymanyvectors. Onereasonforstickingtonite-dimensionalspacesissothattherepresentation ofavectorwithrespecttoabasisisanitely-tallvector,andsocanbeeasily written.Fromnowonwestudyonlynite-dimensionalvectorspaces.Weshall taketheterm`vectorspace'tomean`nite-dimensionalvectorspace'.Other spacesareinterestingandimportant,buttheylieoutsideofourscope. Toprovethemaintheoremweshalluseatechnicalresult. 2.2LemmaExchangeLemma Assumethat B = h ~ 1 ;:::; ~ n i isabasis foravectorspace,andthatforthevector ~v therelationship ~v = c 1 ~ 1 + c 2 ~ 2 + + c n ~ n has c i 6 =0.Thenexchanging ~ i for ~v yieldsanotherbasisforthe space. Proof Calltheoutcomeoftheexchange ^ B = h ~ 1 ;:::; ~ i )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ;~v; ~ i +1 ;:::; ~ n i Werstshowthat ^ B islinearlyindependent.Anyrelationship d 1 ~ 1 + + d i ~v + + d n ~ n = ~ 0amongthemembersof ^ B ,aftersubstitutionfor ~v d 1 ~ 1 + + d i c 1 ~ 1 + + c i ~ i + + c n ~ n + + d n ~ n = ~ 0 givesalinearrelationshipamongthemembersof B .Thebasis B islinearly independent,sothecoecient d i c i of ~ i iszero.Because c i isassumedtobe nonzero, d i =0.Usingthisinequation abovegivesthatalloftheother d 's arealsozero.Therefore ^ B islinearlyindependent. Wenishbyshowingthat ^ B hasthesamespanas B .Halfofthisargument, that[ ^ B ] [ B ],iseasy;anymember d 1 ~ 1 + + d i ~v + + d n ~ n of[ ^ B ]can bewritten d 1 ~ 1 + + d i c 1 ~ 1 + + c n ~ n + + d n ~ n ,whichisalinear combinationoflinearcombinationsofmembersof B ,andhenceisin[ B ].For the[ B ] [ ^ B ]halfoftheargument,recallthatwhen ~v = c 1 ~ 1 + + c n ~ n with c i 6 =0,thentheequationcanberearrangedto ~ i = )]TJ/F11 9.9626 Tf 7.749 0 Td [(c 1 =c i ~ 1 + + =c i ~v + + )]TJ/F11 9.9626 Tf 7.749 0 Td [(c n =c i ~ n .Now,consideranymember d 1 ~ 1 + + d i ~ i + + d n ~ n of[ B ],substitutefor ~ i itsexpressionasalinearcombinationofthemembers of ^ B ,andrecognizeasinthersthalfofthisargumentthattheresultisa linearcombinationoflinearcombinations,ofmembersof ^ B ,andhenceisin [ ^ B ]. QED 2.3Theorem Inanynite-dimensionalvectorspace,allofthebaseshave thesamenumberofelements. Proof Fixavectorspacewithatleastonenitebasis.Choose,fromamong allofthisspace'sbases,one B = h ~ 1 ;:::; ~ n i ofminimalsize.Wewillshow

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120 ChapterTwo.VectorSpaces thatanyotherbasis D = h ~ 1 ; ~ 2 ;::: i alsohasthesamenumberofmembers, n Because B hasminimalsize, D hasnofewerthan n vectors.Wewillarguethat itcannothavemorethan n vectors. Thebasis B spansthespaceand ~ 1 isinthespace,so ~ 1 isanontriviallinear combinationofelementsof B .BytheExchangeLemma, ~ 1 canbeswappedfor avectorfrom B ,resultinginabasis B 1 ,whereoneelementis ~ andallofthe n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1otherelementsare ~ 's. Thepriorparagraphformsthebasisstepforaninductionargument.The inductivestepstartswithabasis B k for1 k
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SectionIII.BasisandDimension 121 subspaceshavebaseswithonemember,andthetrivialsubspacehasabasis withzeromembers.Whenwesawthatdiagramwecouldnotshowthatthese aretheonlysubspacesthatthisspacehas.Wecanshowitnow.Theprior corollaryprovesthattheonlysubspacesof R 3 areeitherthree-,two-,one-,or zero-dimensional.Therefore,thediagramindicatesallofthesubspaces.There arenosubspacessomehow,say,betweenlinesandplanes. 2.10Corollary Anylinearlyindependentsetcanbeexpandedtomakeabasis. Proof Ifalinearlyindependentsetisnotalreadyabasisthenitmustnot spanthespace.Addingtoitavectorthatisnotinthespanpreserveslinear independence.Keepadding,untiltheresultingsetdoesspanthespace,which thepriorcorollaryshowswillhappenafteronlyanitenumberofsteps. QED 2.11Corollary Anyspanningsetcanbeshrunktoabasis. Proof Callthespanningset S .If S isemptythenitisalreadyabasisthe spacemustbeatrivialspace.If S = f ~ 0 g thenitcanbeshrunktotheempty basis,therebymakingitlinearlyindependent,withoutchangingitsspan. Otherwise, S containsavector ~s 1 with ~s 1 6 = ~ 0andwecanformabasis B 1 = h ~s 1 i .If[ B 1 ]=[ S ]thenwearedone. Ifnotthenthereisa ~s 2 2 [ S ]suchthat ~s 2 62 [ B 1 ].Let B 2 = h ~s 1 ;~s 2 i ;if [ B 2 ]=[ S ]thenwearedone. Wecanrepeatthisprocessuntilthespansareequal,whichmusthappenin atmostnitelymanysteps. QED 2.12Corollary Inan n -dimensionalspace,asetof n vectorsislinearlyindependentifandonlyifitspansthespace. Proof Firstwewillshowthatasubsetwith n vectorsislinearlyindependent ifandonlyifitisabasis.`If'istriviallytrue|basesarelinearlyindependent. `Onlyif'holdsbecausealinearlyindependentsetcanbeexpandedtoabasis, butabasishas n elements,sothisexpansionisactuallythesetthatwebegan with. Tonish,wewillshowthatanysubsetwith n vectorsspansthespaceifand onlyifitisabasis.Again,`if'istrivial.`Onlyif'holdsbecauseanyspanning setcanbeshrunktoabasis,butabasishas n elementsandsothisshrunken setisjusttheonewestartedwith. QED Themainresultofthissubsection,thatallofthebasesinanite-dimensional vectorspacehavethesamenumberofelements,isthesinglemostimportant resultinthisbookbecause,asExample2.9shows,itdescribeswhatvector spacesandsubspacestherecanbe.Wewillseemoreinthenextchapter. 2.13Remark Thecaseofinnite-dimensionalvectorspacesissomewhatcontroversial.Thestatement`anyinnite-dimensionalvectorspacehasabasis' isknowntobeequivalenttoastatementcalledtheAxiomofChoicesee [Blass1984].Mathematiciansdierphilosophicallyonwhethertoacceptor rejectthisstatementasanaxiomonwhichtobasemathematicsalthough,the

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122 ChapterTwo.VectorSpaces greatmajorityseemtoacceptit.Consequentlythequestionaboutinnitedimensionalvectorspacesisstillsomewhatupintheair.Adiscussionofthe AxiomofChoicecanbefoundintheFrequentlyAskedQuestionslistforthe Usenetgroup sci.math .Anotheraccessiblereferenceis[Rucker]. Exercises Assumethatallspacesarenite-dimensionalunlessotherwisestated. X 2.14 Findabasisfor,andthedimensionof, P 2 2.15 Findabasisfor,andthedimensionof,thesolutionsetofthissystem. x 1 )]TJ/F29 8.9664 Tf 9.215 0 Td [(4 x 2 +3 x 3 )]TJ/F32 8.9664 Tf 13.823 0 Td [(x 4 =0 2 x 1 )]TJ/F29 8.9664 Tf 9.215 0 Td [(8 x 2 +6 x 3 )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 x 4 =0 X 2.16 Findabasisfor,andthedimensionof, M 2 2 ,thevectorspaceof2 2matrices. 2.17 Findthedimensionofthevectorspaceofmatrices ab cd subjecttoeachcondition. a a;b;c;d 2 R b a )]TJ/F32 8.9664 Tf 9.216 0 Td [(b +2 c =0and d 2 R c a + b + c =0, a + b )]TJ/F32 8.9664 Tf 9.215 0 Td [(c =0,and d 2 R X 2.18 Findthedimensionofeach. a Thespaceofcubicpolynomials p x suchthat p =0 b Thespaceofcubicpolynomials p x suchthat p =0and p =0 c Thespaceofcubicpolynomials p x suchthat p =0, p =0,and p = 0 d Thespaceofcubicpolynomials p x suchthat p =0, p =0, p =0, and p =0 2.19 Whatisthedimensionofthespanoftheset f cos 2 ; sin 2 ; cos2 ; sin2 g ?This spanisasubspaceofthespaceofallreal-valuedfunctionsofonerealvariable. 2.20 Findthedimensionof C 47 ,thevectorspaceof47-tuplesofcomplexnumbers. 2.21 Whatisthedimensionofthevectorspace M 3 5 of3 5matrices? X 2.22 Showthatthisisabasisfor R 4 h 0 B @ 1 0 0 0 1 C A ; 0 B @ 1 1 0 0 1 C A ; 0 B @ 1 1 1 0 1 C A ; 0 B @ 1 1 1 1 1 C A i Theresultsofthissubsectioncanbeusedtosimplifythisjob. 2.23 RefertoExample2.9. a Sketchasimilarsubspacediagramfor P 2 b Sketchonefor M 2 2 X 2.24 Where S isaset,thefunctions f : S R formavectorspaceunderthe naturaloperations:thesum f + g isthefunctiongivenby f + g s = f s + g s andthescalarproductisgivenby r f s = r f s .Whatisthedimensionofthe spaceresultingforeachdomain?

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SectionIII.BasisandDimension 123 a S = f 1 g b S = f 1 ; 2 g c S = f 1 ;:::;n g 2.25 SeeExercise24.Provethatthisisaninnite-dimensionalspace:thesetof allfunctions f : R R underthenaturaloperations. 2.26 SeeExercise24.Whatisthedimensionofthevectorspaceoffunctions f : S R ,underthenaturaloperations,wherethedomain S istheemptyset? 2.27 Showthatanysetoffourvectorsin R 2 islinearlydependent. 2.28 Showthat h ~ 1 ;~ 2 ;~ 3 i R 3 isabasisifandonlyifthereisnoplanethrough theorigincontainingallthreevectors. 2.29a Provethatanysubspaceofanitedimensionalspacehasabasis. b Provethatanysubspaceofanitedimensionalspaceisnitedimensional. 2.30 Whereisthenitenessof B usedinTheorem2.3? X 2.31 Provethatif U and W areboththree-dimensionalsubspacesof R 5 then U W isnon-trivial.Generalize. 2.32 Abasisforaspaceconsistsofelementsofthatspace.Sowearenaturally ledtohowtheproperty`isabasis'interactswithoperations and and [ .Of course,abasisisactuallyasequenceinthatitisordered,butthereisanatural extensionoftheseoperations. a Considerrsthowbasesmightberelatedby .Assumethat U;W are subspacesofsomevectorspaceandthat U W .Canthereexistbases B U for U and B W for W suchthat B U B W ?Mustsuchbasesexist? Foranybasis B U for U ,musttherebeabasis B W for W suchthat B U B W ? Foranybasis B W for W ,musttherebeabasis B U for U suchthat B U B W ? Foranybases B U ;B W for U and W ,must B U beasubsetof B W ? b Isthe ofbasesabasis?Forwhatspace? c Isthe [ ofbasesabasis?Forwhatspace? d Whataboutthecomplementoperation? Hint. Testanyconjecturesagainstsomesubspacesof R 3 X 2.33 Considerhow`dimension'interactswith`subset'.Assume U and W areboth subspacesofsomevectorspace,andthat U W a Provethatdim U dim W b Provethatequalityofdimensionholdsifandonlyif U = W c Showthattheprioritemdoesnotholdiftheyareinnite-dimensional. ? 2.34 Foranyvector ~v in R n andanypermutation ofthenumbers1,2,..., n thatis, isarearrangementofthosenumbersintoaneworder,dene ~v tobethevectorwhosecomponentsare v v ,...,and v n where is therstnumberintherearrangement,etc..Nowx ~v andlet V bethespanof f ~v permutes1,..., n g .Whatarethepossibilitiesforthedimensionof V ? [Wohascumno.47] III.3VectorSpacesandLinearSystems WewillnowreconsiderlinearsystemsandGauss'method,aidedbythetools andtermsofthischapter.Wewillmakethreepoints. Fortherstpoint,recalltherstchapter'sLinearCombinationLemmaand itscorollary:iftwomatricesarerelatedbyrowoperations A )167(!)167(! B then

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124 ChapterTwo.VectorSpaces eachrowof B isalinearcombinationoftherowsof A .Thatis,Gauss'method worksbytakinglinearcombinationsofrows.Therefore,therightsettingin whichtostudyrowoperationsingeneral,andGauss'methodinparticular,is thefollowingvectorspace. 3.1Denition The rowspace ofamatrixisthespanofthesetofitsrows.The rowrank isthedimensionoftherowspace,thenumberoflinearlyindependent rows. 3.2Example If A = 23 46 thenRowspace A isthissubspaceofthespaceoftwo-componentrowvectors. f c 1 )]TJ/F8 9.9626 Tf 4.566 -7.97 Td [(23 + c 2 )]TJ/F8 9.9626 Tf 4.566 -7.97 Td [(46 c 1 ;c 2 2 R g Thelineardependenceofthesecondontherstisobviousandsowecansimplify thisdescriptionto f c )]TJ/F8 9.9626 Tf 4.567 -7.97 Td [(23 c 2 R g 3.3Lemma Ifthematrices A and B arerelatedbyarowoperation A i $ j )167(! B or A k i )167(! B or A k i + j )167(! B for i 6 = j and k 6 =0thentheirrowspacesareequal.Hence,row-equivalent matriceshavethesamerowspace,andhencealso,thesamerowrank. Proof BytheLinearCombinationLemma'scorollary,eachrowof B isinthe rowspaceof A .Further,Rowspace B Rowspace A becauseamemberof thesetRowspace B isalinearcombinationoftherowsof B ,whichmeansit isacombinationofacombinationoftherowsof A ,andhence,bytheLinear CombinationLemma,isalsoamemberofRowspace A Fortheothercontainment,recallthatrowoperationsarereversible: A )167(! B ifandonlyif B )167(! A .Withthat,Rowspace A Rowspace B alsofollows fromthepriorparagraph,andsothetwosetsareequal. QED Thus,rowoperationsleavetherowspaceunchanged.Butofcourse,Gauss' methodperformstherowoperationssystematically,withaspecicgoalinmind, echelonform. 3.4Lemma Thenonzerorowsofanechelonformmatrixmakeupalinearly independentset. Proof Aresultintherstchapter,LemmaIII.2.5,statesthatinanechelon formmatrix,nononzerorowisalinearcombinationoftheotherrows.Thisis arestatementofthatresultintonewterminology. QED Thus,inthelanguageofthischapter,Gaussianreductionworksbyeliminatinglineardependencesamongrows,leavingthespanunchanged,untilno nontriviallinearrelationshipsremainamongthenonzerorows.Thatis,Gauss' methodproducesabasisfortherowspace.

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SectionIII.BasisandDimension 125 3.5Example Fromanymatrix,wecanproduceabasisfortherowspaceby performingGauss'methodandtakingthenonzerorowsoftheresultingechelon formmatrix.Forinstance, 0 @ 131 141 205 1 A )]TJ/F10 6.9738 Tf 6.226 0 Td [( 1 + 2 )167(! )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 1 + 3 6 2 + 3 )167(! 0 @ 131 010 003 1 A producesthebasis h )]TJ/F8 9.9626 Tf 4.567 -7.97 Td [(131 ; )]TJ/F8 9.9626 Tf 4.567 -7.97 Td [(010 ; )]TJ/F8 9.9626 Tf 4.567 -7.97 Td [(003 i fortherowspace.This isabasisfortherowspaceofboththestartingandendingmatrices,sincethe tworowspacesareequal. Usingthistechnique,wecanalsondbasesforspansnotdirectlyinvolving rowvectors. 3.6Denition The columnspace ofamatrixisthespanofthesetofits columns.The columnrank isthedimensionofthecolumnspace,thenumber oflinearlyindependentcolumns. Ourinterestincolumnspacesstemsfromourstudyoflinearsystems.An exampleisthatthissystem c 1 +3 c 2 +7 c 3 = d 1 2 c 1 +3 c 2 +8 c 3 = d 2 c 2 +2 c 3 = d 3 4 c 1 +4 c 3 = d 4 hasasolutionifandonlyifthevectorof d 'sisalinearcombinationoftheother columnvectors, c 1 0 B B @ 1 2 0 4 1 C C A + c 2 0 B B @ 3 3 1 0 1 C C A + c 3 0 B B @ 7 8 2 4 1 C C A = 0 B B @ d 1 d 2 d 3 d 4 1 C C A meaningthatthevectorof d 'sisinthecolumnspaceofthematrixofcoecients. 3.7Example Giventhismatrix, 0 B B @ 137 238 012 404 1 C C A togetabasisforthecolumnspace,temporarilyturnthecolumnsintorowsand reduce. 0 @ 1204 3310 7824 1 A )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 1 + 2 )167(! )]TJ/F7 6.9738 Tf 6.227 0 Td [(7 1 + 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 2 + 3 )167(! 0 @ 1204 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(31 )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 0000 1 A

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126 ChapterTwo.VectorSpaces Nowturntherowsbacktocolumns. h 0 B B @ 1 2 0 4 1 C C A ; 0 B B @ 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 1 C C A i Theresultisabasisforthecolumnspaceofthegivenmatrix. 3.8Denition The transpose ofamatrixistheresultofinterchangingthe rowsandcolumnsofthatmatrix.Thatis,column j ofthematrix A isrow j of A trans ,andviceversa. Sotheinstructionsforthepriorexamplearetranspose,reduce,andtranspose back". Wecaneven,atthepriceoftoleratingtheas-yet-vagueideaofvectorspaces beingthesame",useGauss'methodtondbasesforspansinothertypesof vectorspaces. 3.9Example Togetabasisforthespanof f x 2 + x 4 ; 2 x 2 +3 x 4 ; )]TJ/F11 9.9626 Tf 7.749 0 Td [(x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x 4 g inthespace P 4 ,thinkofthesethreepolynomialsasthesame"astherow vectors )]TJ/F8 9.9626 Tf 4.566 -7.97 Td [(00101 )]TJ/F8 9.9626 Tf 4.566 -7.97 Td [(00203 ,and )]TJ/F8 9.9626 Tf 4.567 -7.97 Td [(00 )]TJ/F8 9.9626 Tf 7.748 0 Td [(10 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ,apply Gauss'method 0 @ 00101 00203 00 )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 1 A )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 1 + 2 )167(! 1 + 3 2 2 + 3 )167(! 0 @ 00101 00001 00000 1 A andtranslatebacktogetthebasis h x 2 + x 4 ;x 4 i .Asmentionedearlier,wewill makethephrasethesame"preciseatthestartofthenextchapter. Thus,ourrstpointinthissubsectionisthatthetoolsofthischaptergive usamoreconceptualunderstandingofGaussianreduction. Forthesecondpointofthissubsection,considertheeectonthecolumn spaceofthisrowreduction. 12 24 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 1 + 2 )167(! 12 00 Thecolumnspaceoftheleft-handmatrixcontainsvectorswithasecondcomponentthatisnonzero.Butthecolumnspaceoftheright-handmatrixisdierent becauseitcontainsonlyvectorswhosesecondcomponentiszero.Itisthis knowledgethatrowoperationscanchangethecolumnspacethatmakesnext resultsurprising. 3.10Lemma Rowoperationsdonotchangethecolumnrank. Proof Restated,if A reducesto B thenthecolumnrankof B equalsthe columnrankof A .

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SectionIII.BasisandDimension 127 Wewillbedoneifwecanshowthatrowoperationsdonotaectlinear relationshipsamongcolumnsbecausethecolumnrankisjustthesizeofthe largestsetofunrelatedcolumns.Thatis,wewillshowthatarelationship existsamongcolumnssuchasthatthefthcolumnistwicethesecondplus thefourthifandonlyifthatrelationshipexistsaftertherowoperation.But thisisexactlythersttheoremofthisbook:inarelationshipamongcolumns, c 1 0 B B B @ a 1 ; 1 a 2 ; 1 a m; 1 1 C C C A + + c n 0 B B B @ a 1 ;n a 2 ;n a m;n 1 C C C A = 0 B B B @ 0 0 0 1 C C C A rowoperationsleaveunchangedthesetofsolutions c 1 ;:::;c n QED Anotherway,besidesthepriorresult,tostatethatGauss'methodhassomethingtosayaboutthecolumnspaceaswellasabouttherowspaceistoconsider againGauss-Jordanreduction.Recallthatitendswiththereducedechelonform ofamatrix,ashere. 0 @ 1316 26316 1316 1 A )167(!)167(! 0 @ 1302 0014 0000 1 A Considertherowspaceandthecolumnspaceofthisresult.Ourrstpoint madeabovesaysthatabasisfortherowspaceiseasytoget:simplycollect togetheralloftherowswithleadingentries.However,becausethisisareduced echelonformmatrix,abasisforthecolumnspaceisjustaseasy:takethe columnscontainingtheleadingentries,thatis, h ~e 1 ;~e 2 i .Linearindependence isobvious.Theothercolumnsareinthespanofthisset,sincetheyallhavea thirdcomponentofzero.Thus,forareducedechelonformmatrix,basesfor therowandcolumnspacescanbefoundinessentiallythesameway|bytaking thepartsofthematrix,therowsorcolumns,containingtheleadingentries. 3.11Theorem Therowrankandcolumnrankofamatrixareequal. Proof Firstbringthematrixtoreducedechelonform.Atthatpoint,the rowrankequalsthenumberofleadingentriessinceeachequalsthenumber ofnonzerorows.Alsoatthatpoint,thenumberofleadingentriesequalsthe columnrankbecausethesetofcolumnscontainingleadingentriesconsistsof someofthe ~e i 'sfromastandardbasis,andthatsetislinearlyindependentand spansthesetofcolumns.Hence,inthereducedechelonformmatrix,therow rankequalsthecolumnrank,becauseeachequalsthenumberofleadingentries. ButLemma3.3andLemma3.10showthattherowrankandcolumnrank arenotchangedbyusingrowoperationstogettoreducedechelonform.Thus therowrankandthecolumnrankoftheoriginalmatrixarealsoequal. QED 3.12Denition The rank ofamatrixisitsrowrankorcolumnrank.

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128 ChapterTwo.VectorSpaces Sooursecondpointinthissubsectionisthatthecolumnspaceandrow spaceofamatrixhavethesamedimension.Ourthirdandnalpointisthat theconceptsthatwe'veseenarisingnaturallyinthestudyofvectorspacesare exactlytheonesthatwehavestudiedwithlinearsystems. 3.13Theorem Forlinearsystemswith n unknownsandwithmatrixofcoecients A ,thestatements therankof A is r thespaceofsolutionsoftheassociatedhomogeneoussystemhasdimension n )]TJ/F11 9.9626 Tf 9.963 0 Td [(r areequivalent. Soifthesystemhasatleastoneparticularsolutionthenforthesetofsolutions, thenumberofparametersequals n )]TJ/F11 9.9626 Tf 9.464 0 Td [(r ,thenumberofvariablesminustherank ofthematrixofcoecients. Proof Therankof A is r ifandonlyifGaussianreductionon A endswith r nonzerorows.That'strueifandonlyifechelonformmatricesrowequivalent to A have r -manyleadingvariables.Thatinturnholdsifandonlyifthereare n )]TJ/F11 9.9626 Tf 9.963 0 Td [(r freevariables. QED 3.14Remark [Munkres]Sometimesthatresultismistakenlyrememberedto saythatthegeneralsolutionofan n unknownsystemof m equationsuses n )]TJ/F11 9.9626 Tf 8.796 0 Td [(m parameters.Thenumberofequationsisnottherelevantgure,rather,what mattersisthenumberofindependentequationsthenumberofequationsin amaximalindependentset.Wherethereare r independentequations,the generalsolutioninvolves n )]TJ/F11 9.9626 Tf 9.963 0 Td [(r parameters. 3.15Corollary Wherethematrix A is n n ,thestatements therankof A is n A isnonsingular therowsof A formalinearlyindependentset thecolumnsof A formalinearlyindependentset anylinearsystemwhosematrixofcoecientsis A hasoneandonlyone solution areequivalent. Proof Clearly .Thelast, ,holds becauseasetof n columnvectorsislinearlyindependentifandonlyifitisa basisfor R n ,butthesystem c 1 0 B B B @ a 1 ; 1 a 2 ; 1 a m; 1 1 C C C A + + c n 0 B B B @ a 1 ;n a 2 ;n a m;n 1 C C C A = 0 B B B @ d 1 d 2 d m 1 C C C A hasauniquesolutionforallchoicesof d 1 ;:::;d n 2 R ifandonlyifthevectors of a 'sformabasis. QED

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SectionIII.BasisandDimension 129 Exercises 3.16 Transposeeach. a 21 31 b 21 13 c 143 678 d 0 0 0 e )]TJ/F34 8.9664 Tf 4.566 -7.771 Td [()]TJ/F29 8.9664 Tf 7.168 0 Td [(1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(2 X 3.17 Decideifthevectorisintherowspaceofthematrix. a 21 31 )]TJ/F29 8.9664 Tf 4.566 -7.771 Td [(10 b 013 )]TJ/F29 8.9664 Tf 7.167 0 Td [(101 )]TJ/F29 8.9664 Tf 7.167 0 Td [(127 )]TJ/F29 8.9664 Tf 4.566 -7.771 Td [(111 X 3.18 Decideifthevectorisinthecolumnspace. a 11 11 1 3 b 131 204 1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(3 )]TJ/F29 8.9664 Tf 7.168 0 Td [(3 1 0 0 X 3.19 Findabasisfortherowspaceofthismatrix. 0 B @ 2034 011 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 3102 10 )]TJ/F29 8.9664 Tf 7.167 0 Td [(4 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 1 C A X 3.20 Findtherankofeachmatrix. a 213 1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(12 103 b 1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(12 3 )]TJ/F29 8.9664 Tf 7.167 0 Td [(36 )]TJ/F29 8.9664 Tf 7.167 0 Td [(22 )]TJ/F29 8.9664 Tf 7.168 0 Td [(4 c 132 511 643 d 000 000 000 X 3.21 Findabasisforthespanofeachset. a f )]TJ/F29 8.9664 Tf 4.567 -7.771 Td [(13 ; )]TJ/F34 8.9664 Tf 4.566 -7.771 Td [()]TJ/F29 8.9664 Tf 7.167 0 Td [(13 ; )]TJ/F29 8.9664 Tf 4.566 -7.771 Td [(14 ; )]TJ/F29 8.9664 Tf 4.566 -7.771 Td [(21 gM 1 2 b f 1 2 1 ; 3 1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 ; 1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(3 )]TJ/F29 8.9664 Tf 7.167 0 Td [(3 g R 3 c f 1+ x; 1 )]TJ/F32 8.9664 Tf 9.215 0 Td [(x 2 ; 3+2 x )]TJ/F32 8.9664 Tf 9.215 0 Td [(x 2 gP 3 d f 101 31 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 ; 103 214 ; )]TJ/F29 8.9664 Tf 7.167 0 Td [(10 )]TJ/F29 8.9664 Tf 7.167 0 Td [(5 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(9 gM 2 3 3.22 Whichmatriceshaverankzero?Rankone? X 3.23 Given a;b;c 2 R ,whatchoiceof d willcausethismatrixtohavetherankof one? ab cd 3.24 Findthecolumnrankofthismatrix. 13 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1504 201041 3.25 Showthatalinearsystemwithatleastonesolutionhasatmostonesolution ifandonlyifthematrixofcoecientshasrankequaltothenumberofitscolumns. X 3.26 Ifamatrixis5 9,whichsetmustbedependent,itssetofrowsoritssetof columns?

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130 ChapterTwo.VectorSpaces 3.27 Giveanexampletoshowthat,despitethattheyhavethesamedimension, therowspaceandcolumnspaceofamatrixneednotbeequal.Aretheyever equal? 3.28 Showthattheset f ; )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 ; 2 ; )]TJ/F29 8.9664 Tf 7.168 0 Td [(3 ; ; 1 ; 2 ; 0 ; ; )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 ; 6 ; )]TJ/F29 8.9664 Tf 7.168 0 Td [(6 g doesnothavethe samespanas f ; 0 ; 1 ; 0 ; ; 2 ; 0 ; 3 g .What,bytheway,isthevectorspace? X 3.29 Showthatthissetofcolumnvectors d 1 d 2 d 3 thereare x y ,and z suchthat 3 x +2 y +4 z = d 1 x )]TJ/F32 8.9664 Tf 13.823 0 Td [(z = d 2 2 x +2 y +5 z = d 3 isasubspaceof R 3 .Findabasis. 3.30 Showthatthetransposeoperationis linear : rA + sB trans = rA trans + sB trans for r;s 2 R and A;B 2M m n X 3.31 InthissubsectionwehaveshownthatGaussianreductionndsabasisfor therowspace. a Showthatthisbasisisnotunique|dierentreductionsmayyielddierent bases. b Producematriceswithequalrowspacesbutunequalnumbersofrows. c ProvethattwomatriceshaveequalrowspacesifandonlyifafterGaussJordanreductiontheyhavethesamenonzerorows. 3.32 WhyistherenotaproblemwithRemark3.14inthecasethat r isbigger than n ? 3.33 Showthattherowrankofan m n matrixisatmost m .Isthereabetter bound? X 3.34 Showthattherankofamatrixequalstherankofitstranspose. 3.35 Trueorfalse:thecolumnspaceofamatrixequalstherowspaceofitstranspose. X 3.36 Wehaveseenthatarowoperationmaychangethecolumnspace.Mustit? 3.37 Provethatalinearsystemhasasolutionifandonlyifthatsystem'smatrix ofcoecientshasthesamerankasitsaugmentedmatrix. 3.38 An m n matrixhas fullrowrank ifitsrowrankis m ,andithas fullcolumn rank ifitscolumnrankis n a Showthatamatrixcanhavebothfullrowrankandfullcolumnrankonly ifitissquare. b Provethatthelinearsystemwithmatrixofcoecients A hasasolutionfor any d 1 ,..., d n 'sontherightsideifandonlyif A hasfullrowrank. c Provethatahomogeneoussystemhasauniquesolutionifandonlyifits matrixofcoecients A hasfullcolumnrank. d Provethatthestatementifasystemwithmatrixofcoecients A hasany solutionthenithasauniquesolution"holdsifandonlyif A hasfullcolumn rank. 3.39 HowwouldtheconclusionofLemma3.3changeifGauss'methodischanged toallowmultiplyingarowbyzero? X 3.40 Whatistherelationshipbetweenrank A andrank )]TJ/F32 8.9664 Tf 7.168 0 Td [(A ?Betweenrank A andrank kA ?What,ifany,istherelationshipbetweenrank A ,rank B ,and rank A + B ?

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SectionIII.BasisandDimension 131 III.4CombiningSubspaces Thissubsectionisoptional.ItisrequiredonlyforthelastsectionsofChapter ThreeandChapterFiveandforoccasionalexercises,andcanbepassedover withoutlossofcontinuity. Thischapteropenedwiththedenitionofavectorspace,andthemiddleconsistedofarstanalysisoftheidea.Thissubsectionclosesthechapter bynishingtheanalysis,inthesensethat`analysis'meansmethodofdeterminingthe...essentialfeaturesofsomethingbyseparatingitintoparts" [MacmillanDictionary]. Acommonwaytounderstandthingsistoseehowtheycanbebuiltfrom componentparts.Forinstance,wethinkof R 3 asputtogether,insomeway, fromthe x -axis,the y -axis,and z -axis.Inthissubsectionwewillmakethis precise;wewilldescribehowtodecomposeavectorspaceintoacombinationof someofitssubspaces.Indevelopingthisideaofsubspacecombination,wewill keepthe R 3 exampleinmindasabenchmarkmodel. Subspacesaresubsetsandsetscombineviaunion.Buttakingthecombinationoperationforsubspacestobethesimpleunionoperationisn'twhatwe want.Foronething,theunionofthe x -axis,the y -axis,and z -axisisnotallof R 3 ,sothebenchmarkmodelwouldbeleftout.Besides,unionisallwrongfor thisreason:aunionofsubspacesneednotbeasubspaceitneednotbeclosed; forinstance,this R 3 vector 0 @ 1 0 0 1 A + 0 @ 0 1 0 1 A + 0 @ 0 0 1 1 A = 0 @ 1 1 1 1 A isinnoneofthethreeaxesandhenceisnotintheunion.Inadditionto themembersofthesubspaces,wemustatleastalsoincludeallofthelinear combinations. 4.1Denition Where W 1 ;:::;W k aresubspacesofavectorspace,their sum isthespanoftheirunion W 1 + W 2 + + W k =[ W 1 [ W 2 [ :::W k ]. Thenotation,writingthe`+'betweensetsinadditiontousingitbetween vectors,tswiththepracticeofusingthissymbolforanynaturalaccumulation operation. 4.2Example The R 3 modeltswiththisoperation.Anyvector ~w 2 R 3 can bewrittenasalinearcombination c 1 ~v 1 + c 2 ~v 2 + c 3 ~v 3 where ~v 1 isamemberof the x -axis,etc.,inthisway 0 @ w 1 w 2 w 3 1 A =1 0 @ w 1 0 0 1 A +1 0 @ 0 w 2 0 1 A +1 0 @ 0 0 w 3 1 A andso R 3 = x -axis+ y -axis+ z -axis.

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132 ChapterTwo.VectorSpaces 4.3Example Asumofsubspacescanbelessthantheentirespace.Insideof P 4 ,let L bethesubspaceoflinearpolynomials f a + bx a;b 2 R g andlet C be thesubspaceofpurely-cubicpolynomials f cx 3 c 2 R g .Then L + C isnotall of P 4 .Instead,itisthesubspace L + C = f a + bx + cx 3 a;b;c 2 R g 4.4Example Aspacecanbedescribedasacombinationofsubspacesinmore thanoneway.Besidesthedecomposition R 3 = x -axis+ y -axis+ z -axis,wecan alsowrite R 3 = xy -plane+ yz -plane.Tocheckthis,notethatany ~w 2 R 3 can bewrittenasalinearcombinationofamemberofthe xy -planeandamember ofthe yz -plane;herearetwosuchcombinations. 0 @ w 1 w 2 w 3 1 A =1 0 @ w 1 w 2 0 1 A +1 0 @ 0 0 w 3 1 A 0 @ w 1 w 2 w 3 1 A =1 0 @ w 1 w 2 = 2 0 1 A +1 0 @ 0 w 2 = 2 w 3 1 A Theabovedenitiongivesonewayinwhichaspacecanbethoughtofasa combinationofsomeofitsparts.However,thepriorexampleshowsthatthereis atleastoneinterestingpropertyofourbenchmarkmodelthatisnotcapturedby thedenitionofthesumofsubspaces.Inthefamiliardecompositionof R 3 ,we oftenspeakofavector's` x part'or` y part'or` z part'.Thatis,inthismodel, eachvectorhasauniquedecompositionintopartsthatcomefromtheparts makingupthewholespace.ButinthedecompositionusedinExample4.4,we cannotrefertothe xy part"ofavector|thesethreesums 0 @ 1 2 3 1 A = 0 @ 1 2 0 1 A + 0 @ 0 0 3 1 A = 0 @ 1 0 0 1 A + 0 @ 0 2 3 1 A = 0 @ 1 1 0 1 A + 0 @ 0 1 3 1 A alldescribethevectorascomprisedofsomethingfromtherstplaneplussomethingfromthesecondplane,butthe xy part"isdierentineach. Thatis,whenweconsiderhow R 3 isputtogetherfromthethreeaxesin someway",wemightmeaninsuchawaythateveryvectorhasatleastone decomposition",andthatleadstothedenitionabove.Butifwetakeitto meaninsuchawaythateveryvectorhasoneandonlyonedecomposition" thenweneedanotherconditiononcombinations.Toseewhatthiscondition is,recallthatvectorsareuniquelyrepresentedintermsofabasis.Wecanuse thistobreakaspaceintoasumofsubspacessuchthatanyvectorinthespace breaksuniquelyintoasumofmembersofthosesubspaces. 4.5Example Thebenchmarkis R 3 withitsstandardbasis E 3 = h ~e 1 ;~e 2 ;~e 3 i Thesubspacewiththebasis B 1 = h ~e 1 i isthe x -axis.Thesubspacewiththe basis B 2 = h ~e 2 i isthe y -axis.Thesubspacewiththebasis B 3 = h ~e 3 i isthe z -axis.Thefactthatanymemberof R 3 isexpressibleasasumofvectorsfrom thesesubspaces 0 @ x y z 1 A = 0 @ x 0 0 1 A + 0 @ 0 y 0 1 A + 0 @ 0 0 z 1 A

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SectionIII.BasisandDimension 133 isareectionofthefactthat E 3 spansthespace|thisequation 0 @ x y z 1 A = c 1 0 @ 1 0 0 1 A + c 2 0 @ 0 1 0 1 A + c 3 0 @ 0 0 1 1 A hasasolutionforany x;y;z 2 R .And,thefactthateachsuchexpressionis uniquereectsthatfactthat E 3 islinearlyindependent|anyequationlikethe oneabovehasauniquesolution. 4.6Example Wedon'thavetotakethebasisvectorsoneatatime,thesame ideaworksifweconglomeratethemintolargersequences.Consideragainthe space R 3 andthevectorsfromthestandardbasis E 3 .Thesubspacewiththe basis B 1 = h ~e 1 ;~e 3 i isthe xz -plane.Thesubspacewiththebasis B 2 = h ~e 2 i is the y -axis.Asinthepriorexample,thefactthatanymemberofthespaceisa sumofmembersofthetwosubspacesinoneandonlyoneway 0 @ x y z 1 A = 0 @ x 0 z 1 A + 0 @ 0 y 0 1 A isareectionofthefactthatthesevectorsformabasis|thissystem 0 @ x y z 1 A = c 1 0 @ 1 0 0 1 A + c 3 0 @ 0 0 1 1 A + c 2 0 @ 0 1 0 1 A hasoneandonlyonesolutionforany x;y;z 2 R Theseexamplesillustrateanaturalwaytodecomposeaspaceintoasum ofsubspacesinsuchawaythateachvectordecomposesuniquelyintoasumof vectorsfromtheparts.Thenextresultsaysthatthiswayistheonlyway. 4.7Denition The concatenation ofthesequences B 1 = h ~ 1 ; 1 ;:::; ~ 1 ;n 1 i ..., B k = h ~ k; 1 ;:::; ~ k;n k i istheiradjoinment. B 1 B 2 B k = h ~ 1 ; 1 ;:::; ~ 1 ;n 1 ; ~ 2 ; 1 ;:::; ~ k;n k i 4.8Lemma Let V beavectorspacethatisthesumofsomeofitssubspaces V = W 1 + + W k .Let B 1 ,..., B k beanybasesforthesesubspaces.Then thefollowingareequivalent. Forevery ~v 2 V ,theexpression ~v = ~w 1 + + ~w k with ~w i 2 W i is unique. Theconcatenation B 1 B k isabasisfor V Thenonzeromembersof f ~w 1 ;:::;~w k g with ~w i 2 W i formalinearly independentset|amongnonzerovectorsfromdierent W i 's,everylinear relationshipistrivial.

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134 ChapterTwo.VectorSpaces Proof Wewillshowthat= ,that= ,andnallythat = .Forthesearguments,observethatwecanpassfromacombination of ~w 'stoacombinationof ~ 's d 1 ~w 1 + + d k ~w k = d 1 c 1 ; 1 ~ 1 ; 1 + + c 1 ;n 1 ~ 1 ;n 1 + + d k c k; 1 ~ k; 1 + + c k;n k ~ k;n k = d 1 c 1 ; 1 ~ 1 ; 1 + + d k c k;n k ~ k;n k andviceversa. For= ,assumethatalldecompositionsareunique.Wewillshow that B 1 B k spansthespaceandislinearlyindependent.Itspansthe spacebecausetheassumptionthat V = W 1 + + W k meansthatevery ~v canbeexpressedas ~v = ~w 1 + + ~w k ,whichtranslatesbyequation toan expressionof ~v asalinearcombinationofthe ~ 'sfromtheconcatenation.For linearindependence,considerthislinearrelationship. ~ 0= c 1 ; 1 ~ 1 ; 1 + + c k;n k ~ k;n k Regroupasin thatis,take d 1 ,..., d k tobe1andmovefrombottomto toptogetthedecomposition ~ 0= ~w 1 + + ~w k .Becauseoftheassumption thatdecompositionsareunique,andbecausethezerovectorobviouslyhasthe decomposition ~ 0= ~ 0+ + ~ 0,wenowhavethateach ~w i isthezerovector.This meansthat c i; 1 ~ i; 1 + + c i;n i ~ i;n i = ~ 0.Thus,sinceeach B i isabasis,wehave thedesiredconclusionthatallofthe c 'sarezero. For= ,assumethat B 1 B k isabasisforthespace.Consider alinearrelationshipamongnonzerovectorsfromdierent W i 's, ~ 0= + d i ~w i + inordertoshowthatitistrivial.Therelationshipiswritteninthisway becauseweareconsideringacombinationofnonzerovectorsfromonlysomeof the W i 's;forinstance,theremightnotbea ~w 1 inthiscombination.Asin ~ 0= + d i c i; 1 ~ i; 1 + + c i;n i ~ i;n i + = + d i c i; 1 ~ i; 1 + + d i c i;n i ~ i;n i + andthelinearindependenceof B 1 B k givesthateachcoecient d i c i;j is zero.Now, ~w i isanonzerovector,soatleastoneofthe c i;j 'sisnotzero,and thus d i iszero.Thisholdsforeach d i ,andthereforethelinearrelationshipis trivial. Finally,for= ,assumethat,amongnonzerovectorsfromdierent W i 's,anylinearrelationshipistrivial.Considertwodecompositionsofavector ~v = ~w 1 + + ~w k and ~v = ~u 1 + + ~u k inordertoshowthatthetwoarethe same.Wehave ~ 0= ~w 1 + + ~w k )]TJ/F8 9.9626 Tf 9.963 0 Td [( ~u 1 + + ~u k = ~w 1 )]TJ/F11 9.9626 Tf 9.834 0 Td [(~u 1 + + ~w k )]TJ/F11 9.9626 Tf 9.834 0 Td [(~u k whichviolatestheassumptionunlesseach ~w i )]TJ/F11 9.9626 Tf 10.352 0 Td [(~u i isthezerovector.Hence, decompositionsareunique. QED

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SectionIII.BasisandDimension 135 4.9Denition Acollectionofsubspaces f W 1 ;:::;W k g is independent ifno nonzerovectorfromany W i isalinearcombinationofvectorsfromtheother subspaces W 1 ;:::;W i )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 ;W i +1 ;:::;W k 4.10Denition Avectorspace V isthe directsum or internaldirectsum ofitssubspaces W 1 ;:::;W k if V = W 1 + W 2 + + W k andthecollection f W 1 ;:::;W k g isindependent.Wewrite V = W 1 W 2 ::: W k 4.11Example Thebenchmarkmodelts: R 3 = x -axis y -axis z -axis. 4.12Example Thespaceof2 2matricesisthisdirectsum. f a 0 0 d a;d 2 R gf 0 b 00 b 2 R gf 00 c 0 c 2 R g Itisthedirectsumofsubspacesinmanyotherwaysaswell;directsumdecompositionsarenotunique. 4.13Corollary Thedimensionofadirectsumisthesumofthedimensions ofitssummands. Proof InLemma4.8,thenumberofbasisvectorsintheconcatenationequals thesumofthenumberofvectorsinthesubbasesthatmakeuptheconcatenation. QED Thespecialcaseoftwosubspacesisworthmentioningseparately. 4.14Denition Whenavectorspaceisthedirectsumoftwoofitssubspaces, thentheyaresaidtobe complements 4.15Lemma Avectorspace V isthedirectsumoftwoofitssubspaces W 1 and W 2 ifandonlyifitisthesumofthetwo V = W 1 + W 2 andtheirintersection istrivial W 1 W 2 = f ~ 0 g Proof Supposerstthat V = W 1 W 2 .Bydenition, V isthesumofthe two.Toshowthatthetwohaveatrivialintersection,let ~v beavectorfrom W 1 W 2 andconsidertheequation ~v = ~v .Ontheleftsideofthatequation isamemberof W 1 ,andontherightsideisalinearcombinationofmembers actually,ofonlyonememberof W 2 .Buttheindependenceofthespacesthen impliesthat ~v = ~ 0,asdesired. Fortheotherdirection,supposethat V isthesumoftwospaceswithatrivial intersection.Toshowthat V isadirectsumofthetwo,weneedonlyshow thatthespacesareindependent|nononzeromemberoftherstisexpressible asalinearcombinationofmembersofthesecond,andviceversa.Thisis truebecauseanyrelationship ~w 1 = c 1 ~w 2 ; 1 + + d k ~w 2 ;k with ~w 1 2 W 1 and ~w 2 ;j 2 W 2 forall j showsthatthevectorontheleftisalsoin W 2 ,sincethe rightsideisacombinationofmembersof W 2 .Theintersectionofthesetwo spacesistrivial,so ~w 1 = ~ 0.Thesameargumentworksforany ~w 2 QED

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136 ChapterTwo.VectorSpaces 4.16Example Inthespace R 2 ,the x -axisandthe y -axisarecomplements,that is, R 2 = x -axis y -axis.Aspacecanhavemorethanonepairofcomplementary subspaces;anotherpairherearethesubspacesconsistingofthelines y = x and y =2 x 4.17Example Inthespace F = f a cos + b sin a;b 2 R g ,thesubspaces W 1 = f a cos a 2 R g and W 2 = f b sin b 2 R g arecomplements.Inaddition tothefactthataspacelike F canhavemorethanonepairofcomplementary subspaces,insideofthespaceasinglesubspacelike W 1 canhavemorethanone complement|anothercomplementof W 1 is W 3 = f b sin + b cos b 2 R g 4.18Example In R 3 ,the xy -planeandthe yz -planesarenotcomplements, whichisthepointofthediscussionfollowingExample4.4.Onecomplementof the xy -planeisthe z -axis.Acomplementofthe yz -planeisthelinethrough ; 1 ; 1. 4.19Example FollowingLemma4.15,hereisanaturalquestion:isthesimple sum V = W 1 + + W k alsoadirectsumifandonlyiftheintersectionofthe subspacesistrivial?Theansweristhatiftherearemorethantwosubspaces thenhavingatrivialintersectionisnotenoughtoguaranteeuniquedecompositioni.e.,isnotenoughtoensurethatthespacesareindependent.In R 3 ,let W 1 bethe x -axis,let W 2 bethe y -axis,andlet W 3 bethis. W 3 = f 0 @ q q r 1 A q;r 2 R g Thecheckthat R 3 = W 1 + W 2 + W 3 iseasy.Theintersection W 1 W 2 W 3 is trivial,butdecompositionsaren'tunique. 0 @ x y z 1 A = 0 @ 0 0 0 1 A + 0 @ 0 y )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 0 1 A + 0 @ x x z 1 A = 0 @ x )]TJ/F11 9.9626 Tf 9.963 0 Td [(y 0 0 1 A + 0 @ 0 0 0 1 A + 0 @ y y z 1 A Thisexamplealsoshowsthatthisrequirementisalsonotenough:thatall pairwiseintersectionsofthesubspacesbetrivial.SeeExercise30. Inthissubsectionwehaveseentwowaystoregardaspaceasbuiltupfrom componentparts.Bothareuseful;inparticular,inthisbookthedirectsum denitionisneededtodotheJordanFormconstructioninthefthchapter. Exercises X 4.20 Decideif R 2 isthedirectsumofeach W 1 and W 2 a W 1 = f x 0 x 2 R g W 2 = f x x x 2 R g b W 1 = f s s s 2 R g W 2 = f s 1 : 1 s s 2 R g c W 1 = R 2 W 2 = f ~ 0 g

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SectionIII.BasisandDimension 137 d W 1 = W 2 = f t t t 2 R g e W 1 = f 1 0 + x 0 x 2 R g W 2 = f )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 0 + 0 y y 2 R g X 4.21 Showthat R 3 isthedirectsumofthe xy -planewitheachofthese. a the z -axis b theline f z z z z 2 R g 4.22 Is P 2 thedirectsumof f a + bx 2 a;b 2 R g and f cx c 2 R g ? X 4.23 In P n ,the even polynomialsarethemembersofthisset E = f p 2P n p )]TJ/F32 8.9664 Tf 7.167 0 Td [(x = p x forall x g andthe odd polynomialsarethemembersofthisset. O = f p 2P n p )]TJ/F32 8.9664 Tf 7.168 0 Td [(x = )]TJ/F32 8.9664 Tf 7.168 0 Td [(p x forall x g Showthatthesearecomplementarysubspaces. 4.24 Whichofthesesubspacesof R 3 W 1 :the x -axis, W 2 :the y -axis, W 3 :the z -axis, W 4 :theplane x + y + z =0, W 5 :the yz -plane canbecombinedto a sumto R 3 ? b directsumto R 3 ? X 4.25 Showthat P n = f a 0 a 0 2 R g ::: f a n x n a n 2 R g 4.26 Whatis W 1 + W 2 if W 1 W 2 ? 4.27 DoesExample4.5generalize?Thatis,isthistrueorfalse:ifavectorspace V hasabasis h ~ 1 ;:::; ~ n i thenitisthedirectsumofthespansoftheone-dimensional subspaces V =[ f ~ 1 g ] ::: [ f ~ n g ]? 4.28 Can R 4 bedecomposedasadirectsumintwodierentways?Can R 1 ? 4.29 Thisexercisemakesthenotationofwriting`+'betweensetsmorenatural. Provethat,where W 1 ;:::;W k aresubspacesofavectorspace, W 1 + + W k = f ~w 1 + ~w 2 + + ~w k ~w 1 2 W 1 ;:::;~w k 2 W k g ; andsothesumofsubspacesisthesubspaceofallsums. 4.30 RefertoExample4.19.Thisexerciseshowsthattherequirementthatpariwiseintersectionsbetrivialisgenuinelystrongerthantherequirementonlythat theintersectionofallofthesubspacesbetrivial.Giveavectorspaceandthree subspaces W 1 W 2 ,and W 3 suchthatthespaceisthesumofthesubspaces,the intersectionofallthreesubspaces W 1 W 2 W 3 istrivial,butthepairwiseintersections W 1 W 2 W 1 W 3 ,and W 2 W 3 arenontrivial. X 4.31 Provethatif V = W 1 ::: W k then W i W j istrivialwhenever i 6 = j .This showsthatthersthalfoftheproofofLemma4.15extendstothecaseofmore thantwosubspaces.Example4.19showsthatthisimplicationdoesnotreverse; theotherhalfdoesnotextend. 4.32 Recallthatnolinearlyindependentsetcontainsthezerovector.Canan independentsetofsubspacescontainthetrivialsubspace? X 4.33 Doeseverysubspacehaveacomplement? X 4.34 Let W 1 ;W 2 besubspacesofavectorspace. a Assumethattheset S 1 spans W 1 ,andthattheset S 2 spans W 2 .Can S 1 [ S 2 span W 1 + W 2 ?Mustit?

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138 ChapterTwo.VectorSpaces b Assumethat S 1 isalinearlyindependentsubsetof W 1 andthat S 2 isa linearlyindependentsubsetof W 2 .Can S 1 [ S 2 bealinearlyindependentsubset of W 1 + W 2 ?Mustit? 4.35 Whenavectorspaceisdecomposedasadirectsum,thedimensionsofthe subspacesaddtothedimensionofthespace.Thesituationwithaspacethatis givenasthesumofitssubspacesisnotassimple.Thisexerciseconsidersthe two-subspacespecialcase. a Forthesesubspacesof M 2 2 nd W 1 W 2 ,dim W 1 W 2 W 1 + W 2 ,and dim W 1 + W 2 W 1 = f 00 cd c;d 2 R g W 2 = f 0 b c 0 b;c 2 R g b Supposethat U and W aresubspacesofavectorspace.Supposethatthe sequence h ~ 1 ;:::; ~ k i isabasisfor U W .Finally,supposethattheprior sequencehasbeenexpandedtogiveasequence h ~ 1 ;:::;~ j ; ~ 1 ;:::; ~ k i thatisa basisfor U ,andasequence h ~ 1 ;:::; ~ k ;~! 1 ;:::;~! p i thatisabasisfor W .Prove thatthissequence h ~ 1 ;:::;~ j ; ~ 1 ;:::; ~ k ;~! 1 ;:::;~! p i isabasisforforthesum U + W c Concludethatdim U + W =dim U +dim W )]TJ/F29 8.9664 Tf 9.216 0 Td [(dim U W d Let W 1 and W 2 beeight-dimensionalsubspacesofaten-dimensionalspace. Listallvaluespossiblefordim W 1 W 2 4.36 Let V = W 1 ::: W k andforeachindex i supposethat S i isalinearly independentsubsetof W i .Provethattheunionofthe S i 'sislinearlyindependent. 4.37 Amatrixis symmetric ifforeachpairofindices i and j ,the i;j entryequals the j;i entry.Amatrixis antisymmetric ifeach i;j entryisthenegativeofthe j;i entry. a Giveasymmetric2 2matrixandanantisymmetric2 2matrix. Remark. Forthesecondone,becarefulabouttheentriesonthediagional. b Whatistherelationshipbetweenasquaresymmetricmatrixanditstranspose?Betweenasquareantisymmetricmatrixanditstranspose? c Showthat M n n isthedirectsumofthespaceofsymmetricmatricesand thespaceofantisymmetricmatrices. 4.38 Let W 1 ;W 2 ;W 3 besubspacesofavectorspace.Provethat W 1 W 2 + W 1 W 3 W 1 W 2 + W 3 .Doestheinclusionreverse? 4.39 Theexampleofthe x -axisandthe y -axisin R 2 showsthat W 1 W 2 = V does notimplythat W 1 [ W 2 = V .Can W 1 W 2 = V and W 1 [ W 2 = V happen? X 4.40 ConsiderCorollary4.13.Doesitworkbothways|thatis,supposingthat V = W 1 + + W k ,is V = W 1 ::: W k ifandonlyifdim V =dim W 1 + +dim W k ? 4.41 Weknowthatif V = W 1 W 2 thenthereisabasisfor V thatsplitsintoa basisfor W 1 andabasisfor W 2 .Canwemakethestrongerstatementthatevery basisfor V splitsintoabasisfor W 1 andabasisfor W 2 ? 4.42 Wecanaskaboutthealgebraofthe`+'operation. a Isitcommutative;is W 1 + W 2 = W 2 + W 1 ? b Isitassociative;is W 1 + W 2 + W 3 = W 1 + W 2 + W 3 ? c Let W beasubspaceofsomevectorspace.Showthat W + W = W d Musttherebeanidentityelement,asubspace I suchthat I + W = W + I = W forallsubspaces W ?

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SectionIII.BasisandDimension 139 e Doesleft-cancelationhold:if W 1 + W 2 = W 1 + W 3 then W 2 = W 3 ?Right cancelation? 4.43 Considerthealgebraicpropertiesofthedirectsumoperation. a Doesdirectsumcommute:does V = W 1 W 2 implythat V = W 2 W 1 ? b Provethatdirectsumisassociative: W 1 W 2 W 3 = W 1 W 2 W 3 c Showthat R 3 isthedirectsumofthethreeaxestherelevancehereisthatby thepreviousitem,weneedn'tspecifywhichtwoofthethreeeaxesarecombined rst. d Doesthedirectsumoperationleft-cancel:does W 1 W 2 = W 1 W 3 imply W 2 = W 3 ?Doesitright-cancel? e Thereisanidentityelementwithrespecttothisoperation.Findit. f Dosome,orall,subspaceshaveinverseswithrespecttothisoperation:is thereasubspace W ofsomevectorspacesuchthatthereisasubspace U with thepropertythat U W equalstheidentityelementfromtheprioritem?

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140 ChapterTwo.VectorSpaces Topic:Fields Linearcombinationsinvolvingonlyfractionsoronlyintegersaremucheasier forcomputationsthancombinationsinvolvingrealnumbers,becausecomputing withirrationalnumbersisawkward.Couldothernumbersystems,likethe rationalsortheintegers,workintheplaceof R inthedenitionofavector space? Yesandno.Ifwetakework"tomeanthattheresultsofthischapter remaintruethenananalysisofwhichpropertiesoftherealswehaveusedin thischaptergivesthefollowinglistofconditionsanalgebraicsystemneedsin ordertowork"intheplaceof R Denition. A eld isaset F withtwooperations`+'and` 'suchthat forany a;b 2F theresultof a + b isin F and a + b = b + a if c 2F then a + b + c = a + b + c forany a;b 2F theresultof a b isin F and a b = b a if c 2F then a b c = a b c if a;b;c 2F then a b + c = a b + a c thereisanelement0 2F suchthat if a 2F then a +0= a foreach a 2F thereisanelement )]TJ/F11 9.9626 Tf 7.749 0 Td [(a 2F suchthat )]TJ/F11 9.9626 Tf 7.749 0 Td [(a + a =0 thereisanelement1 2F suchthat if a 2F then a 1= a foreachelement a 6 =0of F thereisanelement a )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 2F suchthat a )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 a =1. Thenumbersystemconsistingofthesetofrealnumbersalongwiththeusual additionandmultiplicationoperationisaeld,naturally.Anothereldisthe setofrationalnumberswithitsusualadditionandmultiplicationoperations. Anexampleofanalgebraicstructurethatisnotaeldistheintegernumber system|itfailsthenalcondition. Someexamplesaresurprising.Theset f 0 ; 1 g undertheseoperations: + 01 0 01 1 10 01 0 00 1 01 isaeldseeExercise4.

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Topic:Fields 141 WecoulddevelopLinearAlgebraasthetheoryofvectorspaceswithscalars fromanarbitraryeld,insteadofstickingtotakingthescalarsonlyfrom R .In thatcase,almostallofthestatementsinthisbookwouldcarryoverbyreplacing ` R 'with` F ',andthusbytakingcoecients,vectorentries,andmatrixentries tobeelementsof F almost"becausestatementsinvolvingdistancesorangles areexceptions.Herearesomeexamples;eachappliestoavectorspace V over aeld F Forany ~v 2 V and a 2F ,i0 ~v = ~ 0,andii )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ~v + ~v = ~ 0,and iii a ~ 0= ~ 0. Thespanthesetoflinearcombinationsofasubsetof V isasubspace of V Anysubsetofalinearlyindependentsetisalsolinearlyindependent. Inanite-dimensionalvectorspace,anytwobaseshavethesamenumber ofelements. Evenstatementsthatdon'texplicitlymention F useeldpropertiesintheir proof. Wewon'tdevelopvectorspacesinthismoregeneralsettingbecausethe additionalabstractioncanbeadistraction.Theideaswewanttobringout alreadyappearwhenwesticktothereals. TheonlyexceptionisinChapterFive.Inthatchapterwemustfactor polynomials,sowewillswitchtoconsideringvectorspacesovertheeldof complexnumbers.Wewilldiscussthismore,includingabriefreviewofcomplex arithmetic,whenwegetthere. Exercises 1 Showthattherealnumbersformaeld. 2 Provethattheseareelds. a Therationalnumbers Q b Thecomplexnumbers C 3 Giveanexamplethatshowsthattheintegernumbersystemisnotaeld. 4 Considertheset f 0 ; 1 g subjecttotheoperationsgivenabove.Showthatitisa eld. 5 Givesuitableoperationstomaketheset f 0 ; 1 ; 2 g aeld.

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142 ChapterTwo.VectorSpaces Topic:Crystals Everyonehasnoticedthattablesaltcomesinlittlecubes. Remarkably,theexplanationforthecubicalexternalshapeisthesimplest one:theinternalshape,thewaytheatomslie,isalsocubical.Theinternal structureispicturedbelow.Saltissodiumcloride,andthesmallspheresshown aresodiumwhilethebigonesarecloride.Tosimplifytheview,itonlyshows thesodiumsandcloridesonthefront,top,andright. Thespecksofsaltthatweseewhenwespreadalittleoutonthetableconsistof manyrepetitionsofthisfundamentalunit.Thatis,thesecubesofatomsstack uptomakethelargercubicalstructurethatwesee.Asolid,suchastablesalt, witharegularinternalstructureisa crystal Wecanrestrictourattentiontothefrontface.There,wehavethesquare repeatedmanytimes. Thedistancebetweenthecornersofthesquarecellisabout3 : 34 Angstroms an Angstromis10 )]TJ/F7 6.9738 Tf 6.226 0 Td [(10 meters.Obviouslythatunitisunwieldly.Instead,the thingtodoistotakeasaunitthelengthofeachsideofthesquare.Thatis, wenaturallyadoptthisbasis. h 3 : 34 0 ; 0 3 : 34 i Thenwecandescribe,say,thecornerintheupperrightofthepictureaboveas 3 ~ 1 +2 ~ 2 .

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Topic:Crystals 143 Anothercrystalfromeverydayexperienceispencillead.Itisgraphite, formedfromcarbonatomsarrangedinthisshape. Thisisasingleplaneofgraphite.Apieceofgraphiteconsistsofmanyofthese planeslayeredinastack.Thechemicalbondsbetweentheplanesaremuch weakerthanthebondsinsidetheplanes,whichexplainswhypencilswrite|the graphitecanbeshearedsothattheplanesslideoandareleftonthepaper. Wecangetaconvienentunitoflengthbydecomposingthehexagonalringinto threeregionsthatarerotationsofthis unitcell Thenanaturalbasisconsistsofthevectorsthatformthesidesofthatunitcell. Thedistancealongthebottomandslantis1 : 42 Angstroms,sothis h 1 : 42 0 ; 1 : 23 : 71 i isagoodbasis. Theselectionofconvienentbasesextendstothreedimensions.Another familiarcrystalformedfromcarbonisdiamond.Liketablesalt,itisbuiltfrom cubes,butthestructureinsideeachcubeismorecomplicatedthansalt's.In additiontocarbonsateachcorner, therearecarbonsinthemiddleofeachface.

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144 ChapterTwo.VectorSpaces Toshowtheaddedfacecarbonsclearly,thecornercarbonshavebeenreduced todots.Therearealsofourmorecarbonsinsidethecube,twothatarea quarterofthewayupfromthebottomandtwothatareaquarteroftheway downfromthetop. Asbefore,carbonsshownearlierhavebeenreducedheretodots.Thedistancealonganyedgeofthecubeis2 : 18 Angstroms.Thus,anaturalbasisfor describingthelocationsofthecarbons,andthebondsbetweenthem,isthis. h 0 @ 2 : 18 0 0 1 A ; 0 @ 0 2 : 18 0 1 A ; 0 @ 0 0 2 : 18 1 A i Eventhefewexamplesgivenhereshowthatthestructuresofcrystalsiscomplicatedenoughthatsomeorganizedsystemtogivethelocationsoftheatoms, andhowtheyarechemicallybound,isneeded.Onetoolforthatorganization isaconvienentbasis.Thisapplicationofbasesissimple,butitshowsacontext wheretheideaarisesnaturally.Theworkinthischapterjusttakesthissimple ideaanddevelopsit. Exercises 1 Howmanyfundamentalregionsarethereinonefaceofaspeckofsalt?Witha ruler,wecanestimatethatfaceisasquarethatis0 : 1cmonaside. 2 Inthegraphitepicture,imaginethatweareinterestedinapoint5 : 67 Angstroms overand3 : 14 Angstromsupfromtheorigin. a Expressthatpointintermsofthebasisgivenforgraphite. b Howmanyhexagonalshapesawayisthispointfromtheorigin? c Expressthatpointintermsofasecondbasis,wheretherstbasisvectoris thesame,butthesecondisperpendiculartotherstgoinguptheplaneand ofthesamelength. 3 Givethelocationsoftheatomsinthediamondcubebothintermsofthebasis, andin Angstroms. 4 Thisillustrateshowthedimensionsofaunitcellcouldbecomputedfromthe shapeinwhichasubstancecrystalizes[Ebbing],p.462. a Recallthatthereare6 : 022 10 23 atomsinamolethisisAvagadro'snumber. Fromthat,andthefactthatplatinumhasamassof195 : 08gramspermole, calculatethemassofeachatom. b Platinumcrystalizesinaface-centeredcubiclatticewithatomsateachlattice point,thatis,itlookslikethemiddlepicturegivenaboveforthediamondcrystal. Findthenumberofplatinumsperunitcellhint:sumthefractionsofplatinums thatareinsideofasinglecell. c Fromthat,ndthemassofaunitcell. d Platinumcrystalhasadensityof21 : 45gramspercubiccentimeter.From this,andthemassofaunitcell,calculatethevolumeofaunitcell.

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Topic:Crystals 145 e Findthelengthofeachedge. f Describeanaturalthree-dimensionalbasis.

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146 ChapterTwo.VectorSpaces Topic:VotingParadoxes ImaginethataPoliticalScienceclassstudyingtheAmericanpresidentialprocessholdsamockelection.Membersoftheclassareaskedtorank,frommost preferredtoleastpreferred,thenomineesfromtheDemocraticParty,theRepublicanParty,andtheThirdParty,andthisistheresult > means`ispreferred to'. preferenceorder numberwith thatpreference Democrat > Republican > Third 5 Democrat > Third > Republican 4 Republican > Democrat > Third 2 Republican > Third > Democrat 8 Third > Democrat > Republican 8 Third > Republican > Democrat 2 total 29 Whatisthepreferenceofthegroupasawhole? Overall,thegrouppreferstheDemocrattotheRepublicanbyvevotes; seventeenvotersrankedtheDemocratabovetheRepublicanversustwelvethe otherway.And,overall,thegrouppreferstheRepublicantotheThird'snominee,fteentofourteen.But,strangelyenough,thegroupalsopreferstheThird totheDemocrat,eighteentoeleven. Democrat Third Republican 7voters 1voter 5voters Thisisanexampleofa votingparadox ,specically,a majoritycycle Votingparadoxesarestudiedinpartbecauseoftheirimplicationsforpracticalpolitics.Forinstance,theinstructorcanmanipulatetheclassintochoosing theDemocratastheoverallwinnerbyrstaskingtheclasstochoosebetween theRepublicanandtheThird,andthenaskingtheclasstochoosebetweenthe winnerofthatcontest,theRepublican,andtheDemocrat.Bysimilarmanipulations,anyoftheothertwocandidatescanbemadetocomeoutasthewinner. InthisTopicwewillsticktothree-candidateelections,butsimilarresultsapply tolargerelections. Votingparadoxesarealsostudiedsimplybecausetheyaremathematically interesting.Oneinterestingaspectisthatthegroup'soverallmajoritycycle occursdespitethateachsinglevoters'spreferencelistis rational |inastraightlineorder.Thatis,themajoritycycleseemstoariseintheaggregate,without beingpresentintheelementsofthataggregate,thepreferencelists.Recently,

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Topic:VotingParadoxes 147 however,linearalgebrahasbeenused[Zwicker]toarguethatatendencytoward cyclicpreferenceisactuallypresentineachvoter'slist,andthatitsurfaceswhen thereismoreaddingofthetendencythancancelling. Forthisargument,abbreviatingthechoicesas D R ,and T ,wecandescribe howavoterwithpreferenceorder D>R>T contributestotheabovecycle. D T R )]TJ/F7 6.9738 Tf 6.227 0 Td [(1voter 1voter 1voter Thenegativesignisherebecausethearrowdescribes T aspreferredto D ,but thisvoterlikesthemtheotherway.Thedescriptionsfortheotherpreference listsareinthetableonpage148. Now,toconducttheelectionwelinearlycombinethesedescriptions;for instance,thePoliticalSciencemockelection 5 D T R )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 1 1 +4 D T R )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 1 + +2 D T R 1 )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 yieldsthecirculargrouppreferenceshownearlier. Ofcourse,takinglinearcombinationsislinearalgebra.Theabovecyclenotationissuggestivebutinconvienent,sowetemporarilyswitchtousingcolumn vectorsbystartingatthe D andtakingthenumbersfromthecycleincounterclockwiseorder.Thus,themockelectionandasingle D>R>T voteare representedinthisway. 0 @ 7 1 5 1 A and 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 1 1 A Wewilldecomposevotevectorsintotwoparts,onecyclicandtheotheracyclic. Fortherstpart,wesaythatavectoris purelycyclic ifitisinthissubspace of R 3 C = f 0 @ k k k 1 A k 2 R g = f k 0 @ 1 1 1 1 A k 2 R g Forthesecondpart,considerthesubspaceseeExercise6ofvectorsthatare perpendiculartoallofthevectorsin C C ? = f 0 @ c 1 c 2 c 3 1 A 0 @ c 1 c 2 c 3 1 A 0 @ k k k 1 A =0forall k 2 R g = f 0 @ c 1 c 2 c 3 1 A c 1 + c 2 + c 3 =0 g = f c 2 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 0 1 A + c 3 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 0 1 1 A c 2 ;c 3 2 R g

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148 ChapterTwo.VectorSpaces Readthataloudas C perp".Soweareledtothisbasisfor R 3 h 0 @ 1 1 1 1 A ; 0 @ )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 1 0 1 A ; 0 @ )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 0 1 1 A i Wecanrepresentvoteswithrespecttothisbasis,andtherebydecomposethem intoacyclicpartandanacyclicpart. Noteforreaderswhohavecoveredthe optionalsectioninthischapter:thatis,thespaceisthedirectsumof C and C ? Forexample,considerthe D>R>T voterdiscussedabove.Therepresentationintermsofthebasisiseasilyfound, c 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(c 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(c 3 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 c 1 + c 2 =1 c 1 + c 3 =1 )]TJ/F10 6.9738 Tf 6.226 0 Td [( 1 + 2 )167(! )]TJ/F10 6.9738 Tf 6.226 0 Td [( 1 + 3 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 2 2 + 3 )167(! c 1 )]TJ/F11 9.9626 Tf 14.944 0 Td [(c 2 )]TJ/F11 9.9626 Tf 32.655 0 Td [(c 3 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 2 c 2 + c 3 =2 = 2 c 3 =1 sothat c 1 =1 = 3, c 2 =2 = 3,and c 3 =2 = 3.Then 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 1 1 A = 1 3 0 @ 1 1 1 1 A + 2 3 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 0 1 A + 2 3 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 0 1 1 A = 0 @ 1 = 3 1 = 3 1 = 3 1 A + 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 = 3 2 = 3 2 = 3 1 A givesthedesireddecompositionintoacyclicpartandandanacyclicpart. D T R )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 1 1 = D T R 1 = 3 1 = 3 1 = 3 + D T R )]TJ/F6 4.9813 Tf 5.397 0 Td [(4 = 3 2 = 3 2 = 3 Thus,this D>R>T voter'srationalpreferencelistcanindeedbeseento haveacyclicpart. The T>R>D voterisoppositetotheonejustconsideredinthatthe` > symbolsarereversed.Thisvoter'sdecomposition D T R 1 )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 = D T R )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 = 3 )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 = 3 )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 = 3 + D T R 4 = 3 )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 = 3 )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 = 3 showsthattheseoppositepreferenceshavedecompositionsthatareopposite. Wesaythattherstvoterhaspositive spin sincethecyclepartiswiththe directionwehavechosenforthearrows,whilethesecondvoter'sspinisnegative. Thefactthatthattheseoppositevoterscanceleachotherisreectedinthe factthattheirvotevectorsaddtozero.Thissuggestsanalternatewaytotally anelection.Wecouldrstcancelasmanyoppositepreferencelistsaspossible, andthendeterminetheoutcomebyaddingtheremaininglists. Therowsofthetablebelowcontainthethreepairsofoppositepreference lists.Thecolumnsgroupthosepairsbyspin.Forinstance,therstrowcontains thetwovotersjustconsidered.

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Topic:VotingParadoxes 149 positivespin negativespin Democrat > Republican > Third D T R )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 1 1 = D T R 1 = 3 1 = 3 1 = 3 + D T R )]TJ/F6 4.9813 Tf 5.396 0 Td [(4 = 3 2 = 3 2 = 3 Third > Republican > Democrat D T R 1 )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 = D T R )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 = 3 )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 = 3 )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 = 3 + D T R 4 = 3 )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 = 3 )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 = 3 Republican > Third > Democrat D T R 1 1 )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 = D T R 1 = 3 1 = 3 1 = 3 + D T R 2 = 3 2 = 3 )]TJ/F6 4.9813 Tf 5.397 0 Td [(4 = 3 Democrat > Third > Republican D T R )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 1 = D T R )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 = 3 )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 = 3 )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 = 3 + D T R )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 = 3 )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 = 3 4 = 3 Third > Democrat > Republican D T R 1 )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 1 = D T R 1 = 3 1 = 3 1 = 3 + D T R 2 = 3 )]TJ/F6 4.9813 Tf 5.396 0 Td [(4 = 3 2 = 3 Republican > Democrat > Third D T R )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 1 )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 = D T R )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 = 3 )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 = 3 )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 = 3 + D T R )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 = 3 4 = 3 )]TJ/F6 4.9813 Tf 5.397 0 Td [(2 = 3 Ifweconducttheelectionasjustdescribedthenafterthecancellationofasmany oppositepairsofvotersaspossible,therewillbeleftthreesetsofpreference lists,onesetfromtherstrow,onesetfromthesecondrow,andonesetfrom thethirdrow.Wewillnishbyprovingthatavotingparadoxcanhappen onlyifthespinsofthesethreesetsareinthesamedirection.Thatis,fora votingparadoxtooccur,thethreeremainingsetsmustallcomefromtheleft ofthetableorallcomefromtherightseeExercise3.Thisshowsthatthere issomeconnectionbetweenthemajoritycycleandthedecompositionthatwe areusing|avotingparadoxcanhappenonlywhenthetendenciestowardcyclic preferencereinforceeachother. Fortheproof,assumethatoppositepreferenceordershavebeencancelled, andweareleftwithonesetofpreferencelistsfromeachofthethreerows. Considerthesumofthesethreehere,thenumbers a b ,and c couldbepositive, negative,orzero. D T R )]TJ/F9 4.9813 Tf 5.397 0 Td [(a a a + D T R b b )]TJ/F9 4.9813 Tf 5.396 0 Td [(b + D T R c )]TJ/F9 4.9813 Tf 5.396 0 Td [(c c = D T R )]TJ/F9 4.9813 Tf 5.396 0 Td [(a + b + c a + b )]TJ/F9 4.9813 Tf 7.027 0 Td [(c a )]TJ/F9 4.9813 Tf 7.026 0 Td [(b + c Avotingparadoxoccurswhenthethreenumbersontheright, a )]TJ/F11 9.9626 Tf 10.372 0 Td [(b + c and a + b )]TJ/F11 9.9626 Tf 10.374 0 Td [(c and )]TJ/F11 9.9626 Tf 7.749 0 Td [(a + b + c ,areallnonnegativeorallnonpositive.Ontheleft, atleasttwoofthethreenumbers, a and b and c ,arebothnonnegativeorboth nonpositive.Wecanassumethattheyare a and b .Thatmakesfourcases:the cycleisnonnegativeand a and b arenonnegative,thecycleisnonpositiveand a and b arenonpositive,etc.Wewilldoonlytherstcase,sincethesecondis similarandtheothertwoarealsoeasy. Soassumethatthecycleisnonnegativeandthat a and b arenonnegative. Theconditions0 a )]TJ/F11 9.9626 Tf 9.686 0 Td [(b + c and0 )]TJ/F11 9.9626 Tf 18.265 0 Td [(a + b + c addtogivethat0 2 c ,which impliesthat c isalsononnegative,asdesired.Thatendstheproof. Thisresultsaysonlythathavingallthreespininthesamedirectionisa necessaryconditionforamajoritycycle.Itisnotsucient;seeExercise4.

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150 ChapterTwo.VectorSpaces Votingtheoryandassociatedtopicsarethesubjectofcurrentresearch. Therearemanyintriguingresults,mostnotablytheoneproducedbyK.Arrow [Arrow],whowontheNobelPrizeinpartforthiswork,showingthatnovoting systemisentirelyfairforareasonabledenitionoffair".Formoreinformation,somegoodintroductoryarticlesare[Gardner,1970],[Gardner,1974], [Gardner,1980],and[Neimi&Riker].Aquitereadablerecentbookis[Taylor]. ThelonglistofcasesfromrecentAmericanpoliticalhistorygivenin[Poundstone] showthatmanipulationoftheseparadoxesisroutineinpracticeandtheauthor proposesasolution. ThisTopicislargelydrawnfrom[Zwicker]. Author'sNote:Iwouldliketo thankProfessorZwickerforhiskindandilluminatingdiscussions. Exercises 1 Hereisareasonablewayinwhichavotercouldhaveacyclicpreference.Suppose thatthisvoterrankseachcandidateoneachofthreecriteria. a Drawupatablewiththerowslabelled`Democrat',`Republican',and`Third', andthecolumnslabelled`character',`experience',and`policies'.Insideeach column,ranksomecandidateasmostpreferred,rankanotherasinthemiddle, andranktheremainingoneasleastpreferred. b Inthisranking,istheDemocratpreferredtotheRepublicaninatleasttwo outofthreecriteria,orviceversa?IstheRepublicanpreferredtotheThird? c Doesthetablethatwasjustconstructedhaveacyclicpreferenceorder?If not,makeonethatdoes. Soitispossibleforavotertohaveacyclicpreferenceamongcandidates.The paradoxdescribedabove,however,isthatevenifeachvoterhasastraight-line preferencelist,acyclicpreferencecanstillarisefortheentiregroup. 2 Computethevaluesinthetableofdecompositions. 3 DothecancellationsofoppositepreferenceordersforthePoliticalScienceclass's mockelection.Arealltheremainingpreferencesfromtheleftthreerowsofthe tableorfromtheright? 4 Thenecessaryconditionthatisprovedabove|avotingparadoxcanhappenonly ifallthreepreferencelistsremainingaftercancellationhavethesamespin|isnot alsosucient. a Continuingthepositivecyclecaseconsideredintheproof,usethetwoinequalities0 a )]TJ/F32 8.9664 Tf 9.215 0 Td [(b + c and0 )]TJ/F32 8.9664 Tf 16.895 0 Td [(a + b + c toshowthat j a )]TJ/F32 8.9664 Tf 9.216 0 Td [(b j c b Alsoshowthat c a + b ,andhencethat j a )]TJ/F32 8.9664 Tf 9.216 0 Td [(b j c a + b c Giveanexampleofavotewherethereisamajoritycycle,andadditionof onemorevoterwiththesamespincausesthecycletogoaway. d Cantheoppositehappen;canadditionofonevoterwithawrong"spin causeacycletoappear? e Giveaconditionthatisbothnecessaryandsucienttogetamajoritycycle. 5 Aone-voterelectioncannothaveamajoritycyclebecauseoftherequirement thatwe'veimposedthatthevoter'slistmustberational. a Showthatatwo-voterelectionmayhaveamajoritycycle.Weconsiderthe grouppreferenceamajoritycycleifallthreegrouptotalsarenonnegativeorif allthreearenonpositive|thatis,weallowsomezero'sinthegrouppreference. b Showthatforanynumberofvotersgreaterthanone,thereisanelection involvingthatmanyvotersthatresultsinamajoritycycle.

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Topic:VotingParadoxes 151 6 Let U beasubspaceof R 3 .Provethattheset U ? = f ~v ~v ~u =0forall ~u 2 U g ofvectorsthatareperpendiculartoeachvectorin U isalsoasubspaceof R 3 .

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152 ChapterTwo.VectorSpaces Topic:DimensionalAnalysis Youcan'taddapplesandoranges,"theoldsayinggoes.Itreectsourexperiencethatinapplicationsthequantitieshaveunitsandkeepingtrackofthose unitsisworthwhile.Everyonehasdonecalculationssuchasthisonethatuse theunitsasacheck. 60 sec min 60 min hr 24 hr day 365 day year =31536000 sec year However,theideaofincludingtheunitscanbetakenbeyondbookkeeping.It canbeusedtodrawconclusionsaboutwhatrelationshipsarepossibleamong thephysicalquantities. Tostart,considerthephysicsequation:distance=16 time 2 .Ifthe distanceisinfeetandthetimeisinsecondsthenthisisatruestatementabout fallingbodies.Howeveritisnotcorrectinotherunitsystems;forinstance,it isnotcorrectinthemeter-secondsystem.Wecanxthatbymakingthe16a dimensionalconstant dist=16 ft sec 2 time 2 Forinstance,theaboveequationholdsintheyard-secondsystem. distanceinyards=16 = 3yd sec 2 timeinsec 2 = 16 3 yd sec 2 timeinsec 2 Soourrstpointisthatbyincludingtheunits"wemeanthatwearerestricting ourattentiontoequationsthatusedimensionalconstants. Byusingdimensionalconstants,wecanbevagueaboutunitsandsayonly thatallquantitiesaremeasuredincombinationsofsomeunitsoflength L mass M ,andtime T .Weshallrefertothesethreeas dimensions thesearethe onlythreedimensionsthatweshallneedinthisTopic.Forinstance,velocity couldbemeasuredinfeet = secondorfathoms = hour,butinalleventsitinvolves someunitoflengthdividedbysomeunitoftimesothe dimensionalformula ofvelocityis L=T .Similarly,thedimensionalformulaofdensityis M=L 3 .We shallpreferusingnegativeexponentsoverthefractionbarsandweshallinclude thedimensionswithazeroexponent,thatis,weshallwritethedimensional formulaofvelocityas L 1 M 0 T )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 andthatofdensityas L )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 M 1 T 0 Inthiscontext,Youcan'taddapplestooranges"becomestheadviceto checkthatallofanequation'stermshavethesamedimensionalformula.Anexampleisthisversionofthefallingbodyequation: d )]TJ/F11 9.9626 Tf 9.374 0 Td [(gt 2 =0.Thedimensional formulaofthe d termis L 1 M 0 T 0 .Fortheotherterm,thedimensionalformulaof g is L 1 M 0 T )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 g isthedimensionalconstantgivenaboveas16ft = sec 2 andthedimensionalformulaof t is L 0 M 0 T 1 ,sothatoftheentire gt 2 termis L 1 M 0 T )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 L 0 M 0 T 1 2 = L 1 M 0 T 0 .Thusthetwotermshavethesamedimensionalformula.Anequationwiththispropertyis dimensionallyhomogeneous Quantitieswithdimensionalformula L 0 M 0 T 0 are dimensionless .Forexample,wemeasureananglebytakingtheratioofthesubtendedarctothe radius

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Topic:DimensionalAnalysis 153 r arc whichistheratioofalengthtoalength L 1 M 0 T 0 =L 1 M 0 T 0 andthusangles havethedimensionalformula L 0 M 0 T 0 Theclassicexampleofusingtheunitsformorethanbookkeeping,using themtodrawconclusions,considerstheformulafortheperiodofapendulum. p ={someexpressioninvolvingthelengthofthestring,etc.{ Theperiodisinunitsoftime L 0 M 0 T 1 .Sothequantitiesontheothersideof theequationmusthavedimensionalformulasthatcombineinsuchawaythat their L 'sand M 'scancelandonlyasingle T remains.Thetableonpage154has thequantitiesthatanexperiencedinvestigatorwouldconsiderpossiblyrelevant. Theonlydimensionalformulasinvolving L areforthelengthofthestringand theaccelerationduetogravity.Forthe L 'softhesetwotocancel,whenthey appearintheequationtheymustbeinratio,e.g.,as `=g 2 ,orascos `=g ,or as `=g )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 .Thereforetheperiodisafunctionof `=g Thisisaremarkableresult:withapencilandpaperanalysis,beforeweever tookoutthependulumandmademeasurements,wehavedeterminedsomething abouttherelationshipamongthequantities. Tododimensionalanalysissystematically,weneedtoknowtwothingsargumentsforthesearein[Bridgman],ChapterIIandIV.Therstisthateach equationrelatingphysicalquantitiesthatweshallseeinvolvesasumofterms, whereeachtermhastheform m p 1 1 m p 2 2 m p k k fornumbers m 1 ,..., m k thatmeasurethequantities. Forthesecond,observethataneasywaytoconstructadimensionallyhomogeneousexpressionisbytakingaproductofdimensionlessquantitiesor byaddingsuchdimensionlessterms.Buckingham'sTheoremstatesthatany completerelationshipamongquantitieswithdimensionalformulascanbealgebraicallymanipulatedintoaformwherethereissomefunction f suchthat f 1 ;:::; n =0 foracompleteset f 1 ;:::; n g ofdimensionlessproducts.Therstexample belowdescribeswhatmakesasetofdimensionlessproducts`complete'.We usuallywanttoexpressoneofthequantities, m 1 forinstance,intermsofthe others,andforthatwewillassumethattheaboveequalitycanberewritten m 1 = m )]TJ/F10 6.9738 Tf 6.226 0 Td [(p 2 2 m )]TJ/F10 6.9738 Tf 6.227 0 Td [(p k k ^ f 2 ;:::; n where 1 = m 1 m p 2 2 m p k k isdimensionlessandtheproducts 2 ,..., n don't involve m 1 aswith f ,here ^ f isjustsomefunction,thistimeof n )]TJ/F8 9.9626 Tf 8.214 0 Td [(1arguments. Thus,tododimensionalanalysisweshouldndwhichdimensionlessproducts arepossible. Forexample,consideragaintheformulaforapendulum'speriod.

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154 ChapterTwo.VectorSpaces quantity dimensional formula period p L 0 M 0 T 1 lengthofstring ` L 1 M 0 T 0 massofbob m L 0 M 1 T 0 accelerationduetogravity g L 1 M 0 T )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 arcofswing L 0 M 0 T 0 Bytherstfactcitedabove,weexpecttheformulatohavepossiblysums oftermsoftheform p p 1 ` p 2 m p 3 g p 4 p 5 .Tousethesecondfact,tondwhich combinationsofthepowers p 1 ,..., p 5 yielddimensionlessproducts,consider thisequation. L 0 M 0 T 1 p 1 L 1 M 0 T 0 p 2 L 0 M 1 T 0 p 3 L 1 M 0 T )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 p 4 L 0 M 0 T 0 p 5 = L 0 M 0 T 0 Itgivesthreeconditionsonthepowers. p 2 + p 4 =0 p 3 =0 p 1 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 p 4 =0 Notethat p 3 is0andsothemassofthebobdoesnotaecttheperiod.Gaussian reductionandparametrizationofthatsystemgivesthis f 0 B B B B @ p 1 p 2 p 3 p 4 p 5 1 C C C C A = 0 B B B B @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 0 1 = 2 0 1 C C C C A p 1 + 0 B B B B @ 0 0 0 0 1 1 C C C C A p 5 p 1 ;p 5 2 R g we'vetaken p 1 asoneoftheparametersinordertoexpresstheperiodinterms oftheotherquantities. Hereisthelinearalgebra.Thesetofdimensionlessproductscontainsall terms p p 1 ` p 2 m p 3 a p 4 p 5 subjecttotheconditionsabove.Thissetformsavector spaceunderthe`+'operationofmultiplyingtwosuchproductsandthe` operationofraisingsuchaproducttothepowerofthescalarseeExercise5. Theterm`completesetofdimensionlessproducts'inBuckingham'sTheorem meansabasisforthisvectorspace. Wecangetabasisbyrsttaking p 1 =1, p 5 =0andthen p 1 =0, p 5 =1.The associateddimensionlessproductsare 1 = p` )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 2 g 1 = 2 and 2 = .Because theset f 1 ; 2 g iscomplete,Buckingham'sTheoremsaysthat p = ` 1 = 2 g )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = 2 ^ f = p `=g ^ f where ^ f isafunctionthatwecannotdeterminefromthisanalysisarstyear physicstextwillshowbyothermeansthatforsmallanglesitisapproximately theconstantfunction ^ f =2 .

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Topic:DimensionalAnalysis 155 Thus,analysisoftherelationshipsthatarepossiblebetweenthequantities withthegivendimensionalformulashasproducedafairamountofinformation:apendulum'sperioddoesnotdependonthemassofthebob,anditrises withthesquarerootofthelengthofthestring. Forthenextexamplewetrytodeterminetheperiodofrevolutionoftwo bodiesinspaceorbitingeachotherundermutualgravitationalattraction.An experiencedinvestigatorcouldexpectthatthesearetherelevantquantities. quantity dimensional formula period p L 0 M 0 T 1 meanseparation r L 1 M 0 T 0 rstmass m 1 L 0 M 1 T 0 secondmass m 2 L 0 M 1 T 0 grav.constant G L 3 M )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 T )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 Togetthecompletesetofdimensionlessproductsweconsidertheequation L 0 M 0 T 1 p 1 L 1 M 0 T 0 p 2 L 0 M 1 T 0 p 3 L 0 M 1 T 0 p 4 L 3 M )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 T )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 p 5 = L 0 M 0 T 0 whichresultsinasystem p 2 +3 p 5 =0 p 3 + p 4 )]TJ/F11 9.9626 Tf 14.944 0 Td [(p 5 =0 p 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 p 5 =0 withthissolution. f 0 B B B B @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 = 2 1 = 2 0 1 = 2 1 C C C C A p 1 + 0 B B B B @ 0 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 0 1 C C C C A p 4 p 1 ;p 4 2 R g Asearlier,thelinearalgebrahereisthatthesetofdimensionlessproductsofthesequantitiesformsavectorspace,andwewanttoproduceabasis forthatspace,a`complete'setofdimensionlessproducts.Onesuchset,gottenfromsetting p 1 =1and p 4 =0,andalsosetting p 1 =0and p 4 =1 is f 1 = pr )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 = 2 m 1 = 2 1 G 1 = 2 ; 2 = m )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1 m 2 g .Withthat,Buckingham'sTheorem saysthatanycompleterelationshipamongthesequantitiesisstateablethis form. p = r 3 = 2 m )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 2 1 G )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 2 ^ f m )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1 m 2 = r 3 = 2 p Gm 1 ^ f m 2 =m 1 Remark. Animportantapplicationofthepriorformulaiswhen m 1 isthe massofthesunand m 2 isthemassofaplanet.Because m 1 isverymuchgreater than m 2 ,theargumentto ^ f isapproximately0,andwecanwonderwhether thispartoftheformularemainsapproximatelyconstantas m 2 varies.Oneway toseethatitdoesisthis.Thesunissomuchlargerthantheplanetthatthe

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156 ChapterTwo.VectorSpaces mutualrotationisapproximatelyaboutthesun'scenter.Ifwevarytheplanet's mass m 2 byafactorof x e.g.,Venus'smassis x =0 : 815timesEarth'smass, thentheforceofattractionismultipliedby x ,and x timestheforceactingon x timesthemassgives,since F = ma ,thesameacceleration,aboutthesame centerapproximately.Hence,theorbitwillbethesameandsoitsperiod willbethesame,andthustherightsideoftheaboveequationalsoremains unchangedapproximately.Therefore, ^ f m 2 =m 1 isapproximatelyconstant as m 2 varies.ThisisKepler'sThirdLaw:thesquareoftheperiodofaplanet isproportionaltothecubeofthemeanradiusofitsorbitaboutthesun. Thenalexamplewasoneoftherstexplicitapplicationsofdimensional analysis.LordRaleighconsideredthespeedofawaveindeepwaterandsuggestedtheseastherelevantquantities. quantity dimensional formula velocityofthewave v L 1 M 0 T )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 densityofthewater d L )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 M 1 T 0 accelerationduetogravity g L 1 M 0 T )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 wavelength L 1 M 0 T 0 Theequation L 1 M 0 T )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 p 1 L )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 M 1 T 0 p 2 L 1 M 0 T )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 p 3 L 1 M 0 T 0 p 4 = L 0 M 0 T 0 givesthissystem p 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 p 2 + p 3 + p 4 =0 p 2 =0 )]TJ/F11 9.9626 Tf 7.749 0 Td [(p 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 p 3 =0 withthissolutionspace f 0 B B @ 1 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 1 C C A p 1 p 1 2 R g asinthependulumexample,oneofthequantities d turnsoutnottobeinvolved intherelationship.Thereisonedimensionlessproduct, 1 = vg )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 2 ,and so v is p g timesaconstant ^ f isconstantsinceitisafunctionofnoarguments. Asthethreeexamplesaboveshow,dimensionalanalysiscanbringusfar towardexpressingtherelationshipamongthequantities.Forfurtherreading, theclassicreferenceis[Bridgman]|thisbriefbookisdelightful.Anothersource is[Giordano,Wells,Wilde].Adescriptionofdimensionalanalysis'splacein modelingisin[Giordano,Jaye,Weir]. Exercises 1 Consideraprojectile,launchedwithinitialvelocity v 0 ,atanangle .Aninvestigationofthismotionmightstartwiththeguessthatthesearetherelevant

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Topic:DimensionalAnalysis 157 quantities.[deMestre] quantity dimensional formula horizontalposition x L 1 M 0 T 0 verticalposition y L 1 M 0 T 0 initialspeed v 0 L 1 M 0 T )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 angleoflaunch L 0 M 0 T 0 accelerationduetogravity g L 1 M 0 T )]TJ/F31 5.9776 Tf 5.756 0 Td [(2 time t L 0 M 0 T 1 a Showthat f gt=v 0 ;gx=v 2 0 ;gy=v 2 0 ; g isacompletesetofdimensionlessproducts. Hint. Thiscanbedonebyndingtheappropriatefreevariablesinthe linearsystemthatarises,butthereisashortcutthatusesthepropertiesofa basis. b Thesetwoequationsofmotionforprojectilesarefamiliar: x = v 0 cos t and y = v 0 sin t )]TJ/F29 8.9664 Tf 9.306 0 Td [( g= 2 t 2 .Manipulateeachtorewriteitasarelationshipamong thedimensionlessproductsoftheprioritem. 2 [Einstein]conjecturedthattheinfraredcharacteristicfrequenciesofasolidmay bedeterminedbythesameforcesbetweenatomsasdeterminethesolid'sordanary elasticbehavior.Therelevantquantitiesare quantity dimensional formula characteristicfrequency L 0 M 0 T )]TJ/F31 5.9776 Tf 5.757 0 Td [(1 compressibility k L 1 M )]TJ/F31 5.9776 Tf 5.757 0 Td [(1 T 2 numberofatomspercubiccm N L )]TJ/F31 5.9776 Tf 5.756 0 Td [(3 M 0 T 0 massofanatom m L 0 M 1 T 0 Showthatthereisonedimensionlessproduct.Concludethat,inanycomplete relationshipamongquantitieswiththesedimensionalformulas, k isaconstant times )]TJ/F31 5.9776 Tf 5.756 0 Td [(2 N )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 = 3 m )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 .Thisconclusionplayedanimportantroleintheearlystudy ofquantumphenomena. 3 Thetorqueproducedbyanenginehasdimensionalformula L 2 M 1 T )]TJ/F31 5.9776 Tf 5.756 0 Td [(2 .Wemay rstguessthatitdependsontheengine'srotationratewithdimensionalformula L 0 M 0 T )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 ,andthevolumeofairdisplacedwithdimensionalformula L 3 M 0 T 0 [Giordano,Wells,Wilde] a Trytondacompletesetofdimensionlessproducts.Whatgoeswrong? b Adjusttheguessbyaddingthedensityoftheairwithdimensionalformula L )]TJ/F31 5.9776 Tf 5.756 0 Td [(3 M 1 T 0 .Nowndacompletesetofdimensionlessproducts. 4 Dominoesfallingmakeawave.Wemayconjecturethatthewavespeed v depends onthethespacing d betweenthedominoes,theheight h ofeachdomino,andthe accelerationduetogravity g .[Tilley] a Findthedimensionalformulaforeachofthefourquantities. b Showthat f 1 = h=d; 2 = dg=v 2 g isacompletesetofdimensionlessproducts. c Showthatif h=d isxedthenthepropagationspeedisproportionaltothe squarerootof d 5 Provethatthedimensionlessproductsformavectorspaceunderthe ~ +operation ofmultiplyingtwosuchproductsandthe ~ operationofraisingsuchtheproduct tothepowerofthescalar.Thevectorarrowsareaprecautionagainstconfusion. Thatis,provethat,foranyparticularhomogeneoussystem,thissetofproducts

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158 ChapterTwo.VectorSpaces ofpowersof m 1 ,..., m k f m p 1 1 :::m p k k p 1 ,..., p k satisfythesystem g isavectorspaceunder: m p 1 1 :::m p k k ~ + m q 1 1 :::m q k k = m p 1 + q 1 1 :::m p k + q k k and r ~ m p 1 1 :::m p k k = m rp 1 1 :::m rp k k assumethatallvariablesrepresentrealnumbers. 6 Theadviceaboutapplesandorangesisnotright.Considerthefamiliarequations foracircle C =2 r and A = r 2 a Checkthat C and A havedierentdimensionalformulas. b Produceanequationthatisnotdimensionallyhomogeneousi.e.,itadds applesandorangesbutisnonethelesstrueofanycircle. c Theprioritemasksforanequationthatiscompletebutnotdimensionally homogeneous.Produceanequationthatisdimensionallyhomogeneousbutnot complete. Justbecausetheoldsayingisn'tstrictlyright,doesn'tkeepitfrombeingauseful strategy.Dimensionalhomogeneityisoftenusedasacheckontheplausibility ofequationsusedinmodels.Foranargumentthatanycompleteequationcan easilybemadedimensionallyhomogeneous,see[Bridgman],ChapterI,especially page15.

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ChapterThree MapsBetweenSpaces IIsomorphisms Intheexamplesfollowingthedenitionofavectorspacewedevelopedthe intuitionthatsomespacesarethesame"asothers.Forinstance,thespace oftwo-tallcolumnvectorsandthespaceoftwo-widerowvectorsarenotequal becausetheirelements|columnvectorsandrowvectors|arenotequal,but wehavetheideathatthesespacesdieronlyinhowtheirelementsappear.We willnowmakethisideaprecise. Thissectionillustratesacommonaspectofamathematicalinvestigation. Withthehelpofsomeexamples,we'vegottenanidea.Wewillnextgiveaformal denition,andthenwewillproducesomeresultsbackingourcontentionthat thedenitioncapturestheidea.We'veseenthishappenalready,forinstance,in therstsectionoftheVectorSpacechapter.There,thestudyoflinearsystems ledustoconsidercollectionsclosedunderlinearcombinations.Wedenedsuch acollectionasavectorspace,andwefolloweditwithsomesupportingresults. Ofcourse,thatdenitionwasn'tanendpoint,insteaditledtonewinsights suchastheideaofabasis.Heretoo,afterproducingadenition,andsupporting it,wewillgettwosurprisespleasantones.First,wewillndthatthedenition appliestosomeunforeseen,andinteresting,cases.Second,thestudyofthe denitionwillleadtonewideas.Inthisway,ourinvestigationwillbuilda momentum. I.1DefinitionandExamples Westartwithtwoexamplesthatsuggesttherightdenition. 1.1Example Considertheexamplementionedabove,thespaceoftwo-wide rowvectorsandthespaceoftwo-tallcolumnvectors.Theyarethesame"in thatifweassociatethevectorsthathavethesamecomponents,e.g., )]TJ/F8 9.9626 Tf 4.566 -7.97 Td [(12 1 2 159

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160 ChapterThree.MapsBetweenSpaces thenthiscorrespondencepreservestheoperations,forinstancethisaddition )]TJ/F8 9.9626 Tf 4.566 -7.97 Td [(12 + )]TJ/F8 9.9626 Tf 4.567 -7.97 Td [(34 = )]TJ/F8 9.9626 Tf 4.566 -7.97 Td [(46 1 2 + 3 4 = 4 6 andthisscalarmultiplication. 5 )]TJ/F8 9.9626 Tf 4.567 -7.97 Td [(12 = )]TJ/F8 9.9626 Tf 4.566 -7.97 Td [(510 5 1 2 = 5 10 Moregenerallystated,underthecorrespondence )]TJ/F11 9.9626 Tf 4.566 -7.97 Td [(a 0 a 1 a 0 a 1 bothoperationsarepreserved: )]TJ/F11 9.9626 Tf 4.566 -7.97 Td [(a 0 a 1 + )]TJ/F11 9.9626 Tf 4.566 -7.97 Td [(b 0 b 1 = )]TJ/F11 9.9626 Tf 4.566 -7.97 Td [(a 0 + b 0 a 1 + b 1 a 0 a 1 + b 0 b 1 = a 0 + b 0 a 1 + b 1 and r )]TJ/F11 9.9626 Tf 4.566 -7.97 Td [(a 0 a 1 = )]TJ/F11 9.9626 Tf 4.566 -7.97 Td [(ra 0 ra 1 r a 0 a 1 = ra 0 ra 1 allofthevariablesarerealnumbers. 1.2Example Anothertwospaceswecanthinkofasthesame"are P 2 ,the spaceofquadraticpolynomials,and R 3 .Anaturalcorrespondenceisthis. a 0 + a 1 x + a 2 x 2 0 @ a 0 a 1 a 2 1 A e.g.,1+2 x +3 x 2 0 @ 1 2 3 1 A Thestructureispreserved:correspondingelementsaddinacorrespondingway a 0 + a 1 x + a 2 x 2 + b 0 + b 1 x + b 2 x 2 a 0 + b 0 + a 1 + b 1 x + a 2 + b 2 x 2 0 @ a 0 a 1 a 2 1 A + 0 @ b 0 b 1 b 2 1 A = 0 @ a 0 + b 0 a 1 + b 1 a 2 + b 2 1 A andscalarmultiplicationcorrespondsalso. r a 0 + a 1 x + a 2 x 2 = ra 0 + ra 1 x + ra 2 x 2 r 0 @ a 0 a 1 a 2 1 A = 0 @ ra 0 ra 1 ra 2 1 A

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SectionI.Isomorphisms 161 1.3Denition An isomorphism betweentwovectorspaces V and W isa map f : V W that isacorrespondence: f isone-to-oneandonto; preservesstructure: if ~v 1 ;~v 2 2 V then f ~v 1 + ~v 2 = f ~v 1 + f ~v 2 andif ~v 2 V and r 2 R then f r~v = rf ~v wewrite V = W ,read V isisomorphicto W ",whensuchamapexists. Morphism"meansmap,soisomorphism"meansamapexpressingsameness. 1.4Example Thevectorspace G = f c 1 cos + c 2 sin c 1 ;c 2 2 R g offunctionsof isisomorphictothevectorspace R 2 underthismap. c 1 cos + c 2 sin f 7)167(! c 1 c 2 Wewillcheckthisbygoingthroughtheconditionsinthedenition. Wewillrstverifycondition,thatthemapisacorrespondencebetween thesetsunderlyingthespaces. Toestablishthat f isone-to-one,wemustprovethat f ~a = f ~ b onlywhen ~a = ~ b .If f a 1 cos + a 2 sin = f b 1 cos + b 2 sin then,bythedenitionof f a 1 a 2 = b 1 b 2 fromwhichwecanconcludethat a 1 = b 1 and a 2 = b 2 becausecolumnvectorsare equalonlywhentheyhaveequalcomponents.We'veprovedthat f ~a = f ~ b impliesthat ~a = ~ b ,whichshowsthat f isone-to-one. Tocheckthat f isontowemustcheckthatanymemberofthecodomain R 2 istheimageofsomememberofthedomain G .Butthat'sclear|any x y 2 R 2 istheimageunder f of x cos + y sin 2 G Nextwewillverifycondition,that f preservesstructure. Moreinformationonone-to-oneandontomapsisintheappendix.

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162 ChapterThree.MapsBetweenSpaces Thiscomputationshowsthat f preservesaddition. f )]TJ/F8 9.9626 Tf 6.227 -8.07 Td [( a 1 cos + a 2 sin + b 1 cos + b 2 sin = f )]TJ/F8 9.9626 Tf 6.226 -8.07 Td [( a 1 + b 1 cos + a 2 + b 2 sin = a 1 + b 1 a 2 + b 2 = a 1 a 2 + b 1 b 2 = f a 1 cos + a 2 sin + f b 1 cos + b 2 sin Asimilarcomputationshowsthat f preservesscalarmultiplication. f )]TJ/F11 9.9626 Tf 6.227 -8.07 Td [(r a 1 cos + a 2 sin = f ra 1 cos + ra 2 sin = ra 1 ra 2 = r a 1 a 2 = r f a 1 cos + a 2 sin Withthat,conditionsandareveried,soweknowthat f isan isomorphismandwecansaythatthespacesareisomorphic G = R 2 1.5Example Let V bethespace f c 1 x + c 2 y + c 3 z c 1 ;c 2 ;c 3 2 R g oflinear combinationsofthreevariables x y ,and z ,underthenaturaladditionand scalarmultiplicationoperations.Then V isisomorphicto P 2 ,thespaceof quadraticpolynomials. Toshowthiswewillproduceanisomorphismmap.Thereismorethanone possibility;forinstance,herearefour. c 1 x + c 2 y + c 3 z f 1 7)167(! c 1 + c 2 x + c 3 x 2 f 2 7)167(! c 2 + c 3 x + c 1 x 2 f 3 7)167(!)]TJ/F11 9.9626 Tf 35.755 0 Td [(c 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(c 2 x )]TJ/F11 9.9626 Tf 9.963 0 Td [(c 3 x 2 f 4 7)167(! c 1 + c 1 + c 2 x + c 1 + c 3 x 2 Therstmapisthemorenaturalcorrespondenceinthatitjustcarriesthe coecientsover.However,belowweshallverifythatthesecondoneisanisomorphism,tounderlinethatthereareisomorphismsotherthanjusttheobvious oneshowingthat f 1 isanisomorphismisExercise12. Toshowthat f 2 isone-to-one,wewillprovethatif f 2 c 1 x + c 2 y + c 3 z = f 2 d 1 x + d 2 y + d 3 z then c 1 x + c 2 y + c 3 z = d 1 x + d 2 y + d 3 z .Theassumption that f 2 c 1 x + c 2 y + c 3 z = f 2 d 1 x + d 2 y + d 3 z gives,bythedenitionof f 2 ,that c 2 + c 3 x + c 1 x 2 = d 2 + d 3 x + d 1 x 2 .Equalpolynomialshaveequalcoecients,so c 2 = d 2 c 3 = d 3 ,and c 1 = d 1 .Thus f 2 c 1 x + c 2 y + c 3 z = f 2 d 1 x + d 2 y + d 3 z impliesthat c 1 x + c 2 y + c 3 z = d 1 x + d 2 y + d 3 z andtherefore f 2 isone-to-one.

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SectionI.Isomorphisms 163 Themap f 2 isontobecauseanymember a + bx + cx 2 ofthecodomainisthe imageofsomememberofthedomain,namelyitistheimageof cx + ay + bz Forinstance,2+3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x 2 is f 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 x +2 y +3 z Thecomputationsforstructurepreservationarelikethoseinthepriorexample.Thismappreservesaddition f 2 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [( c 1 x + c 2 y + c 3 z + d 1 x + d 2 y + d 3 z = f 2 )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [( c 1 + d 1 x + c 2 + d 2 y + c 3 + d 3 z = c 2 + d 2 + c 3 + d 3 x + c 1 + d 1 x 2 = c 2 + c 3 x + c 1 x 2 + d 2 + d 3 x + d 1 x 2 = f 2 c 1 x + c 2 y + c 3 z + f 2 d 1 x + d 2 y + d 3 z andscalarmultiplication. f 2 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(r c 1 x + c 2 y + c 3 z = f 2 rc 1 x + rc 2 y + rc 3 z = rc 2 + rc 3 x + rc 1 x 2 = r c 2 + c 3 x + c 1 x 2 = r f 2 c 1 x + c 2 y + c 3 z Thus f 2 isanisomorphismandwewrite V = P 2 Wearesometimesinterestedinanisomorphismofaspacewithitself,called an automorphism .Anidentitymapisanautomorphism.Thenexttwoexamples showthatthereareothers. 1.6Example A dilation map d s : R 2 R 2 thatmultipliesallvectorsbya nonzeroscalar s isanautomorphismof R 2 ~u ~v d 1 : 5 ~u d 1 : 5 ~v d 1 : 5 )167(! A rotation or turningmap t : R 2 R 2 thatrotatesallvectorsthroughanangle isanautomorphism. ~u t = 6 ~u t = 6 )167(! Athirdtypeofautomorphismof R 2 isamap f ` : R 2 R 2 that ips or reects allvectorsoveraline ` throughtheorigin.

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164 ChapterThree.MapsBetweenSpaces ~u f ` ~u f ` )167(! SeeExercise29. 1.7Example Considerthespace P 5 ofpolynomialsofdegree5orlessandthe map f thatsendsapolynomial p x to p x )]TJ/F8 9.9626 Tf 10.025 0 Td [(1.Forinstance,underthismap x 2 7! x )]TJ/F8 9.9626 Tf 8.627 0 Td [(1 2 = x 2 )]TJ/F8 9.9626 Tf 8.627 0 Td [(2 x +1and x 3 +2 x 7! x )]TJ/F8 9.9626 Tf 8.626 0 Td [(1 3 +2 x )]TJ/F8 9.9626 Tf 8.626 0 Td [(1= x 3 )]TJ/F8 9.9626 Tf 8.627 0 Td [(3 x 2 +5 x )]TJ/F8 9.9626 Tf 8.627 0 Td [(3. Thismapisanautomorphismofthisspace;thecheckisExercise21. Thisisomorphismof P 5 withitselfdoesmorethanjusttellusthatthespace isthesame"asitself.Itgivesussomeinsightintothespace'sstructure.For instance,belowisshownafamilyofparabolas,graphsofmembersof P 5 .Each hasavertexat y = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1,andtheleft-mostonehaszeroesat )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 : 25and )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 : 75, thenextonehaszeroesat )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 : 25and )]TJ/F8 9.9626 Tf 7.748 0 Td [(0 : 75,etc. p 0 p 1 Geometrically,thesubstitutionof x )]TJ/F8 9.9626 Tf 9.899 0 Td [(1for x inanyfunction'sargumentshifts itsgraphtotherightbyone.Thus, f p 0 = p 1 and f 'sactionistoshiftallof theparabolastotherightbyone.Noticethatthepicturebefore f isappliedis thesameasthepictureafter f isapplied,becausewhileeachparabolamovesto theright,anotheronecomesinfromthelefttotakeitsplace.Thisalsoholds trueforcubics,etc.Sotheautomorphism f givesustheinsightthat P 5 hasa certainhorizontal-homogeneity;thisspacelooksthesamenear x =1asnear x =0. Asdescribedinthepreambletothissection,wewillnextproducesome resultssupportingthecontentionthatthedenitionofisomorphismabovecapturesourintuitionofvectorspacesbeingthesame. Ofcoursethedenitionitselfispersuasive:avectorspaceconsistsoftwo components,asetandsomestructure,andthedenitionsimplyrequiresthat thesetscorrespondandthatthestructurescorrespondalso.Alsopersuasiveare theexamplesabove.Inparticular,Example1.1,whichgivesanisomorphism betweenthespaceoftwo-widerowvectorsandthespaceoftwo-tallcolumn vectors,dramatizesourintuitionthatisomorphicspacesarethesameinall relevantrespects.Sometimespeoplesay,where V = W ,that W isjust V paintedgreen"|anydierencesaremerelycosmetic. Furthersupportforthedenition,incaseitisneeded,isprovidedbythe followingresultsthat,takentogether,suggestthatallthethingsofinterestina

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SectionI.Isomorphisms 165 vectorspacecorrespondunderanisomorphism.Sincewestudiedvectorspaces tostudylinearcombinations,ofinterest"meanspertainingtolinearcombinations".Notofinterestisthewaythatthevectorsarepresentedtypographically ortheircolor!. Asanexample,althoughthedenitionofisomorphismdoesn'texplicitlysay thatthezerovectorsmustcorrespond,itisaconsequenceofthatdenition. 1.8Lemma Anisomorphismmapsazerovectortoazerovector. Proof Where f : V W isanisomorphism,xany ~v 2 V .Then f ~ 0 V = f ~v =0 f ~v = ~ 0 W QED Thedenitionofisomorphismrequiresthatsumsoftwovectorscorrespond andthatsodoscalarmultiples.Wecanextendthattosaythatalllinear combinationscorrespond. 1.9Lemma Foranymap f : V W betweenvectorspacesthesestatements areequivalent. f preservesstructure f ~v 1 + ~v 2 = f ~v 1 + f ~v 2 and f c~v = cf ~v f preserveslinearcombinationsoftwovectors f c 1 ~v 1 + c 2 ~v 2 = c 1 f ~v 1 + c 2 f ~v 2 f preserveslinearcombinationsofanynitenumberofvectors f c 1 ~v 1 + + c n ~v n = c 1 f ~v 1 + + c n f ~v n Proof Sincetheimplications= and= areclear,weneed onlyshowthat= .Assumestatement.Wewillprovestatement byinductiononthenumberofsummands n Theone-summandbasecase,that f c~v 1 = cf ~v 1 ,iscoveredbytheassumptionofstatement. Fortheinductivestepassumethatstatementholdswheneverthereare k orfewersummands,thatis,whenever n =1,or n =2,...,or n = k .Consider the k +1-summandcase.Thersthalfofgives f c 1 ~v 1 + + c k ~v k + c k +1 ~v k +1 = f c 1 ~v 1 + + c k ~v k + f c k +1 ~v k +1 bybreakingthesumalongthenal`+'.Thentheinductivehypothesisletsus breakupthe k -termsum. = f c 1 ~v 1 + + f c k ~v k + f c k +1 ~v k +1 Finally,thesecondhalfofstatementgives = c 1 f ~v 1 + + c k f ~v k + c k +1 f ~v k +1 whenapplied k +1times. QED

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166 ChapterThree.MapsBetweenSpaces Inadditiontoaddingtotheintuitionthatthedenitionofisomorphismdoes indeedpreservethethingsofinterestinavectorspace,thatlemma'sseconditem isanespeciallyhandywayofcheckingthatamappreservesstructure. Weclosewithasummary.Thematerialinthissectionaugmentsthechapter onVectorSpaces.There,aftergivingthedenitionofavectorspace,weinformallylookedatwhatdierentthingscanhappen.Here,wedenedtherelation ` = 'betweenvectorspacesandwehavearguedthatitistherightwaytosplitthe collectionofvectorspacesintocasesbecauseitpreservesthefeaturesofinterest inavectorspace|inparticular,itpreserveslinearcombinations.Thatis,we havenowsaidpreciselywhatwemeanby`thesame',andby`dierent',andso wehavepreciselyclassiedthevectorspaces. Exercises X 1.10 Verify,usingExample1.4asamodel,thatthetwocorrespondencesgiven beforethedenitionareisomorphisms. a Example1.1 b Example1.2 X 1.11 Forthemap f : P 1 R 2 givenby a + bx f 7)171(! a )]TJ/F32 8.9664 Tf 9.215 0 Td [(b b Findtheimageofeachoftheseelementsofthedomain. a 3 )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 x b 2+2 x c x Showthatthismapisanisomorphism. 1.12 Showthatthenaturalmap f 1 fromExample1.5isanisomorphism. X 1.13 Decidewhethereachmapisanisomorphismifitisanisomorphismthen proveitandifitisn'tthenstateaconditionthatitfailstosatisfy. a f : M 2 2 R givenby ab cd 7! ad )]TJ/F32 8.9664 Tf 9.216 0 Td [(bc b f : M 2 2 R 4 givenby ab cd 7! 0 B @ a + b + c + d a + b + c a + b a 1 C A c f : M 2 2 !P 3 givenby ab cd 7! c + d + c x + b + a x 2 + ax 3 d f : M 2 2 !P 3 givenby ab cd 7! c + d + c x + b + a +1 x 2 + ax 3 1.14 Showthatthemap f : R 1 R 1 givenby f x = x 3 isone-to-oneandonto. Isitanisomorphism? X 1.15 RefertoExample1.1.Producetwomoreisomorphismsofcourse,thatthey satisfytheconditionsinthedenitionofisomorphismmustbeveried. 1.16 RefertoExample1.2.Producetwomoreisomorphismsandverifythatthey satisfytheconditions.

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SectionI.Isomorphisms 167 X 1.17 Showthat,although R 2 isnotitselfasubspaceof R 3 ,itisisomorphictothe xy -planesubspaceof R 3 1.18 Findtwoisomorphismsbetween R 16 and M 4 4 X 1.19 Forwhat k is M m n isomorphicto R k ? 1.20 Forwhat k is P k isomorphicto R n ? 1.21 ProvethatthemapinExample1.7,from P 5 to P 5 givenby p x 7! p x )]TJ/F29 8.9664 Tf 8.875 0 Td [(1, isavectorspaceisomorphism. 1.22 Why,inLemma1.8,musttherebea ~v 2 V ?Thatis,whymust V be nonempty? 1.23 Areanytwotrivialspacesisomorphic? 1.24 IntheproofofLemma1.9,whataboutthezero-summandscasethatis,if n iszero? 1.25 Showthatanyisomorphism f : P 0 R 1 hastheform a 7! ka forsomenonzero realnumber k X 1.26 Theseprovethatisomorphismisanequivalencerelation. a Showthattheidentitymapid: V V isanisomorphism.Thus,anyvector spaceisisomorphictoitself. b Showthatif f : V W isanisomorphismthensoisitsinverse f )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 : W V Thus,if V isisomorphicto W thenalso W isisomorphicto V c Showthatacompositionofisomorphismsisanisomorphism:if f : V W is anisomorphismand g : W U isanisomorphismthensoalsois g f : V U Thus,if V isisomorphicto W and W isisomorphicto U ,thenalso V isisomorphicto U 1.27 Supposethat f : V W preservesstructure.Showthat f isone-to-oneifand onlyiftheuniquememberof V mappedby f to ~ 0 W is ~ 0 V 1.28 Supposethat f : V W isanisomorphism.Provethattheset f ~v 1 ;:::;~v k g V islinearlydependentifandonlyifthesetofimages f f ~v 1 ;:::;f ~v k g W is linearlydependent. X 1.29 ShowthateachtypeofmapfromExample1.6isanautomorphism. a Dilation d s byanonzeroscalar s b Rotation t throughanangle c Reection f ` overalinethroughtheorigin. Hint. Forthesecondandthirditems,polarcoordinatesareuseful. 1.30 Produceanautomorphismof P 2 otherthantheidentitymap,andotherthan ashiftmap p x 7! p x )]TJ/F32 8.9664 Tf 9.215 0 Td [(k 1.31a Showthatafunction f : R 1 R 1 isanautomorphismifandonlyifit hastheform x 7! kx forsome k 6 =0. b Let f beanautomorphismof R 1 suchthat f =7.Find f )]TJ/F29 8.9664 Tf 7.168 0 Td [(2. c Showthatafunction f : R 2 R 2 isanautomorphismifandonlyifithas theform x y 7! ax + by cx + dy forsome a;b;c;d 2 R with ad )]TJ/F32 8.9664 Tf 9.692 0 Td [(bc 6 =0. Hint. Exercisesinpriorsubsections haveshownthat b d isnotamultipleof a c ifandonlyif ad )]TJ/F32 8.9664 Tf 9.215 0 Td [(bc 6 =0.

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168 ChapterThree.MapsBetweenSpaces d Let f beanautomorphismof R 2 with f 1 3 = 2 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 and f 1 4 = 0 1 : Find f 0 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 : 1.32 RefertoLemma1.8andLemma1.9.Findtwomorethingspreservedby isomorphism. 1.33 Weshowthatisomorphismscanbetailoredtotinthat,sometimes,given vectorsinthedomainandintherangewecanproduceanisomorphismassociating thosevectors. a Let B = h ~ 1 ; ~ 2 ; ~ 3 i beabasisfor P 2 sothatany ~p 2P 2 hasaunique representationas ~p = c 1 ~ 1 + c 2 ~ 2 + c 3 ~ 3 ,whichwedenoteinthisway. Rep B ~p = c 1 c 2 c 3 ShowthattheRep B operationisafunctionfrom P 2 to R 3 thisentailsshowing thatwitheverydomainvector ~v 2P 2 thereisanassociatedimagevectorin R 3 andfurther,thatwitheverydomainvector ~v 2P 2 thereisatmostoneassociated imagevector. b ShowthatthisRep B functionisone-to-oneandonto. c Showthatitpreservesstructure. d Produceanisomorphismfrom P 2 to R 3 thattsthesespecications. x + x 2 7! 1 0 0 and1 )]TJ/F32 8.9664 Tf 9.216 0 Td [(x 7! 0 1 0 1.34 Provethataspaceis n -dimensionalifandonlyifitisisomorphicto R n Hint. Fixabasis B forthespaceandconsiderthemapsendingavectoroverto itsrepresentationwithrespectto B 1.35 RequiresthesubsectiononCombiningSubspaces,whichisoptional. Let U and W bevectorspaces.Deneanewvectorspace,consistingoftheset U W = f ~u;~w ~u 2 U and ~w 2 W g alongwiththeseoperations. ~u 1 ;~w 1 + ~u 2 ;~w 2 = ~u 1 + ~u 2 ;~w 1 + ~w 2 and r ~u;~w = r~u;r~w Thisisavectorspace,the externaldirectsum of U and W a Checkthatitisavectorspace. b Findabasisfor,andthedimensionof,theexternaldirectsum P 2 R 2 c Whatistherelationshipamongdim U ,dim W ,anddim U W ? d Supposethat U and W aresubspacesofavectorspace V suchthat V = U W inthiscasewesaythat V isthe internaldirectsum of U and W .Show thatthemap f : U W V givenby ~u;~w f 7)171(! ~u + ~w isanisomorphism.Thusiftheinternaldirectsumisdenedthentheinternal andexternaldirectsumsareisomorphic.

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SectionI.Isomorphisms 169 I.2DimensionCharacterizesIsomorphism Inthepriorsubsection,afterstatingthedenitionofanisomorphism,we gavesomeresultssupportingtheintuitionthatsuchamapdescribesspacesas thesame".Herewewillformalizethisintuition.Whiletwospacesthatare isomorphicarenotequal,wethinkofthemasalmostequal|asequivalent. Inthissubsectionweshallshowthattherelationship`isisomorphicto'isan equivalencerelation. 2.1Theorem Isomorphismisanequivalencerelationbetweenvectorspaces. Proof Wemustprovethatthisrelationhasthethreepropertiesofbeingsymmetric,reexive,andtransitive.Foreachofthethreewewilluseitem ofLemma1.9andshowthatthemappreservesstructurebyshowingthatit preserveslinearcombinationsoftwomembersofthedomain. Tocheckreexivity,thatanyspaceisisomorphictoitself,considertheidentitymap.Itisclearlyone-to-oneandonto.Thecalculationshowingthatit preserveslinearcombinationsiseasy. id c 1 ~v 1 + c 2 ~v 2 = c 1 ~v 1 + c 2 ~v 2 = c 1 id ~v 1 + c 2 id ~v 2 Tochecksymmetry,thatif V isisomorphicto W viasomemap f : V W thenthereisanisomorphismgoingtheotherway,considertheinversemap f )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 : W V .Asstatedintheappendix,suchaninversefunctionexistsandit isalsoacorrespondence.Thuswehavereducedthesymmetryissuetochecking that,because f preserveslinearcombinations,soalsodoes f )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 .Assumethat ~w 1 = f ~v 1 and ~w 2 = f ~v 2 ,i.e.,that f )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ~w 1 = ~v 1 and f )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ~w 2 = ~v 2 f )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 c 1 ~w 1 + c 2 ~w 2 = f )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F11 9.9626 Tf 6.227 -8.07 Td [(c 1 f ~v 1 + c 2 f ~v 2 = f )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 f )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(c 1 ~v 1 + c 2 ~v 2 = c 1 ~v 1 + c 2 ~v 2 = c 1 f )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ~w 1 + c 2 f )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ~w 2 Finally,wemustchecktransitivity,thatif V isisomorphicto W viasome map f andif W isisomorphicto U viasomemap g thenalso V isisomorphic to U .Considerthecomposition g f : V U .Theappendixnotesthatthe compositionoftwocorrespondencesisacorrespondence,soweneedonlycheck thatthecompositionpreserveslinearcombinations. g f )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(c 1 ~v 1 + c 2 ~v 2 = g )]TJ/F11 9.9626 Tf 6.226 -8.07 Td [(f c 1 ~v 1 + c 2 ~v 2 = g )]TJ/F11 9.9626 Tf 6.226 -8.07 Td [(c 1 f ~v 1 + c 2 f ~v 2 = c 1 g )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(f ~v 1 + c 2 g f ~v 2 = c 1 g f ~v 1 + c 2 g f ~v 2 Thus g f : V U isanisomorphism. QED Moreinformationonequivalencerelationsisintheappendix.

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170 ChapterThree.MapsBetweenSpaces Asaconsequenceofthatresult,weknowthattheuniverseofvectorspaces ispartitionedintoclasses:everyspaceisinoneandonlyoneisomorphismclass. Allnitedimensional vectorspaces: ... V W V = W 2.2Theorem Vectorspacesareisomorphicifandonlyiftheyhavethesame dimension. Thisfollowsfromthenexttwolemmas. 2.3Lemma Ifspacesareisomorphicthentheyhavethesamedimension. Proof Weshallshowthatanisomorphismoftwospacesgivesacorrespondence betweentheirbases.Thatis,where f : V W isanisomorphismandabasis forthedomain V is B = h ~ 1 ;:::; ~ n i ,thentheimageset D = h f ~ 1 ;:::;f ~ n i isabasisforthecodomain W .Theotherhalfofthecorrespondence|that foranybasisof W theinverseimageisabasisfor V |followsonrecallingthat if f isanisomorphismthen f )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 isalsoanisomorphism,andapplyingtheprior sentenceto f )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Toseethat D spans W ,xany ~w 2 W ,notethat f isontoandsothereis a ~v 2 V with ~w = f ~v ,andexpand ~v asacombinationofbasisvectors. ~w = f ~v = f v 1 ~ 1 + + v n ~ n = v 1 f ~ 1 + + v n f ~ n Forlinearindependenceof D ,if ~ 0 W = c 1 f ~ 1 + + c n f ~ n = f c 1 ~ 1 + + c n ~ n then,since f isone-to-oneandsotheonlyvectorsentto ~ 0 W is ~ 0 V ,wehave that ~ 0 V = c 1 ~ 1 + + c n ~ n ,implyingthatallofthe c 'sarezero. QED 2.4Lemma Ifspaceshavethesamedimensionthentheyareisomorphic. Proof Toshowthatanytwospacesofdimension n areisomorphic,wecan simplyshowthatanyoneisisomorphicto R n .Thenwewillhaveshownthat theyareisomorphictoeachother,bythetransitivityofisomorphismwhich wasestablishedinTheorem2.1. Let V be n -dimensional.Fixabasis B = h ~ 1 ;:::; ~ n i forthedomain V Considertherepresentationofthemembersofthatdomainwithrespecttothe basisasafunctionfrom V to R n ~v = v 1 ~ 1 + + v n ~ n Rep B 7)167(! 0 B @ v 1 v n 1 C A

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SectionI.Isomorphisms 171 itiswell-dened sinceevery ~v hasoneandonlyonesuchrepresentation|see Remark2.5below. Thisfunctionisone-to-onebecauseif Rep B u 1 ~ 1 + + u n ~ n =Rep B v 1 ~ 1 + + v n ~ n then 0 B @ u 1 u n 1 C A = 0 B @ v 1 v n 1 C A andso u 1 = v 1 ,..., u n = v n ,andthereforetheoriginalarguments u 1 ~ 1 + + u n ~ n and v 1 ~ 1 + + v n ~ n areequal. Thisfunctionisonto;any n -tallvector ~w = 0 B @ w 1 w n 1 C A istheimageofsome ~v 2 V ,namely ~w =Rep B w 1 ~ 1 + + w n ~ n Finally,thisfunctionpreservesstructure. Rep B r ~u + s ~v =Rep B ru 1 + sv 1 ~ 1 + + ru n + sv n ~ n = 0 B @ ru 1 + sv 1 ru n + sv n 1 C A = r 0 B @ u 1 u n 1 C A + s 0 B @ v 1 v n 1 C A = r Rep B ~u + s Rep B ~v ThustheRep B functionisanisomorphismandthusany n -dimensionalspaceis isomorphictothe n -dimensionalspace R n .Consequently,anytwospaceswith thesamedimensionareisomorphic. QED 2.5Remark Theparentheticalcommentinthatproofabouttheroleplayed bythe`oneandonlyonerepresentation'resultrequiressomeexplanation.We needtoshowthatforaxed B eachvectorinthedomainisassociatedby Rep B withoneandonlyonevectorinthecodomain. Acontrastingexample,whereanassociationdoesn'thavethisproperty,is illuminating.Considerthissubsetof P 2 ,whichisnotabasis. A = f 1+0 x +0 x 2 ; 0+1 x +0 x 2 ; 0+0 x +1 x 2 ; 1+1 x +2 x 2 g Moreinformationonwell-denednessisintheappendix.

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172 ChapterThree.MapsBetweenSpaces Callthosefourpolynomials ~ 1 ,..., ~ 4 .If,mimicingaboveproof,wetryto writethemembersof P 2 as ~p = c 1 ~ 1 + c 2 ~ 2 + c 3 ~ 3 + c 4 ~ 4 ,andassociate ~p with thefour-tallvectorwithcomponents c 1 ,..., c 4 thenthereisaproblem.For, consider ~p x =1+ x + x 2 .Theset A spansthespace P 2 ,sothereisatleast onefour-tallvectorassociatedwith ~p .But A isnotlinearlyindependentandso vectorsdonothaveuniquedecompositions.Inthiscase,both ~p x =1 ~ 1 +1 ~ 2 +1 ~ 3 +0 ~ 4 and ~p x =0 ~ 1 +0 ~ 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 ~ 3 +1 ~ 4 andsothereismorethanonefour-tallvectorassociatedwith ~p 0 B B @ 1 1 1 0 1 C C A and 0 B B @ 0 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 1 C C A Thatis,withinput ~p thisassociationdoesnothaveawell-denedi.e.,single outputvalue. Anymapwhosedenitionappearspossiblyambiguousmustbecheckedto seethatitiswell-dened.ForRep B intheaboveproofthatcheckisExercise18. ThatendstheproofofTheorem2.2.Wesaythattheisomorphismclasses are characterized bydimensionbecausewecandescribeeachclasssimplyby givingthenumberthatisthedimensionofallofthespacesinthatclass. Thissubsection'sresultsgiveusacollectionofrepresentativesoftheisomorphismclasses. 2.6Corollary Anite-dimensionalvectorspaceisisomorphictooneandonly oneofthe R n Theproofsabovepackmanyideasintoasmallspace.Throughtherestof thischapterwe'llconsidertheseideasagain,andllthemout.Foratasteof this,wewillexpandhereontheproofofLemma2.4. 2.7Example Thespace M 2 2 of2 2matricesisisomorphicto R 4 .Withthis basisforthedomain B = h 10 00 ; 01 00 ; 00 10 ; 00 01 i theisomorphismgiveninthelemma,therepresentationmap f 1 =Rep B ,simply carriestheentriesover. ab cd f 1 7)167(! 0 B B @ a b c d 1 C C A Moreinformationonequivalenceclassrepresentativesisintheappendix.

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SectionI.Isomorphisms 173 Onewaytothinkofthemap f 1 is:xthebasis B forthedomainandthebasis E 4 forthecodomain,andassociate ~ 1 with ~e 1 ,and ~ 2 with ~e 2 ,etc.Thenextend thisassociationtoallofthemembersoftwospaces. ab cd = a ~ 1 + b ~ 2 + c ~ 3 + d ~ 4 f 1 7)167(! a~e 1 + b~e 2 + c~e 3 + d~e 4 = 0 B B @ a b c d 1 C C A Wesaythatthemaphasbeen extendedlinearly fromthebasestothespaces. Wecandothesamethingwithdierentbases,forinstance,takingthisbasis forthedomain. A = h 20 00 ; 02 00 ; 00 20 ; 00 02 i Associatingcorrespondingmembersof A and E 4 andextendinglinearly ab cd = a= 2 ~ 1 + b= 2 ~ 2 + c= 2 ~ 3 + d= 2 ~ 4 f 2 7)167(! a= 2 ~e 1 + b= 2 ~e 2 + c= 2 ~e 3 + d= 2 ~e 4 = 0 B B @ a= 2 b= 2 c= 2 d= 2 1 C C A givesrisetoanisomorphismthatisdierentthan f 1 Thepriormaparosebychangingthebasisforthedomain.Wecanalso changethebasisforthecodomain.Startingwith B and D = h 0 B B @ 1 0 0 0 1 C C A ; 0 B B @ 0 1 0 0 1 C C A ; 0 B B @ 0 0 0 1 1 C C A ; 0 B B @ 0 0 1 0 1 C C A i associating ~ 1 with ~ 1 ,etc.,andthenlinearlyextendingthatcorrespondenceto allofthetwospaces ab cd = a ~ 1 + b ~ 2 + c ~ 3 + d ~ 4 f 3 7)167(! a ~ 1 + b ~ 2 + c ~ 3 + d ~ 4 = 0 B B @ a b d c 1 C C A givesstillanotherisomorphism. Sothereisaconnectionbetweenthemapsbetweenspacesandbasesfor thosespaces.Latersectionswillexplorethatconnection. Wewillclosethissectionwithasummary. Recallthatintherstchapterwedenedtwomatricesasrowequivalent iftheycanbederivedfromeachotherbyelementaryrowoperationsthiswas

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174 ChapterThree.MapsBetweenSpaces themeaningofsame-nessthatwasofinterestthere.Weshowedthatisan equivalencerelationandsothecollectionofmatricesispartitionedintoclasses, whereallthematricesthatarerowequivalentfalltogetherintoasingleclass. Then,forinsightintowhichmatricesareineachclass,wegaverepresentatives fortheclasses,thereducedechelonformmatrices. Inthissection,exceptthattheappropriatenotionofsame-nesshereisvector spaceisomorphism,wehavefollowedmuchthesameoutline.Firstwedened isomorphism,sawsomeexamples,andestablishedsomeproperties.Thenwe showedthatitisanequivalencerelation,andnowwehaveasetofclassrepresentatives,therealvectorspaces R 1 R 2 ,etc. Allnitedimensional vectorspaces: ... ? R 2 ? R 0 ? R 3 ? R 1 Onerepresentative perclass Asbefore,thelistofrepresentativeshelpsustounderstandthepartition.Itis simplyaclassicationofspacesbydimension. Inthesecondchapter,withthedenitionofvectorspaces,weseemedto haveopenedupourstudiestomanyexamplesofnewstructuresbesidesthe familiar R n 's.Wenowknowthatisn'tthecase.Anynite-dimensionalvector spaceisactuallythesame"asarealspace.Wearethusconsideringexactly thestructuresthatweneedtoconsider. Therestofthechapterllsouttheworkinthissection.Inparticular, inthenextsectionwewillconsidermapsthatpreservestructure,butarenot necessarilycorrespondences. Exercises X 2.8 Decideifthespacesareisomorphic. a R 2 R 4 b P 5 R 5 c M 2 3 R 6 d P 5 M 2 3 e M 2 k C k X 2.9 ConsidertheisomorphismRep B : P 1 R 2 where B = h 1 ; 1+ x i .Findthe imageofeachoftheseelementsofthedomain. a 3 )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 x ; b 2+2 x ; c x X 2.10 Showthatif m 6 = n then R m 6 = R n X 2.11 Is M m n = M n m ? X 2.12 Areanytwoplanesthroughtheoriginin R 3 isomorphic? 2.13 Findasetofequivalenceclassrepresentativesotherthanthesetof R n 's. 2.14 Trueorfalse:betweenany n -dimensionalspaceand R n thereisexactlyone isomorphism. 2.15 Canavectorspacebeisomorphictooneofitspropersubspaces? X 2.16 Thissubsectionshowsthatforanyisomorphism,theinversemapisalsoanisomorphism.Thissubsectionalsoshowsthatforaxedbasis B ofan n -dimensional vectorspace V ,themapRep B : V R n isanisomorphism.Findtheinverseof thismap. X 2.17 Provethesefactsaboutmatrices. a Therowspaceofamatrixisisomorphictothecolumnspaceofitstranspose.

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SectionI.Isomorphisms 175 b Therowspaceofamatrixisisomorphictoitscolumnspace. 2.18 ShowthatthefunctionfromTheorem2.2iswell-dened. 2.19 IstheproofofTheorem2.2validwhen n =0? 2.20 Foreach,decideifitisasetofisomorphismclassrepresentatives. a f C k k 2 N g b fP k k 2f)]TJ/F29 8.9664 Tf 20.478 0 Td [(1 ; 0 ; 1 ;::: gg c fM m n m;n 2 N g 2.21 Let f beacorrespondencebetweenvectorspaces V and W thatis,amap thatisone-to-oneandonto.Showthatthespaces V and W areisomorphicvia f ifandonlyiftherearebases B V and D W suchthatcorrespondingvectors havethesamecoordinates:Rep B ~v =Rep D f ~v 2.22 ConsidertheisomorphismRep B : P 3 R 4 a Vectorsinarealspaceareorthogonalifandonlyiftheirdotproductiszero. Giveadenitionoforthogonalityforpolynomials. b Thederivativeofamemberof P 3 isin P 3 .Giveadenitionofthederivative ofavectorin R 4 X 2.23 Doeseverycorrespondencebetweenbases,whenextendedtothespaces,give anisomorphism? 2.24 RequiresthesubsectiononCombiningSubspaces,whichisoptional. Suppose that V = V 1 V 2 andthat V isisomorphictothespace U underthemap f .Show that U = f V 1 f U 2 2.25 Showthatthisisnotawell-denedfunctionfromtherationalnumberstothe integers:witheachfraction,associatethevalueofitsnumerator.

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176 ChapterThree.MapsBetweenSpaces IIHomomorphisms Thedenitionofisomorphismhastwoconditions.Inthissectionwewillconsiderthesecondone,thatthemapmustpreservethealgebraicstructureofthe space.Wewillfocusonthisconditionbystudyingmapsthatarerequiredonly topreservestructure;thatis,mapsthatarenotrequiredtobecorrespondences. Experienceshowsthatthiskindofmapistremendouslyusefulinthestudy ofvectorspaces.Foronething,asweshallseeinthesecondsubsectionbelow, whileisomorphismsdescribehowspacesarethesame,thesemapsdescribehow spacescanbethoughtofasalike. II.1Denition 1.1Denition Afunctionbetweenvectorspaces h : V W thatpreserves theoperationsofaddition if ~v 1 ;~v 2 2 V then h ~v 1 + ~v 2 = h ~v 1 + h ~v 2 andscalarmultiplication if ~v 2 V and r 2 R then h r ~v = r h ~v isa homomorphism or linearmap 1.2Example Theprojectionmap : R 3 R 2 0 @ x y z 1 A 7)167(! x y isahomomorphism.Itpreservesaddition 0 @ x 1 y 1 z 1 1 A + 0 @ x 2 y 2 z 2 1 A = 0 @ x 1 + x 2 y 1 + y 2 z 1 + z 2 1 A = x 1 + x 2 y 1 + y 2 = 0 @ x 1 y 1 z 1 1 A + 0 @ x 2 y 2 z 2 1 A andscalarmultiplication. r 0 @ x 1 y 1 z 1 1 A = 0 @ rx 1 ry 1 rz 1 1 A = rx 1 ry 1 = r 0 @ x 1 y 1 z 1 1 A Thismapisnotanisomorphismsinceitisnotone-to-one.Forinstance,both ~ 0and ~e 3 in R 3 aremappedtothezerovectorin R 2 .

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SectionII.Homomorphisms 177 1.3Example Ofcourse,thedomainandcodomainmightbeotherthanspaces ofcolumnvectors.Bothofthesearehomomorphisms;thevericationsare straightforward. f 1 : P 2 !P 3 givenby a 0 + a 1 x + a 2 x 2 7! a 0 x + a 1 = 2 x 2 + a 2 = 3 x 3 f 2 : M 2 2 R givenby ab cd 7! a + d 1.4Example Betweenanytwospacesthereisa zerohomomorphism ,mapping everyvectorinthedomaintothezerovectorinthecodomain. 1.5Example Thesetwosuggestwhyweusetheterm`linearmap'. Themap g : R 3 R givenby 0 @ x y z 1 A g 7)167(! 3 x +2 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 : 5 z islineari.e.,isahomomorphism.Incontrast,themap^ g : R 3 R given by 0 @ x y z 1 A ^ g 7)167(! 3 x +2 y )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 : 5 z +1 isnot;forinstance, ^ g 0 @ 0 0 0 1 A + 0 @ 1 0 0 1 A =4while^ g 0 @ 0 0 0 1 A +^ g 0 @ 1 0 0 1 A =5 toshowthatamapisnotlinearweneedonlyproduceoneexampleofa linearcombinationthatisnotpreserved. Therstofthesetwomaps t 1 ;t 2 : R 3 R 2 islinearwhilethesecondis not. 0 @ x y z 1 A t 1 7)167(! 5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y x + y and 0 @ x y z 1 A t 2 7)167(! 5 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 y xy Findinganexamplethatthesecondfailstopreservestructureiseasy. Whatdistinguishesthehomomorphismsisthatthecoordinatefunctionsare linearcombinationsofthearguments.SeealsoExercise23.

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178 ChapterThree.MapsBetweenSpaces Obviously,anyisomorphismisahomomorphism|anisomorphismisahomomorphismthatisalsoacorrespondence.So,onewaytothinkofthe`homomorphism'ideaisthatitisageneralizationof`isomorphism',motivatedby theobservationthatmanyofthepropertiesofisomorphismshaveonlytodo withthemap'sstructurepreservationpropertyandnottodowithitbeing acorrespondence.Asexamples,thesetworesultsfromthepriorsectiondo notuseone-to-one-nessoronto-nessintheirproof,andthereforeapplytoany homomorphism. 1.6Lemma Ahomomorphismsendsazerovectortoazerovector. 1.7Lemma Eachoftheseisanecessaryandsucientconditionfor f : V W tobeahomomorphism. f c 1 ~v 1 + c 2 ~v 2 = c 1 f ~v 1 + c 2 f ~v 2 forany c 1 ;c 2 2 R and ~v 1 ;~v 2 2 V f c 1 ~v 1 + + c n ~v n = c 1 f ~v 1 + + c n f ~v n forany c 1 ;:::;c n 2 R and ~v 1 ;:::;~v n 2 V Partisoftenusedtocheckthatafunctionislinear. 1.8Example Themap f : R 2 R 4 givenby x y f 7)167(! 0 B B @ x= 2 0 x + y 3 y 1 C C A satisesofthepriorresult 0 B B @ r 1 x 1 = 2+ r 2 x 2 = 2 0 r 1 x 1 + y 1 + r 2 x 2 + y 2 r 1 y 1 + r 2 y 2 1 C C A = r 1 0 B B @ x 1 = 2 0 x 1 + y 1 3 y 1 1 C C A + r 2 0 B B @ x 2 = 2 0 x 2 + y 2 3 y 2 1 C C A andsoitisahomomorphism. However,someoftheresultsthatwehaveseenforisomorphismsfailtohold forhomomorphismsingeneral.Considerthetheoremthatanisomorphismbetweenspacesgivesacorrespondencebetweentheirbases.Homomorphismsdo notgiveanysuchcorrespondence;Example1.2showsthatthereisnosuchcorrespondence,andanotherexampleisthezeromapbetweenanytwonontrivial spaces.Instead,forhomomorphismsaweakerbutstillveryusefulresultholds. 1.9Theorem Ahomomorphismisdeterminedbyitsactiononabasis.That is,if h ~ 1 ;:::; ~ n i isabasisofavectorspace V and ~w 1 ;:::;~w n areperhaps notdistinctelementsofavectorspace W thenthereexistsahomomorphism from V to W sending ~ 1 to ~w 1 ,...,and ~ n to ~w n ,andthathomomorphismis unique.

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SectionII.Homomorphisms 179 Proof Wewilldenethemapbyassociating ~ 1 with ~w 1 ,etc.,andthenextendinglinearlytoallofthedomain.Thatis,where ~v = c 1 ~ 1 + + c n ~ n themap h : V W isgivenby h ~v = c 1 ~w 1 + + c n ~w n .Thisiswell-dened because,withrespecttothebasis,therepresentationofeachdomainvector ~v isunique. Thismapisahomomorphismsinceitpreserveslinearcombinations;where ~v 1 = c 1 ~ 1 + + c n ~ n and ~v 2 = d 1 ~ 1 + + d n ~ n ,wehavethis. h r 1 ~v 1 + r 2 ~v 2 = h r 1 c 1 + r 2 d 1 ~ 1 + + r 1 c n + r 2 d n ~ n = r 1 c 1 + r 2 d 1 ~w 1 + + r 1 c n + r 2 d n ~w n = r 1 h ~v 1 + r 2 h ~v 2 And,thismapisuniquesinceif ^ h : V W isanotherhomomorphismsuch that ^ h ~ i = ~w i foreach i then h and ^ h agreeonallofthevectorsinthedomain. ^ h ~v = ^ h c 1 ~ 1 + + c n ~ n = c 1 ^ h ~ 1 + + c n ^ h ~ n = c 1 ~w 1 + + c n ~w n = h ~v Thus, h and ^ h arethesamemap. QED 1.10Example Thisresultsaysthatwecanconstructahomomorphismby xingabasisforthedomainandspecifyingwherethemapsendsthosebasis vectors.Forinstance,ifwespecifyamap h : R 2 R 2 thatactsonthestandard basis E 2 inthisway h 1 0 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 1 and h 0 1 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 4 thentheactionof h onanyothermemberofthedomainisalsospecied.For instance,thevalueof h onthisargument h 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 = h 1 0 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 0 1 =3 h 1 0 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 h 0 1 = 5 )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 isadirectconsequenceofthevalueof h onthebasisvectors. Laterinthischapterweshalldevelopascheme,usingmatrices,thatis convienentforcomputationslikethisone. Justastheisomorphismsofaspacewithitselfareusefulandinteresting,so tooarethehomomorphismsofaspacewithitself. 1.11Denition Alinearmapfromaspaceintoitself t : V V isa linear transformation

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180 ChapterThree.MapsBetweenSpaces 1.12Remark Inthisbookweuse`lineartransformation'onlyinthecase wherethecodomainequalsthedomain,butitiswidelyusedinothertextsas ageneralsynonymfor`homomorphism'. 1.13Example Themapon R 2 thatprojectsallvectorsdowntothe x -axis x y 7! x 0 isalineartransformation. 1.14Example Thederivativemap d=dx : P n !P n a 0 + a 1 x + + a n x n d=dx 7)167(! a 1 +2 a 2 x +3 a 3 x 2 + + na n x n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 isalineartransformation,asthisresultfromcalculusnotes: d c 1 f + c 2 g =dx = c 1 df=dx + c 2 dg=dx 1.15Example Thematrixtransposemap ab cd 7! ac bd isalineartransformationof M 2 2 .Notethatthistransformationisone-to-one andonto,andsoinfactitisanautomorphism. Wenishthissubsectionaboutmapsbyrecallingthatwecanlinearlycombinemaps.Forinstance,forthesemapsfrom R 2 toitself x y f 7)167(! 2 x 3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 y and x y g 7)167(! 0 5 x thelinearcombination5 f )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 g isalsoamapfrom R 2 toitself. x y 5 f )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 g 7)167(! 10 x 5 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(10 y 1.16Lemma Forvectorspaces V and W ,thesetoflinearfunctionsfrom V to W isitselfavectorspace,asubspaceofthespaceofallfunctionsfrom V to W .Itisdenoted L V;W Proof Thissetisnon-emptybecauseitcontainsthezerohomomorphism.So toshowthatitisasubspaceweneedonlycheckthatitisclosedunderlinear combinations.Let f;g : V W belinear.Thentheirsumislinear f + g c 1 ~v 1 + c 2 ~v 2 = c 1 f ~v 1 + c 2 f ~v 2 + c 1 g ~v 1 + c 2 g ~v 2 = c 1 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(f + g ~v 1 + c 2 )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(f + g ~v 2 andanyscalarmultipleisalsolinear. r f c 1 ~v 1 + c 2 ~v 2 = r c 1 f ~v 1 + c 2 f ~v 2 = c 1 r f ~v 1 + c 2 r f ~v 2 Hence L V;W isasubspace. QED

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SectionII.Homomorphisms 181 Westartedthissectionbyisolatingthestructurepreservationpropertyof isomorphisms.Thatis,wedenedhomomorphismsasageneralizationofisomorphisms.Someofthepropertiesthatwestudiedforisomorphismscarried overunchanged,whileotherswereadaptedtothismoregeneralsetting. Itwouldbeamistake,though,toviewthisnewnotionofhomomorphismas derivedfrom,orsomehowsecondaryto,thatofisomorphism.Intherestofthis chapterweshallworkmostlywithhomomorphisms,partlybecauseanystatementmadeabouthomomorphismsisautomaticallytrueaboutisomorphisms, butmorebecause,whiletheisomorphismconceptisperhapsmorenatural,experienceshowsthatthehomomorphismconceptisactuallymorefruitfuland morecentraltofurtherprogress. Exercises X 1.17 Decideifeach h : R 3 R 2 islinear. a h x y z = x x + y + z b h x y z = 0 0 c h x y z = 1 1 d h x y z = 2 x + y 3 y )]TJ/F29 8.9664 Tf 9.215 0 Td [(4 z X 1.18 Decideifeachmap h : M 2 2 R islinear. a h ab cd = a + d b h ab cd = ad )]TJ/F32 8.9664 Tf 9.216 0 Td [(bc c h ab cd =2 a +3 b + c )]TJ/F32 8.9664 Tf 9.215 0 Td [(d d h ab cd = a 2 + b 2 X 1.19 Showthatthesetwomapsarehomomorphisms. a d=dx : P 3 !P 2 givenby a 0 + a 1 x + a 2 x 2 + a 3 x 3 mapsto a 1 +2 a 2 x +3 a 3 x 2 b R : P 2 !P 3 givenby b 0 + b 1 x + b 2 x 2 mapsto b 0 x + b 1 = 2 x 2 + b 2 = 3 x 3 Arethesemapsinversetoeachother? 1.20 Isperpendicularprojectionfrom R 3 tothe xz -planeahomomorphism?Projectiontothe yz -plane?Tothe x -axis?The y -axis?The z -axis?Projectiontothe origin? 1.21 Showthat,whilethemapsfromExample1.3preservelinearoperations,they arenotisomorphisms. 1.22 Isanidentitymapalineartransformation? X 1.23 Statingthatafunctionis`linear'isdierentthanstatingthatitsgraphisa line. a Thefunction f 1 : R R givenby f 1 x =2 x )]TJ/F29 8.9664 Tf 9.314 0 Td [(1hasagraphthatisaline. Showthatitisnotalinearfunction. b Thefunction f 2 : R 2 R givenby x y 7! x +2 y doesnothaveagraphthatisaline.Showthatitisalinearfunction.

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182 ChapterThree.MapsBetweenSpaces X 1.24 Partofthedenitionofalinearfunctionisthatitrespectsaddition.Doesa linearfunctionrespectsubtraction? 1.25 Assumethat h isalineartransformationof V andthat h ~ 1 ;:::; ~ n i isabasis of V .Proveeachstatement. a If h ~ i = ~ 0foreachbasisvectorthen h isthezeromap. b If h ~ i = ~ i foreachbasisvectorthen h istheidentitymap. c Ifthereisascalar r suchthat h ~ i = r ~ i foreachbasisvectorthen h ~v = r ~v forallvectorsin V X 1.26 Considerthevectorspace R + wherevectoradditionandscalarmultiplication arenottheonesinheritedfrom R butratherarethese: a + b istheproductof a and b ,and r a isthe r -thpowerof a .Thiswasshowntobeavectorspace inanearlierexercise.Verifythatthenaturallogarithmmapln: R + R isa homomorphismbetweenthesetwospaces.Isitanisomorphism? X 1.27 Considerthistransformationof R 2 x y 7! x= 2 y= 3 Findtheimageunderthismapofthisellipse. f x y x 2 = 4+ y 2 = 9=1 g X 1.28 Imaginearopewoundaroundtheearth'sequatorsothatittssnuglysupposethattheearthisasphere.Howmuchextraropemustbeaddedtoraisethe circletoaconstantsixfeetotheground? X 1.29 Verifythatthismap h : R 3 R x y z 7! x y z 3 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 =3 x )]TJ/F32 8.9664 Tf 9.216 0 Td [(y )]TJ/F32 8.9664 Tf 9.215 0 Td [(z islinear.Generalize. 1.30 Showthateveryhomomorphismfrom R 1 to R 1 actsviamultiplicationbya scalar.Concludethateverynontriviallineartransformationof R 1 isanisomorphism.Isthattruefortransformationsof R 2 ? R n ? 1.31a Showthatforanyscalars a 1 ; 1 ;:::;a m;n thismap h : R n R m isahomomorphism. 0 B @ x 1 x n 1 C A 7! 0 B @ a 1 ; 1 x 1 + + a 1 ;n x n a m; 1 x 1 + + a m;n x n 1 C A b Showthatforeach i ,the i -thderivativeoperator d i =dx i isalineartransformationof P n .Concludethatforanyscalars c k ;:::;c 0 thismapisalinear transformationofthatspace. f 7! d k dx k f + c k )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 d k )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 dx k )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 f + + c 1 d dx f + c 0 f 1.32 Lemma1.16showsthatasumoflinearfunctionsislinearandthatascalar multipleofalinearfunctionislinear.Showalsothatacompositionoflinear functionsislinear. X 1.33 Where f : V W islinear,supposethat f ~v 1 = ~w 1 ,..., f ~v n = ~w n for somevectors ~w 1 ,..., ~w n from W a Ifthesetof ~w 'sisindependent,mustthesetof ~v 'salsobeindependent?

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SectionII.Homomorphisms 183 b Ifthesetof ~v 'sisindependent,mustthesetof ~w 'salsobeindependent? c Ifthesetof ~w 'sspans W ,mustthesetof ~v 'sspan V ? d Ifthesetof ~v 'sspans V ,mustthesetof ~w 'sspan W ? 1.34 GeneralizeExample1.15byprovingthatthematrixtransposemapislinear. Whatisthedomainandcodomain? 1.35a Where ~u;~v 2 R n ,thelinesegmentconnectingthemisdenedtobe theset ` = f t ~u + )]TJ/F32 8.9664 Tf 9.215 0 Td [(t ~v t 2 [0 :: 1] g .Showthattheimage,underahomomorphism h ,ofthesegmentbetween ~u and ~v isthesegmentbetween h ~u and h ~v b Asubsetof R n is convex if,foranytwopointsinthatset,thelinesegment joiningthemliesentirelyinthatset.Theinsideofasphereisconvexwhilethe skinofasphereisnot.Provethatlinearmapsfrom R n to R m preservethe propertyofsetconvexity. X 1.36 Let h : R n R m beahomomorphism. a Showthattheimageunder h ofalinein R n isapossiblydegenerateline in R n b Whathappenstoa k -dimensionallinearsurface? 1.37 Provethattherestrictionofahomomorphismtoasubspaceofitsdomainis anotherhomomorphism. 1.38 Assumethat h : V W islinear. a Showthatthe rangespace ofthismap f h ~v ~v 2 V g isasubspaceofthe codomain W b Showthatthe nullspace ofthismap f ~v 2 V h ~v = ~ 0 W g isasubspaceof thedomain V c Showthatif U isasubspaceofthedomain V thenitsimage f h ~u ~u 2 U g isasubspaceofthecodomain W .Thisgeneralizestherstitem. d Generalizetheseconditem. 1.39 Considerthesetofisomorphismsfromavectorspacetoitself.Isthisa subspaceofthespace L V;V ofhomomorphismsfromthespacetoitself? 1.40 DoesTheorem1.9needthat h ~ 1 ;:::; ~ n i isabasis?Thatis,canwestillget awell-denedanduniquehomomorphismifwedropeithertheconditionthatthe setof ~ 'sbelinearlyindependent,ortheconditionthatitspanthedomain? 1.41 Let V beavectorspaceandassumethatthemaps f 1 ;f 2 : V R 1 arelinear. a Deneamap F : V R 2 whosecomponentfunctionsarethegivenlinear ones. ~v 7! f 1 ~v f 2 ~v Showthat F islinear. b Doestheconversehold|isanylinearmapfrom V to R 2 madeupoftwo linearcomponentmapsto R 1 ? c Generalize. II.2RangespaceandNullspace Isomorphismsandhomomorphismsbothpreservestructure.Thedierenceis

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184 ChapterThree.MapsBetweenSpaces thathomomorphismsneedn'tbeontoandneedn'tbeone-to-one.Thismeans thathomomorphismsareamoregeneralkindofmap,subjecttofewerrestrictionsthanisomorphisms.Wewillexaminewhatcanhappenwithhomomorphismsthatispreventedbytheextrarestrictionssatisedbyisomorphisms. Werstconsidertheeectofdroppingtheontorequirement,ofnotrequiringaspartofthedenitionthatahomomorphismbeontoitscodomain.For instance,theinjectionmap : R 2 R 3 x y 7! 0 @ x y 0 1 A isnotanisomorphismbecauseitisnotonto.Ofcourse,beingafunction,a homomorphismisontosomeset,namelyitsrange;themap isontothe xy planesubsetof R 3 2.1Lemma Underahomomorphism,theimageofanysubspaceofthedomain isasubspaceofthecodomain.Inparticular,theimageoftheentirespace,the rangeofthehomomorphism,isasubspaceofthecodomain. Proof Let h : V W belinearandlet S beasubspaceofthedomain V Theimage h S isasubsetofthecodomain W .Itisnonemptybecause S is nonemptyandthustoshowthat h S isasubspaceof W weneedonlyshow thatitisclosedunderlinearcombinationsoftwovectors.If h ~s 1 and h ~s 2 are membersof h S then c 1 h ~s 1 + c 2 h ~s 2 = h c 1 ~s 1 + h c 2 ~s 2 = h c 1 ~s 1 + c 2 ~s 2 isalsoamemberof h S becauseitistheimageof c 1 ~s 1 + c 2 ~s 2 from S QED 2.2Denition The rangespace ofahomomorphism h : V W is R h = f h ~v ~v 2 V g sometimesdenoted h V .Thedimensionoftherangespaceisthemap's rank Weshallsoonseetheconnectionbetweentherankofamapandtherankofa matrix. 2.3Example Recallthatthederivativemap d=dx : P 3 !P 3 givenby a 0 + a 1 x + a 2 x 2 + a 3 x 3 7! a 1 +2 a 2 x +3 a 3 x 2 islinear.Therangespace R d=dx is thesetofquadraticpolynomials f r + sx + tx 2 r;s;t 2 R g .Thus,therankof thismapisthree. 2.4Example Withthishomomorphism h : M 2 2 !P 3 ab cd 7! a + b +2 d +0 x + cx 2 + cx 3 animagevectorintherangecanhaveanyconstantterm,musthavean x coecientofzero,andmusthavethesamecoecientof x 2 asof x 3 .Thatis, therangespaceis R h = f r +0 x + sx 2 + sx 3 r;s 2 R g andsotherankistwo.

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SectionII.Homomorphisms 185 Thepriorresultshowsthat,inpassingfromthedenitionofisomorphismto themoregeneraldenitionofhomomorphism,omittingthe`onto'requirement doesn'tmakeanessentialdierence.Anyhomomorphismisontoitsrangespace. However,omittingthe`one-to-one'conditiondoesmakeadierence.A homomorphismmayhavemanyelementsofthedomainthatmaptooneelement ofthecodomain.Belowisabean"sketchofamany-to-onemapbetween sets. Itshowsthreeelementsofthecodomainthatareeachtheimageofmany membersofthedomain. Recallthatforanyfunction h : V W ,thesetofelementsof V thataremapped to ~w 2 W isthe inverseimage h )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ~w = f ~v 2 V h ~v = ~w g .Above,thethree setsofmanyelementsontheleftareinverseimages. 2.5Example Considertheprojection : R 3 R 2 0 @ x y z 1 A 7)167(! x y whichisahomomorphismthatismany-to-one.Inthisinstance,aninverse imagesetisaverticallineofvectorsinthedomain. R 3 R 2 ~w 2.6Example Thishomomorphism h : R 2 R 1 x y h 7)167(! x + y isalsomany-to-one;foraxed w 2 R 1 ,theinverseimage h )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 w R 2 R 1 w Moreinformationonmany-to-onemapsisintheappendix.

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186 ChapterThree.MapsBetweenSpaces isthesetofplanevectorswhosecomponentsaddto w Theaboveexampleshaveonlytodowiththefactthatweareconsidering functions,specically,many-to-onefunctions.Theyshowtheinverseimagesas setsofvectorsthatarerelatedtotheimagevector ~w .Butthesearemorethan justarbitraryfunctions,theyarehomomorphisms;whatdothetwopreservation conditionssayabouttherelationships? Ingeneralizingfromisomorphismstohomomorphismsbydroppingtheoneto-onecondition,welosethepropertythatwe'vestatedintuitivelyas:the domainisthesameas"therange.Thatis,welosethatthedomaincorresponds perfectlytotherangeinaone-vector-by-one-vectorway.Whatweshallkeep, astheexamplesbelowillustrate,isthatahomomorphismdescribesawayin whichthedomainislike",oranalgousto",therange. 2.7Example Wethinkof R 3 asbeinglike R 2 ,exceptthatvectorshavean extracomponent.Thatis,wethinkofthevectorwithcomponents x y ,and z aslikethevectorwithcomponents x and y .Indeningtheprojectionmap wemakeprecisewhichmembersofthedomainwearethinkingofasrelatedto whichmembersofthecodomain. Understandinginwhatwaythepreservationconditionsinthedenitionof homomorphismshowthatthedomainelementsarelikethecodomainelements iseasiestifwedraw R 2 asthe xy -planeinsideof R 3 .Ofcourse, R 2 isaset oftwo-tallvectorswhilethe xy -planeisasetofthree-tallvectorswithathird componentofzero,butthereisanobviouscorrespondence.Then, ~v isthe shadow"of ~v intheplaneandthepreservationofadditionpropertysaysthat x 1 y 1 z 1 above x 1 y 1 plus x 2 y 2 z 2 above x 2 y 2 equals x 1 + y 1 y 1 + y 2 z 1 + z 2 above x 1 + x 2 y 1 + y 2 Briey,theshadowofasum ~v 1 + ~v 2 equalsthesumoftheshadows ~v 1 + ~v 2 .Preservationofscalarmultiplicationhasasimilarinterpretation. Redrawingbyseparatingthetwospaces,movingthecodomain R 2 tothe right,givesanuglierpicturebutonethatismorefaithfultothebean"sketch. ~w 1 ~w 2 ~w 1 + ~w 2

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SectionII.Homomorphisms 187 Againinthisdrawing,thevectorsthatmapto ~w 1 lieinthedomaininavertical lineonlyonesuchvectorisshown,ingray.Callanysuchmemberofthis inverseimagea ~w 1 vector".Similarly,thereisaverticallineof ~w 2 vectors" andaverticallineof ~w 1 + ~w 2 vectors".Now, hasthepropertythatif ~v 1 = ~w 1 and ~v 2 = ~w 2 then ~v 1 + ~v 2 = ~v 1 + ~v 2 = ~w 1 + ~w 2 Thissaysthatthevectorclassesadd,inthesensethatany ~w 1 vectorplusany ~w 2 vectorequalsa ~w 1 + ~w 2 vector,Asimilarstatementholdsabouttheclasses underscalarmultiplication. Thus,althoughthetwospaces R 3 and R 2 arenotisomorphic, describesa wayinwhichtheyarealike:vectorsin R 3 addasdotheassociatedvectorsin R 2 |vectorsaddastheirshadowsadd. 2.8Example Ahomomorphismcanbeusedtoexpressananalogybetween spacesthatismoresubtlethanthepriorone.Forthemap x y h 7)167(! x + y fromExample2.6xtwonumbers w 1 ;w 2 intherange R .A ~v 1 thatmapsto w 1 hascomponentsthataddto w 1 ,thatis,theinverseimage h )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 w 1 isthe setofvectorswithendpointonthediagonalline x + y = w 1 .Callthesethe w 1 vectors".Similarly,wehavethe w 2 vectors"andthe w 1 + w 2 vectors".Then theadditionpreservationpropertysaysthat ~v 1 ~v 2 ~v 1 + ~v 2 a w 1 vector"plusa w 2 vector"equalsa w 1 + w 2 vector". Restated,ifa w 1 vectorisaddedtoa w 2 vectorthentheresultismappedby h toa w 1 + w 2 vector.Briey,theimageofasumisthesumoftheimages. Evenmorebriey, h ~v 1 + ~v 2 = h ~v 1 + h ~v 2 .Thepreservationofscalar multiplicationconditionhasasimilarrestatement. 2.9Example Theinverseimagescanbestructuresotherthanlines.Forthe linearmap h : R 3 R 2 0 @ x y z 1 A 7! x x theinverseimagesetsareplanes x =0, x =1,etc.,perpendiculartothe x -axis.

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188 ChapterThree.MapsBetweenSpaces Wewon'tdescribehoweveryhomomorphismthatwewilluseisananalogy becausetheformalsensethatwemakeofalikeinthat..."is`ahomomorphism existssuchthat...'.Nonetheless,theideathatahomomorphismbetweentwo spacesexpresseshowthedomain'svectorsfallintoclassesthatactlikethethe range'svectorsisagoodwaytoviewhomomorphisms. Anotherreasonthatwewon'ttreatallofthehomomorphismsthatweseeas aboveisthatmanyvectorspacesarehardtodrawe.g.,aspaceofpolynomials. However,thereisnothingbadaboutgaininginsightsfromthosespacesthatwe areabletodraw,especiallywhenthoseinsightsextendtoallvectorspaces.We derivetwosuchinsightsfromthethreeexamples2.7,2.8,and2.9. First,inallthreeexamples,theinverseimagesarelinesorplanes,thatis, linearsurfaces.Inparticular,theinverseimageoftherange'szerovectorisa lineorplanethroughtheorigin|asubspaceofthedomain. 2.10Lemma Foranyhomomorphism,theinverseimageofasubspaceofthe rangeisasubspaceofthedomain.Inparticular,theinverseimageofthetrivial subspaceoftherangeisasubspaceofthedomain. Proof Let h : V W beahomomorphismandlet S beasubspaceofthe rangespace h .Consider h )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 S = f ~v 2 V h ~v 2 S g ,theinverseimageofthe set S .Itisnonemptybecauseitcontains ~ 0 V ,since h ~ 0 V = ~ 0 W ,whichisan element S ,as S isasubspace.Toshowthat h )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 S isclosedunderlinear combinations,let ~v 1 and ~v 2 beelements,sothat h ~v 1 and h ~v 2 areelements of S ,andthen c 1 ~v 1 + c 2 ~v 2 isalsointheinverseimagebecause h c 1 ~v 1 + c 2 ~v 2 = c 1 h ~v 1 + c 2 h ~v 2 isamemberofthesubspace S QED 2.11Denition The nullspace or kernel ofalinearmap h : V W isthe inverseimageof0 W N h = h )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ~ 0 W = f ~v 2 V h ~v = ~ 0 W g : Thedimensionofthenullspaceisthemap's nullity 0 V 0 W 2.12Example ThemapfromExample2.3hasthisnullspace N d=dx = f a 0 +0 x +0 x 2 +0 x 3 a 0 2 R g 2.13Example ThemapfromExample2.4hasthisnullspace. N h = f ab 0 )]TJ/F8 9.9626 Tf 7.748 0 Td [( a + b = 2 a;b 2 R g

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SectionII.Homomorphisms 189 Nowforthesecondinsightfromtheabovepictures.InExample2.7,each oftheverticallinesissquasheddowntoasinglepoint| ,inpassingfromthe domaintotherange,takesalloftheseone-dimensionalverticallinesandzeroes themout",leavingtherangeonedimensionsmallerthanthedomain.Similarly, inExample2.8,thetwo-dimensionaldomainismappedtoaone-dimensional rangebybreakingthedomainintolineshere,theyarediagonallines,and compressingeachofthoselinestoasinglememberoftherange.Finally,in Example2.9,thedomainbreaksintoplaneswhichgetzeroedout",andsothe mapstartswithathree-dimensionaldomainbutendswithaone-dimensional range|thismapsubtracts"twofromthedimension.Noticethat,inthis thirdexample,thecodomainistwo-dimensionalbuttherangeofthemapis onlyone-dimensional,anditisthedimensionoftherangethatisofinterest. 2.14Theorem Alinearmap'srankplusitsnullityequalsthedimensionof itsdomain. Proof Let h : V W belinearandlet B N = h ~ 1 ;:::; ~ k i beabasisforthe nullspace.Extendthattoabasis B V = h ~ 1 ;:::; ~ k ; ~ k +1 ;:::; ~ n i fortheentiredomain.Weshallshowthat B R = h h ~ k +1 ;:::;h ~ n i isabasisforthe rangespace.Thencountingthesizeofthesebasesgivestheresult. Toseethat B R islinearlyindependent,considertheequation c k +1 h ~ k +1 + + c n h ~ n = ~ 0 W .Thisgivesthat h c k +1 ~ k +1 + + c n ~ n = ~ 0 W andso c k +1 ~ k +1 + + c n ~ n isinthenullspaceof h .As B N isabasisforthisnullspace, therearescalars c 1 ;:::;c k 2 R satisfyingthisrelationship. c 1 ~ 1 + + c k ~ k = c k +1 ~ k +1 + + c n ~ n But B V isabasisfor V soeachscalarequalszero.Therefore B R islinearly independent. Toshowthat B R spanstherangespace,consider h ~v 2 R h andwrite ~v asalinearcombination ~v = c 1 ~ 1 + + c n ~ n ofmembersof B V .Thisgives h ~v = h c 1 ~ 1 + + c n ~ n = c 1 h ~ 1 + + c k h ~ k + c k +1 h ~ k +1 + + c n h ~ n andsince ~ 1 ,..., ~ k areinthenullspace,wehavethat h ~v = ~ 0+ + ~ 0+ c k +1 h ~ k +1 + + c n h ~ n .Thus, h ~v isalinearcombinationofmembersof B R ,andso B R spansthespace. QED 2.15Example Where h : R 3 R 4 is 0 @ x y z 1 A h 7)167(! 0 B B @ x 0 y 0 1 C C A therangespaceandnullspaceare R h = f 0 B B @ a 0 b 0 1 C C A a;b 2 R g and N h = f 0 @ 0 0 z 1 A z 2 R g

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190 ChapterThree.MapsBetweenSpaces andsotherankof h istwowhilethenullityisone. 2.16Example If t : R R isthelineartransformation x 7!)]TJ/F8 9.9626 Tf 21.69 0 Td [(4 x; thenthe rangeis R t = R 1 ,andsotherankof t isoneandthenullityiszero. 2.17Corollary Therankofalinearmapislessthanorequaltothedimension ofthedomain.Equalityholdsifandonlyifthenullityofthemapiszero. Weknowthatanisomorphismexistsbetweentwospacesifandonlyiftheir dimensionsareequal.Hereweseethatforahomomorphismtoexist,the dimensionoftherangemustbelessthanorequaltothedimensionofthe domain.Forinstance,thereisnohomomorphismfrom R 2 onto R 3 .Thereare manyhomomorphismsfrom R 2 into R 3 ,butnoneisontoallofthree-space. Therangespaceofalinearmapcanbeofdimensionstrictlylessthanthe dimensionofthedomainExample2.3'sderivativetransformationon P 3 has adomainofdimensionfourbutarangeofdimensionthree.Thus,undera homomorphism,linearlyindependentsetsinthedomainmaymaptolinearly dependentsetsintherangeforinstance,thederivativesends f 1 ;x;x 2 ;x 3 g to f 0 ; 1 ; 2 x; 3 x 2 g .Thatis,underahomomorphism,independencemaybelost.In contrast,dependencestays. 2.18Lemma Underalinearmap,theimageofalinearlydependentsetis linearlydependent. Proof Supposethat c 1 ~v 1 + + c n ~v n = ~ 0 V ,withsome c i nonzero.Then, because h c 1 ~v 1 + + c n ~v n = c 1 h ~v 1 + + c n h ~v n andbecause h ~ 0 V = ~ 0 W wehavethat c 1 h ~v 1 + + c n h ~v n = ~ 0 W withsomenonzero c i QED Whenisindependencenotlost?Oneobvioussucientconditioniswhen thehomomorphismisanisomorphism.Thisconditionisalsonecessary;see Exercise35.Wewillnishthissubsectioncomparinghomomorphismswith isomorphismsbyobservingthataone-to-onehomomorphismisanisomorphism fromitsdomainontoitsrange. 2.19Denition Alinearmapthatisone-to-oneis nonsingular Inthenextsectionwewillseetheconnectionbetweenthisuseof`nonsingular' formapsanditsfamiliaruseformatrices. 2.20Example Thisnonsingularhomomorphism : R 2 R 3 x y 7)167(! 0 @ x y 0 1 A givestheobviouscorrespondencebetween R 2 andthe xy -planeinsideof R 3 Thepriorobservationallowsustoadaptsomeresultsaboutisomorphisms tothissetting.

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SectionII.Homomorphisms 191 2.21Theorem Inan n -dimensionalvectorspace V ,these: h isnonsingular,thatis,one-to-one h hasalinearinverse N h = f ~ 0 g ,thatis,nullity h =0 rank h = n if h ~ 1 ;:::; ~ n i isabasisfor V then h h ~ 1 ;:::;h ~ n i isabasisfor R h areequivalentstatementsaboutalinearmap h : V W Proof Wewillrstshowthat .Wewillthenshowthat= = = = For= ,supposethatthelinearmap h isone-to-one,andsohasan inverse.Thedomainofthatinverseistherangeof h andsoalinearcombinationoftwomembersofthatdomainhastheform c 1 h ~v 1 + c 2 h ~v 2 .Onthat combination,theinverse h )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 givesthis. h )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 c 1 h ~v 1 + c 2 h ~v 2 = h )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 h c 1 ~v 1 + c 2 ~v 2 = h )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 h c 1 ~v 1 + c 2 ~v 2 = c 1 ~v 1 + c 2 ~v 2 = c 1 h )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 h ~v 1 + c 2 h )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 h ~v 2 = c 1 h )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 h ~v 1 + c 2 h )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 h ~v 2 Thustheinverseofaone-to-onelinearmapisautomaticallylinear.Butthisalso givesthe= implication,becausetheinverseitselfmustbeone-to-one. Oftheremainingimplications,= holdsbecauseanyhomomorphismmaps ~ 0 V to ~ 0 W ,butaone-to-onemapsendsatmostonememberof V to ~ 0 W Next,= istruesincerankplusnullityequalsthedimensionofthe domain. For= ,toshowthat h h ~ 1 ;:::;h ~ n i isabasisfortherangespace weneedonlyshowthatitisaspanningset,becausebyassumptiontherange hasdimension n .Consider h ~v 2 R h .Expressing ~v asalinearcombination ofbasiselementsproduces h ~v = h c 1 ~ 1 + c 2 ~ 2 + + c n ~ n ,whichgivesthat h ~v = c 1 h ~ 1 + + c n h ~ n ,asdesired. Finally,forthe= implication,assumethat h ~ 1 ;:::; ~ n i isabasis for V sothat h h ~ 1 ;:::;h ~ n i isabasisfor R h .Thenevery ~w 2 R h athe uniquerepresentation ~w = c 1 h ~ 1 + + c n h ~ n .Deneamapfrom R h to V by ~w 7! c 1 ~ 1 + c 2 ~ 2 + + c n ~ n uniquenessoftherepresentationmakesthiswell-dened.Checkingthatitis linearandthatitistheinverseof h areeasy. QED We'venowseenthatalinearmapshowshowthestructureofthedomainis likethatoftherange.Suchamapcanbethoughttoorganizethedomainspace intoinverseimagesofpointsintherange.Inthespecialcasethatthemapis

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192 ChapterThree.MapsBetweenSpaces one-to-one,eachinverseimageisasinglepointandthemapisanisomorphism betweenthedomainandtherange. Exercises X 2.22 Let h : P 3 !P 4 begivenby p x 7! x p x .Whichoftheseareinthe nullspace?Whichareintherangespace? a x 3 b 0 c 7 d 12 x )]TJ/F29 8.9664 Tf 9.215 0 Td [(0 : 5 x 3 e 1+3 x 2 )]TJ/F32 8.9664 Tf 9.215 0 Td [(x 3 X 2.23 Findthenullspace,nullity,rangespace,andrankofeachmap. a h : R 2 !P 3 givenby a b 7! a + ax + ax 2 b h : M 2 2 R givenby ab cd 7! a + d c h : M 2 2 !P 2 givenby ab cd 7! a + b + c + dx 2 d thezeromap Z : R 3 R 4 X 2.24 Findthenullityofeachmap. a h : R 5 R 8 ofrankve b h : P 3 !P 3 ofrankone c h : R 6 R 3 ,anontomap d h : M 3 3 !M 3 3 ,onto X 2.25 Whatisthenullspaceofthedierentiationtransformation d=dx : P n !P n ? Whatisthenullspaceofthesecondderivative,asatransformationof P n ?The k -thderivative? 2.26 Example2.7restatestherstconditioninthedenitionofhomomorphismas `theshadowofasumisthesumoftheshadows'.Restatethesecondconditionin thesamestyle. 2.27 Forthehomomorphism h : P 3 !P 3 givenby h a 0 + a 1 x + a 2 x 2 + a 3 x 3 = a 0 + a 0 + a 1 x + a 2 + a 3 x 3 ndthese. a N h b h )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 )]TJ/F32 8.9664 Tf 9.215 0 Td [(x 3 c h )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 + x 2 X 2.28 Forthemap f : R 2 R givenby f x y =2 x + y sketchtheseinverseimagesets: f )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(3, f )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 ,and f )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 X 2.29 Eachofthesetransformationsof P 3 isnonsingular.Findtheinversefunction ofeach. a a 0 + a 1 x + a 2 x 2 + a 3 x 3 7! a 0 + a 1 x +2 a 2 x 2 +3 a 3 x 3 b a 0 + a 1 x + a 2 x 2 + a 3 x 3 7! a 0 + a 2 x + a 1 x 2 + a 3 x 3 c a 0 + a 1 x + a 2 x 2 + a 3 x 3 7! a 1 + a 2 x + a 3 x 2 + a 0 x 3 d a 0 + a 1 x + a 2 x 2 + a 3 x 3 7! a 0 + a 0 + a 1 x + a 0 + a 1 + a 2 x 2 + a 0 + a 1 + a 2 + a 3 x 3 2.30 Describethenullspaceandrangespaceofatransformationgivenby ~v 7! 2 ~v 2.31 Listallpairsrank h ; nullity h thatarepossibleforlinearmapsfrom R 5 to R 3 2.32 Doesthedierentiationmap d=dx : P n !P n haveaninverse?

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SectionII.Homomorphisms 193 X 2.33 Findthenullityofthemap h : P n R givenby a 0 + a 1 x + + a n x n 7! Z x =1 x =0 a 0 + a 1 x + + a n x n dx: 2.34a Provethatahomomorphismisontoifandonlyifitsrankequalsthe dimensionofitscodomain. b Concludethatahomomorphismbetweenvectorspaceswiththesamedimensionisone-to-oneifandonlyifitisonto. 2.35 Showthatalinearmapisnonsingularifandonlyifitpreserveslinearindependence. 2.36 Corollary2.17saysthatfortheretobeanontohomomorphismfromavector space V toavectorspace W ,itisnecessarythatthedimensionof W beless thanorequaltothedimensionof V .Provethatthisconditionisalsosucient; useTheorem1.9toshowthatifthedimensionof W islessthanorequaltothe dimensionof V ,thenthereisahomomorphismfrom V to W thatisonto. 2.37 Let h : V R beahomomorphism,butnotthezerohomomorphism.Prove thatif h ~ 1 ;:::; ~ n i isabasisforthenullspaceandif ~v 2 V isnotinthenullspace then h ~v; ~ 1 ;:::; ~ n i isabasisfortheentiredomain V X 2.38 Recallthatthenullspaceisasubsetofthedomainandtherangespaceisa subsetofthecodomain.Aretheynecessarilydistinct?Isthereahomomorphism thathasanontrivialintersectionofitsnullspaceanditsrangespace? 2.39 Provethattheimageofaspanequalsthespanoftheimages.Thatis,where h : V W islinear,provethatif S isasubsetof V then h [ S ]equals[ h S ].This generalizesLemma2.1sinceitshowsthatif U isanysubspaceof V thenitsimage f h ~u ~u 2 U g isasubspaceof W ,becausethespanoftheset U is U X 2.40a Provethatforanylinearmap h : V W andany ~w 2 W ,theset h )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 ~w hastheform f ~v + ~n ~n 2 N h g for ~v 2 V with h ~v = ~w if h isnotontothenthissetmaybeempty.Sucha setisa coset of N h andisdenoted ~v + N h b Considerthemap t : R 2 R 2 givenby x y t 7)171(! ax + by cx + dy forsomescalars a b c ,and d .Provethat t islinear. c Concludefromthepriortwoitemsthatforanylinearsystemoftheform ax + by = e cx + dy = f thesolutionsetcanbewrittenthevectorsaremembersof R 2 f ~p + ~ h ~ h satisestheassociatedhomogeneoussystem g where ~p isaparticularsolutionofthatlinearsystemifthereisnoparticular solutionthentheabovesetisempty. d Showthatthismap h : R n R m islinear 0 B @ x 1 x n 1 C A 7! 0 B @ a 1 ; 1 x 1 + + a 1 ;n x n a m; 1 x 1 + + a m;n x n 1 C A foranyscalars a 1 ; 1 ,..., a m;n .Extendtheconclusionmadeintheprioritem.

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194 ChapterThree.MapsBetweenSpaces e Showthatthe k -thderivativemapisalineartransformationof P n foreach k .Provethatthismapisalineartransformationofthatspace f 7! d k dx k f + c k )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 d k )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 dx k )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 f + + c 1 d dx f + c 0 f foranyscalars c k ,..., c 0 .Drawaconclusionasabove. 2.41 Provethatforanytransformation t : V V thatisrankone,themapgiven bycomposingtheoperatorwithitself t t : V V satises t t = r t forsome realnumber r 2.42 Showthatforanyspace V ofdimension n ,the dualspace L V; R = f h : V R h islinear g isisomorphicto R n .Itisoftendenoted V .Concludethat V = V 2.43 Showthatanylinearmapisthesumofmapsofrankone. 2.44 Is`ishomomorphicto'anequivalencerelation? Hint: thedicultyisto decideonanappropriatemeaningforthequotedphrase. 2.45 Showthattherangespacesandnullspacesofpowersoflinearmaps t : V V formdescending V R t R t 2 ::: andascending f ~ 0 g N t N t 2 ::: chains.Alsoshowthatif k issuchthat R t k = R t k +1 thenallfollowing rangespacesareequal: R t k = R t k +1 = R t k +2 ::: .Similarly,if N t k = N t k +1 then N t k = N t k +1 = N t k +2 = ::: .

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SectionIII.ComputingLinearMaps 195 IIIComputingLinearMaps Thepriorsectionshowsthatalinearmapisdeterminedbyitsactiononabasis. Infact,theequation h ~v = h c 1 ~ 1 + + c n ~ n = c 1 h ~ 1 + + c n h ~ n showsthat,ifweknowthevalueofthemaponthevectorsinabasis,thenwe cancomputethevalueofthemaponanyvector ~v atall.Wejustneedtond the c 'stoexpress ~v withrespecttothebasis. Thissectiongivestheschemethatcomputes,fromtherepresentationofa vectorinthedomainRep B ~v ,therepresentationofthatvector'simageinthe codomainRep D h ~v ,usingtherepresentationsof h ~ 1 ,..., h ~ n III.1RepresentingLinearMapswithMatrices 1.1Example Consideramap h withdomain R 2 andcodomain R 3 xing B = h 2 0 ; 1 4 i and D = h 0 @ 1 0 0 1 A ; 0 @ 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 0 1 A ; 0 @ 1 0 1 1 A i asthebasesforthesespacesthatisdeterminedbythisactiononthevectors inthedomain'sbasis. 2 0 h 7)167(! 0 @ 1 1 1 1 A 1 4 h 7)167(! 0 @ 1 2 0 1 A Tocomputetheactionofthismaponanyvectoratallfromthedomain,we rstexpress h ~ 1 and h ~ 2 withrespecttothecodomain'sbasis: 0 @ 1 1 1 1 A =0 0 @ 1 0 0 1 A )]TJ/F8 9.9626 Tf 11.158 6.74 Td [(1 2 0 @ 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 0 1 A +1 0 @ 1 0 1 1 A soRep D h ~ 1 = 0 @ 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 1 1 A D and 0 @ 1 2 0 1 A =1 0 @ 1 0 0 1 A )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 0 @ 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 0 1 A +0 0 @ 1 0 1 1 A soRep D h ~ 2 = 0 @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 0 1 A D

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196 ChapterThree.MapsBetweenSpaces theseareeasytocheck.Then,asdescribedinthepreamble,foranymember ~v ofthedomain,wecanexpresstheimage h ~v intermsofthe h ~ 's. h ~v = h c 1 2 0 + c 2 1 4 = c 1 h 2 0 + c 2 h 1 4 = c 1 0 @ 1 0 0 1 A )]TJ/F8 9.9626 Tf 11.158 6.74 Td [(1 2 0 @ 0 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 0 1 A +1 0 @ 1 0 1 1 A + c 2 0 @ 1 0 0 1 A )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 0 @ 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 0 1 A +0 0 @ 1 0 1 1 A = c 1 +1 c 2 0 @ 1 0 0 1 A + )]TJ/F8 9.9626 Tf 8.944 6.74 Td [(1 2 c 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 c 2 0 @ 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 0 1 A + c 1 +0 c 2 0 @ 1 0 1 1 A Thus, withRep B ~v = c 1 c 2 thenRep D h ~v = 0 @ 0 c 1 +1 c 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [( = 2 c 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 c 2 1 c 1 +0 c 2 1 A Forinstance, withRep B 4 8 = 1 2 B thenRep D h 4 8 = 0 @ 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 = 2 1 1 A Wewillexpresscomputationsliketheoneabovewithamatrixnotation. 0 @ 01 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 10 1 A B;D c 1 c 2 B = 0 @ 0 c 1 +1 c 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 c 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 c 2 1 c 1 +0 c 2 1 A D Inthemiddleistheargument ~v tothemap,representedwithrespecttothe domain'sbasis B byacolumnvectorwithcomponents c 1 and c 2 .Ontheright isthevalue h ~v ofthemaponthatargument,representedwithrespecttothe codomain'sbasis D byacolumnvectorwithcomponents0 c 1 +1 c 2 ,etc.The matrixontheleftisthenewthing.Itconsistsofthecoecientsfromthevector ontheright,0and1fromtherstrow, )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 = 2and )]TJ/F8 9.9626 Tf 7.749 0 Td [(1fromthesecondrow,and 1and0fromthethirdrow. Thisnotationsimplybreaksthepartsfromtheright,thecoecientsandthe c 's,outseparatelyontheleft,intoavectorthatrepresentsthemap'sargument andamatrixthatwewilltaketorepresentthemapitself.

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SectionIII.ComputingLinearMaps 197 1.2Denition Supposethat V and W arevectorspacesofdimensions n and m withbases B and D ,andthat h : V W isalinearmap.If Rep D h ~ 1 = 0 B B B @ h 1 ; 1 h 2 ; 1 h m; 1 1 C C C A D ::: Rep D h ~ n = 0 B B B @ h 1 ;n h 2 ;n h m;n 1 C C C A D then Rep B;D h = 0 B B B @ h 1 ; 1 h 1 ; 2 :::h 1 ;n h 2 ; 1 h 2 ; 2 :::h 2 ;n h m; 1 h m; 2 :::h m;n 1 C C C A B;D isthe matrixrepresentationof h withrespectto B;D Briey,thevectorsrepresentingthe h ~ 'sareadjoinedtomakethematrix representingthemap. Rep B;D h = 0 B B @ Rep D h ~ 1 Rep D h ~ n 1 C C A Observethatthenumberofcolumns n ofthematrixisthedimensionofthe domainofthemap,andthenumberofrows m isthedimensionofthecodomain. 1.3Example If h : R 3 !P 1 isgivenby 0 @ a 1 a 2 a 3 1 A h 7)167(! a 1 + a 2 + )]TJ/F11 9.9626 Tf 7.748 0 Td [(a 3 x thenwhere B = h 0 @ 0 0 1 1 A ; 0 @ 0 2 0 1 A ; 0 @ 2 0 0 1 A i and D = h 1+ x; )]TJ/F8 9.9626 Tf 7.749 0 Td [(1+ x i theactionof h on B isgivenby 0 @ 0 0 1 1 A h 7)167(!)]TJ/F11 9.9626 Tf 26.567 0 Td [(x 0 @ 0 2 0 1 A h 7)167(! 2 0 @ 2 0 0 1 A h 7)167(! 4 andasimplecalculationgives Rep D )]TJ/F11 9.9626 Tf 7.749 0 Td [(x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 D Rep D = 1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 D Rep D = 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 D

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198 ChapterThree.MapsBetweenSpaces showingthatthisisthematrixrepresenting h withrespecttothebases. Rep B;D h = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 212 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 B;D Wewilluselowercaselettersforamap,uppercaseforthematrix,and lowercaseagainfortheentriesofthematrix.Thusforthemap h ,thematrix representingitis H ,withentries h i;j 1.4Theorem Assumethat V and W arevectorspacesofdimensions n and m withbases B and D ,andthat h : V W isalinearmap.If h isrepresented by Rep B;D h = 0 B B B @ h 1 ; 1 h 1 ; 2 :::h 1 ;n h 2 ; 1 h 2 ; 2 :::h 2 ;n h m; 1 h m; 2 :::h m;n 1 C C C A B;D and ~v 2 V isrepresentedby Rep B ~v = 0 B B B @ c 1 c 2 c n 1 C C C A B thentherepresentationoftheimageof ~v isthis. Rep D h ~v = 0 B B B @ h 1 ; 1 c 1 + h 1 ; 2 c 2 + + h 1 ;n c n h 2 ; 1 c 1 + h 2 ; 2 c 2 + + h 2 ;n c n h m; 1 c 1 + h m; 2 c 2 + + h m;n c n 1 C C C A D Proof Exercise28. QED WewillthinkofthematrixRep B;D h andthevectorRep B ~v ascombining tomakethevectorRep D h ~v 1.5Denition The matrix-vectorproduct ofa m n matrixanda n 1vector isthis. 0 B B B @ a 1 ; 1 a 1 ; 2 :::a 1 ;n a 2 ; 1 a 2 ; 2 :::a 2 ;n a m; 1 a m; 2 :::a m;n 1 C C C A 0 B @ c 1 c n 1 C A = 0 B B B @ a 1 ; 1 c 1 + a 1 ; 2 c 2 + + a 1 ;n c n a 2 ; 1 c 1 + a 2 ; 2 c 2 + + a 2 ;n c n a m; 1 c 1 + a m; 2 c 2 + + a m;n c n 1 C C C A ThepointofDenition1.2istogeneralizeExample1.1,thatis,thepoint ofthedenitionisTheorem1.4,thatthematrixdescribeshowtogetfrom

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SectionIII.ComputingLinearMaps 199 therepresentationofadomainvectorwithrespecttothedomain'sbasisto therepresentationofitsimageinthecodomainwithrespecttothecodomain's basis.WithDenition1.5,wecanrestatethisas:applicationofalinearmapis representedbythematrix-vectorproductofthemap'srepresentativeandthe vector'srepresentative. 1.6Example WiththematrixfromExample1.3wecancalculatewherethat mapsendsthisvector. ~v = 0 @ 4 1 0 1 A Thisvectorisrepresented,withrespecttothedomainbasis B ,by Rep B ~v = 0 @ 0 1 = 2 2 1 A B andsothisistherepresentationofthevalue h ~v withrespecttothecodomain basis D Rep D h ~v = )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 = 212 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 = 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 B;D 0 @ 0 1 = 2 2 1 A B = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 0+1 = 2+2 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 0 )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 = 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 2 D = 9 = 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 = 2 D Tond h ~v itself,notitsrepresentation,take = 2+ x )]TJ/F8 9.9626 Tf 9.103 0 Td [( = 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1+ x =9. 1.7Example Let : R 3 R 2 beprojectionontothe xy -plane.Togivea matrixrepresentingthismap,werstxbases. B = h 0 @ 1 0 0 1 A ; 0 @ 1 1 0 1 A ; 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 0 1 1 A i D = h 2 1 ; 1 1 i Foreachvectorinthedomain'sbasis,wenditsimageunderthemap. 0 @ 1 0 0 1 A 7)167(! 1 0 0 @ 1 1 0 1 A 7)167(! 1 1 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 0 1 1 A 7)167(! )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 0 Thenwendtherepresentationofeachimagewithrespecttothecodomain's basis Rep D 1 0 = 1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 Rep D 1 1 = 0 1 Rep D )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 0 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 theseareeasilychecked.Finally,adjoiningtheserepresentationsgivesthe matrixrepresenting withrespectto B;D Rep B;D = 10 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(111 B;D

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200 ChapterThree.MapsBetweenSpaces WecanillustrateTheorem1.4bycomputingthematrix-vectorproductrepresentingthefollowingstatementabouttheprojectionmap. 0 @ 2 2 1 1 A = 2 2 Representingthisvectorfromthedomainwithrespecttothedomain'sbasis Rep B 0 @ 2 2 1 1 A = 0 @ 1 2 1 1 A B givesthismatrix-vectorproduct. Rep D 0 @ 2 1 1 1 A = 10 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(111 B;D 0 @ 1 2 1 1 A B = 0 2 D Expandingthisrepresentationintoalinearcombinationofvectorsfrom D 0 2 1 +2 1 1 = 2 2 checksthatthemap'sactionisindeedreectedintheoperationofthematrix. Wewillsometimescompressthesethreedisplayedequationsintoone 0 @ 2 2 1 1 A = 0 @ 1 2 1 1 A B h 7)167(! H 0 2 D = 2 2 inthecourseofacalculation. Wenowhavetwowaystocomputetheeectofprojection,thestraightforwardformulathatdropseachthree-tallvector'sthirdcomponenttomake atwo-tallvector,andtheaboveformulathatusesrepresentationsandmatrixvectormultiplication.Comparedtotherstway,thesecondwaymightseem complicated.However,ithasadvantages.Thenextexampleshowsthatgiving aformulaforsomemapsissimpliedbythisnewscheme. 1.8Example Torepresenta rotation map t : R 2 R 2 thatturnsallvectors intheplanecounterclockwisethroughanangle ~u t = 6 ~u t = 6 )167(!

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SectionIII.ComputingLinearMaps 201 westartbyxingbases.Using E 2 bothasadomainbasisandasacodomain basisisnatural,Now,wendtheimageunderthemapofeachvectorinthe domain'sbasis. 1 0 t 7)167(! cos sin 0 1 t 7)167(! )]TJ/F8 9.9626 Tf 9.41 0 Td [(sin cos Thenwerepresenttheseimageswithrespecttothecodomain'sbasis.Because thisbasisis E 2 ,vectorsarerepresentedbythemselves.Finally,adjoiningthe representationsgivesthematrixrepresentingthemap. Rep E 2 ; E 2 t = cos )]TJ/F8 9.9626 Tf 9.409 0 Td [(sin sin cos Theadvantageofthisschemeisthatjustbyknowinghowtorepresenttheimage ofthetwobasisvectors,wegetaformulathattellsustheimageofanyvector atall;hereavectorrotatedby = = 6. 3 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 t = 6 7)167(! p 3 = 2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 = 2 1 = 2 p 3 = 2 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 3 : 598 )]TJ/F8 9.9626 Tf 7.748 0 Td [(0 : 232 Again,weareusingthefactthat,withrespectto E 2 ,vectorsrepresentthemselves. Wehavealreadyseentheadditionandscalarmultiplicationoperationsof matricesandthedotproductoperationofvectors.Matrix-vectormultiplication isanewoperationinthearithmeticofvectorsandmatrices.NothinginDenition1.5requiresustoviewitintermsofrepresentations.Wecangetsome insightintothisoperationbyturningawayfromwhatisbeingrepresented,and insteadfocusingonhowtheentriescombine. 1.9Example Inthedenitionthewidthofthematrixequalstheheightof thevector.Hence,therstproductbelowisdenedwhilethesecondisnot. 100 431 0 @ 1 0 2 1 A = 1 6 100 431 1 0 Onereasonthatthisproductisnotdenedispurelyformal:thedenition requiresthatthesizesmatch,andthesesizesdon'tmatch.Behindtheformality, though,isareasonwhywewillleaveitundened|thematrixrepresentsamap withathree-dimensionaldomainwhilethevectorrepresentsamemberofatwodimensionalspace. Agoodwaytoviewamatrix-vectorproductisasthedotproductsofthe rowsofthematrixwiththecolumnvector. 0 B B @ a i; 1 a i; 2 :::a i;n 1 C C A 0 B B B @ c 1 c 2 c n 1 C C C A = 0 B B @ a i; 1 c 1 + a i; 2 c 2 + ::: + a i;n c n 1 C C A

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202 ChapterThree.MapsBetweenSpaces Lookedatinthisrow-by-rowway,thisnewoperationgeneralizesdotproduct. Matrix-vectorproductcanalsobeviewedcolumn-by-column. 0 B B B @ h 1 ; 1 h 1 ; 2 :::h 1 ;n h 2 ; 1 h 2 ; 2 :::h 2 ;n h m; 1 h m; 2 :::h m;n 1 C C C A 0 B B B @ c 1 c 2 c n 1 C C C A = 0 B B B @ h 1 ; 1 c 1 + h 1 ; 2 c 2 + + h 1 ;n c n h 2 ; 1 c 1 + h 2 ; 2 c 2 + + h 2 ;n c n h m; 1 c 1 + h m; 2 c 2 + + h m;n c n 1 C C C A = c 1 0 B B B @ h 1 ; 1 h 2 ; 1 h m; 1 1 C C C A + + c n 0 B B B @ h 1 ;n h 2 ;n h m;n 1 C C C A 1.10Example 10 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 203 0 @ 2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 1 1 A =2 1 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 0 0 +1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 3 = 1 7 Theresulthasthecolumnsofthematrixweightedbytheentriesofthe vector.Thiswayoflookingatitbringsusbacktotheobjectivestatedatthe startofthissection,tocompute h c 1 ~ 1 + + c n ~ n as c 1 h ~ 1 + + c n h ~ n Webeganthissectionbynotingthattheequalityofthesetwoenablesus tocomputetheactionof h onanyargumentknowingonly h ~ 1 ,..., h ~ n Wehavedevelopedthisintoaschemetocomputetheactionofthemapby takingthematrix-vectorproductofthematrixrepresentingthemapandthe vectorrepresentingtheargument.Inthisway,anylinearmapisrepresented withrespecttosomebasesbyamatrix.Inthenextsubsection,wewillshow theconverse,thatanymatrixrepresentsalinearmap. Exercises X 1.11 Multiplythematrix 131 0 )]TJ/F29 8.9664 Tf 7.168 0 Td [(12 110 byeachvectororstatenotdened". a 2 1 0 b )]TJ/F29 8.9664 Tf 7.168 0 Td [(2 )]TJ/F29 8.9664 Tf 7.168 0 Td [(2 c 0 0 0 1.12 Perform,ifpossible,eachmatrix-vectormultiplication. a 21 3 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 = 2 4 2 b 110 )]TJ/F29 8.9664 Tf 7.168 0 Td [(210 1 3 1 c 11 )]TJ/F29 8.9664 Tf 7.168 0 Td [(21 1 3 1 X 1.13 Solvethismatrixequation. 211 013 1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(12 x y z = 8 4 4 !

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SectionIII.ComputingLinearMaps 203 X 1.14 Forahomomorphismfrom P 2 to P 3 thatsends 1 7! 1+ x;x 7! 1+2 x; and x 2 7! x )]TJ/F32 8.9664 Tf 9.216 0 Td [(x 3 wheredoes1 )]TJ/F29 8.9664 Tf 9.215 0 Td [(3 x +2 x 2 go? X 1.15 Assumethat h : R 2 R 3 isdeterminedbythisaction. 1 0 7! 2 2 0 0 1 7! 0 1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 Usingthestandardbases,nd a thematrixrepresentingthismap; b ageneralformulafor h ~v X 1.16 Let d=dx : P 3 !P 3 bethederivativetransformation. a Represent d=dx withrespectto B;B where B = h 1 ;x;x 2 ;x 3 i b Represent d=dx withrespectto B;D where D = h 1 ; 2 x; 3 x 2 ; 4 x 3 i X 1.17 Representeachlinearmapwithrespecttoeachpairofbases. a d=dx : P n !P n withrespectto B;B where B = h 1 ;x;:::;x n i ,givenby a 0 + a 1 x + a 2 x 2 + + a n x n 7! a 1 +2 a 2 x + + na n x n )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 b R : P n !P n +1 withrespectto B n ;B n +1 where B i = h 1 ;x;:::;x i i ,givenby a 0 + a 1 x + a 2 x 2 + + a n x n 7! a 0 x + a 1 2 x 2 + + a n n +1 x n +1 c R 1 0 : P n R withrespectto B; E 1 where B = h 1 ;x;:::;x n i and E 1 = h 1 i givenby a 0 + a 1 x + a 2 x 2 + + a n x n 7! a 0 + a 1 2 + + a n n +1 d eval 3 : P n R withrespectto B; E 1 where B = h 1 ;x;:::;x n i and E 1 = h 1 i givenby a 0 + a 1 x + a 2 x 2 + + a n x n 7! a 0 + a 1 3+ a 2 3 2 + + a n 3 n e slide )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 : P n !P n withrespectto B;B where B = h 1 ;x;:::;x n i ,givenby a 0 + a 1 x + a 2 x 2 + + a n x n 7! a 0 + a 1 x +1+ + a n x +1 n 1.18 Representtheidentitymaponanynontrivialspacewithrespectto B;B where B isanybasis. 1.19 Represent,withrespecttothenaturalbasis,thetransposetransformationon thespace M 2 2 of2 2matrices. 1.20 Assumethat B = h ~ 1 ; ~ 2 ; ~ 3 ; ~ 4 i isabasisforavectorspace.Representwith respectto B;B thetransformationthatisdeterminedbyeach. a ~ 1 7! ~ 2 ~ 2 7! ~ 3 ~ 3 7! ~ 4 ~ 4 7! ~ 0 b ~ 1 7! ~ 2 ~ 2 7! ~ 0, ~ 3 7! ~ 4 ~ 4 7! ~ 0 c ~ 1 7! ~ 2 ~ 2 7! ~ 3 ~ 3 7! ~ 0, ~ 4 7! ~ 0 1.21 Example1.8showshowtorepresenttherotationtransformationoftheplane withrespecttothestandardbasis.Expresstheseothertransformationsalsowith respecttothestandardbasis. a the dilation map d s ,whichmultipliesallvectorsbythesamescalar s b the reection map f ` ,whichreectsallallvectorsacrossaline ` through theorigin X 1.22 Consideralineartransformationof R 2 determinedbythesetwo. 1 1 7! 2 0 1 0 7! )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 0 a Representthistransformationwithrespecttothestandardbases.

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204 ChapterThree.MapsBetweenSpaces b Wheredoesthetransformationsendthisvector? 0 5 c Representthistransformationwithrespecttothesebases. B = h 1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 ; 1 1 i D = h 2 2 ; )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 1 i d Using B fromtheprioritem,representthetransformationwithrespectto B;B 1.23 Supposethat h : V W isnonsingularsothatbyTheorem2.21,forany basis B = h ~ 1 ;:::; ~ n i V theimage h B = h h ~ 1 ;:::;h ~ n i isabasisfor W a Representthemap h withrespectto B;h B b Foramember ~v ofthedomain,wheretherepresentationof ~v hascomponents c 1 ,..., c n ,representtheimagevector h ~v withrespecttotheimagebasis h B 1.24 Giveaformulafortheproductofamatrixand ~e i ,thecolumnvectorthatis allzeroesexceptforasingleoneinthe i -thposition. X 1.25 Foreachvectorspaceoffunctionsofonerealvariable,representthederivative transformationwithrespectto B;B a f a cos x + b sin x a;b 2 R g B = h cos x; sin x i b f ae x + be 2 x a;b 2 R g B = h e x ;e 2 x i c f a + bx + ce x + dxe x a;b;c;d 2 R g B = h 1 ;x;e x ;xe x i 1.26 Findtherangeofthelineartransformationof R 2 representedwithrespectto thestandardbasesbyeachmatrix. a 10 00 b 00 32 c amatrixoftheform ab 2 a 2 b X 1.27 Canonematrixrepresenttwodierentlinearmaps?Thatis,canRep B;D h = Rep ^ B; ^ D ^ h ? 1.28 ProveTheorem1.4. X 1.29 Example1.8showshowtorepresentrotationofallvectorsintheplanethrough anangle abouttheorigin,withrespecttothestandardbases. a Rotationofallvectorsinthree-spacethroughanangle aboutthe x -axisisa transformationof R 3 .Representitwithrespecttothestandardbases.Arrange therotationsothattosomeonewhosefeetareattheoriginandwhoseheadis at ; 0 ; 0,themovementappearsclockwise. b Repeattheprioritem,onlyrotateaboutthe y -axisinstead.Puttheperson's headat ~e 2 c Repeat,aboutthe z -axis. d Extendtheprioritemto R 4 Hint: `rotateaboutthe z -axis'canberestated as`rotateparalleltothe xy -plane'. 1.30 Schur'sTriangularizationLemma a Let U beasubspaceof V andxbases B U B V .Whatistherelationship betweentherepresentationofavectorfrom U withrespectto B U andthe representationofthatvectorviewedasamemberof V withrespectto B V ? b Whataboutmaps? c Fixabasis B = h ~ 1 ;:::; ~ n i for V andobservethatthespans [ f ~ 0 g ]= f ~ 0 g [ f ~ 1 g ] [ f ~ 1 ; ~ 2 g ] [ B ]= V

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SectionIII.ComputingLinearMaps 205 formastrictlyincreasingchainofsubspaces.Showthatforanylinearmap h : V W thereisachain W 0 = f ~ 0 g W 1 W m = W ofsubspacesof W suchthat h [ f ~ 1 ;:::; ~ i g ] W i foreach i d Concludethatforeverylinearmap h : V W therearebases B;D sothe matrixrepresenting h withrespectto B;D isupper-triangularthatis,each entry h i;j with i>j iszero. e Isanupper-triangularrepresentationunique? III.2AnyMatrixRepresentsaLinearMap Thepriorsubsectionshowsthattheactionofalinearmap h isdescribedby amatrix H ,withrespecttoappropriatebases,inthisway. ~v = 0 B @ v 1 v n 1 C A B h 7)167(! H 0 B @ h 1 ; 1 v 1 + + h 1 ;n v n h m; 1 v 1 + + h m;n v n 1 C A D = h ~v Inthissubsection,wewillshowtheconverse,thateachmatrixrepresentsa linearmap. Recallthat,inthedenitionofthematrixrepresentationofalinearmap, thenumberofcolumnsofthematrixisthedimensionofthemap'sdomainand thenumberofrowsofthematrixisthedimensionofthemap'scodomain.Thus, forinstance,a2 3matrixcannotrepresentamapfrom R 5 to R 4 .Thenext resultsaysthat,beyondthisrestrictiononthedimensions,therearenoother limitations:the2 3matrixrepresentsamapfromanythree-dimensionalspace toanytwo-dimensionalspace. 2.1Theorem Anymatrixrepresentsahomomorphismbetweenvectorspaces ofappropriatedimensions,withrespecttoanypairofbases. Proof Forthematrix H = 0 B B B @ h 1 ; 1 h 1 ; 2 :::h 1 ;n h 2 ; 1 h 2 ; 2 :::h 2 ;n h m; 1 h m; 2 :::h m;n 1 C C C A xany n -dimensionaldomainspace V andany m -dimensionalcodomainspace W .Alsoxbases B = h ~ 1 ;:::; ~ n i and D = h ~ 1 ;:::; ~ m i forthosespaces. Deneafunction h : V W by:where ~v inthedomainisrepresentedas Rep B ~v = 0 B @ v 1 v n 1 C A B

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206 ChapterThree.MapsBetweenSpaces thenitsimage h ~v isthememberthecodomainrepresentedby Rep D h ~v = 0 B @ h 1 ; 1 v 1 + + h 1 ;n v n h m; 1 v 1 + + h m;n v n 1 C A D thatis, h ~v = h v 1 ~ 1 + + v n ~ n isdenedtobe h 1 ; 1 v 1 + + h 1 ;n v n ~ 1 + + h m; 1 v 1 + + h m;n v n ~ m .Thisiswell-denedbytheuniquenessofthe representationRep B ~v Observethat h hassimplybeendenedtomakeitthemapthatisrepresentedwithrespectto B;D bythematrix H .Sotonish,weneedonlycheck that h islinear.If ~v;~u 2 V aresuchthat Rep B ~v = 0 B @ v 1 v n 1 C A andRep B ~u = 0 B @ u 1 u n 1 C A and c;d 2 R thenthecalculation h c~v + d~u = )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(h 1 ; 1 cv 1 + du 1 + + h 1 ;n cv n + du n ~ 1 + + )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(h m; 1 cv 1 + du 1 + + h m;n cv n + du n ~ m = c h ~v + d h ~u providesthisverication. QED 2.2Example Whichmapthematrixrepresentsdependsonwhichbasesare used.If H = 10 00 ;B 1 = D 1 = h 1 0 ; 0 1 i ; and B 2 = D 2 = h 0 1 ; 1 0 i ; then h 1 : R 2 R 2 representedby H withrespectto B 1 ;D 1 maps c 1 c 2 = c 1 c 2 B 1 7! c 1 0 D 1 = c 1 0 while h 2 : R 2 R 2 representedby H withrespectto B 2 ;D 2 isthismap. c 1 c 2 = c 2 c 1 B 2 7! c 2 0 D 2 = 0 c 2 Thesetwoaredierent.Therstisprojectionontothe x axis,whilethesecond isprojectionontothe y axis. Sonotonlyisanylinearmapdescribedbyamatrixbutanymatrixdescribes alinearmap.Thismeansthatwecan,whenconvenient,handlelinearmaps entirelyasmatrices,simplydoingthecomputations,withouthavetoworrythat

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SectionIII.ComputingLinearMaps 207 amatrixofinterestdoesnotrepresentalinearmaponsomepairofspacesof interest.Inpractice,whenweareworkingwithamatrixbutnospacesor baseshavebeenspecied,wewilloftentakethedomainandcodomaintobe R n and R m andusethestandardbases.Inthiscase,becausetherepresentationis transparent|therepresentationwithrespecttothestandardbasisof ~v is ~v | thecolumnspaceofthematrixequalstherangeofthemap.Consequently,the columnspaceof H isoftendenotedby R H Withthetheorem,wehavecharacterizedlinearmapsasthosemapsthatact inthismatrixway.Eachlinearmapisdescribedbyamatrixandeachmatrix describesalinearmap.Wenishthissectionbyillustratinghowamatrixcan beusedtotellthingsaboutitsmaps. 2.3Theorem Therankofamatrixequalstherankofanymapthatit represents. Proof Supposethatthematrix H is m n .Fixdomainandcodomainspaces V and W ofdimension n and m ,withbases B = h ~ 1 ;:::; ~ n i and D .Then H representssomelinearmap h betweenthosespaceswithrespecttothesebases whoserangespace f h ~v ~v 2 V g = f h c 1 ~ 1 + + c n ~ n c 1 ;:::;c n 2 R g = f c 1 h ~ 1 + + c n h ~ n c 1 ;:::;c n 2 R g isthespan[ f h ~ 1 ;:::;h ~ n g ].Therankof h isthedimensionofthisrangespace. Therankofthematrixisitscolumnrankoritsrowrank;thetwoare equal.Thisisthedimensionofthecolumnspaceofthematrix,whichisthe spanofthesetofcolumnvectors[ f Rep D h ~ 1 ;:::; Rep D h ~ n g ]. Toseethatthetwospanshavethesamedimension,recallthatarepresentationwithrespecttoabasisgivesanisomorphismRep D : W R m .Under thisisomorphism,thereisalinearrelationshipamongmembersoftherangespaceifandonlyifthesamerelationshipholdsinthecolumnspace,e.g, ~ 0= c 1 h ~ 1 + + c n h ~ n ifandonlyif ~ 0= c 1 Rep D h ~ 1 + + c n Rep D h ~ n Hence,asubsetoftherangespaceislinearlyindependentifandonlyifthecorrespondingsubsetofthecolumnspaceislinearlyindependent.Thismeansthat thesizeofthelargestlinearlyindependentsubsetoftherangespaceequalsthe sizeofthelargestlinearlyindependentsubsetofthecolumnspace,andsothe twospaceshavethesamedimension. QED 2.4Example Anymaprepresentedby 0 B B @ 122 121 003 002 1 C C A must,bydenition,befromathree-dimensionaldomaintoafour-dimensional codomain.Inaddition,becausetherankofthismatrixistwowecanspotthis

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208 ChapterThree.MapsBetweenSpaces byeyeorgetitwithGauss'method,anymaprepresentedbythismatrixhas atwo-dimensionalrangespace. 2.5Corollary Let h bealinearmaprepresentedbyamatrix H .Then h isontoifandonlyiftherankof H equalsthenumberofitsrows,and h is one-to-oneifandonlyiftherankof H equalsthenumberofitscolumns. Proof Forthersthalf,thedimensionoftherangespaceof h istherankof h whichequalstherankof H bythetheorem.Sincethedimensionofthecodomain of h isthenumberofrowsin H ,iftherankof H equalsthenumberofrows,then thedimensionoftherangespaceequalsthedimensionofthecodomain.Buta subspacewiththesamedimensionasitssuperspacemustequalthatsuperspace abasisfortherangespaceisalinearlyindependentsubsetofthecodomain, whosesizeisequaltothedimensionofthecodomain,andsothissetisabasis forthecodomain. Forthesecondhalf,alinearmapisone-to-oneifandonlyifitisanisomorphismbetweenitsdomainanditsrange,thatis,ifandonlyifitsdomainhasthe samedimensionasitsrange.Butthenumberofcolumnsin h isthedimension of h 'sdomain,andbythetheoremtherankof H equalsthedimensionof h 's range. QED Theaboveresultsendanyconfusioncausedbyouruseoftheword`rank'to meanapparentlydierentthingswhenappliedtomatricesandwhenappliedto maps.Wecanalsojustifythedualuseof`nonsingular'.We'vedenedamatrix tobenonsingularifitissquareandisthematrixofcoecientsofalinearsystem withauniquesolution,andwe'vedenedalinearmaptobenonsingularifitis one-to-one. 2.6Corollary Asquarematrixrepresentsnonsingularmapsifandonlyifit isanonsingularmatrix.Thus,amatrixrepresentsanisomorphismifandonly ifitissquareandnonsingular. Proof Immediatefromthepriorresult. QED 2.7Example Anymapfrom R 2 to P 1 representedwithrespecttoanypairof basesby 12 03 isnonsingularbecausethismatrixhasranktwo. 2.8Example Anymap g : V W representedby 12 36 isnotnonsingularbecausethismatrixisnotnonsingular.

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SectionIII.ComputingLinearMaps 209 We'venowseenthattherelationshipbetweenmapsandmatricesgoesboth ways:xingbases,anylinearmapisrepresentedbyamatrixandanymatrix describesalinearmap.Thatis,byxingspacesandbaseswegetacorrespondencebetweenmapsandmatrices.Intherestofthischapterwewillexplore thiscorrespondence.Forinstance,we'vedenedforlinearmapstheoperations ofadditionandscalarmultiplicationandweshallseewhatthecorresponding matrixoperationsare.Weshallalsoseethematrixoperationthatrepresent themapoperationofcomposition.And,weshallseehowtondthematrix thatrepresentsamap'sinverse. Exercises X 2.9 Decideifthevectorisinthecolumnspaceofthematrix. a 21 25 1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(3 b 4 )]TJ/F29 8.9664 Tf 7.168 0 Td [(8 2 )]TJ/F29 8.9664 Tf 7.168 0 Td [(4 0 1 c 1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(11 11 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(11 2 0 0 X 2.10 Decideifeachvectorliesintherangeofthemapfrom R 3 to R 2 represented withrespecttothestandardbasesbythematrix. a 113 014 1 3 b 203 406 1 1 X 2.11 Considerthismatrix,representingatransformationof R 2 ,andthesebasesfor thatspace. 1 2 11 )]TJ/F29 8.9664 Tf 7.167 0 Td [(11 B = h 0 1 ; 1 0 i D = h 1 1 ; 1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 i a Towhatvectorinthecodomainistherstmemberof B mapped? b Thesecondmember? c Whereisageneralvectorfromthedomainavectorwithcomponents x and y mapped?Thatis,whattransformationof R 2 isrepresentedwithrespectto B;D bythismatrix? 2.12 Whattransformationof F = f a cos + b sin a;b 2 R g isrepresentedwith respectto B = h cos )]TJ/F29 8.9664 Tf 9.216 0 Td [(sin ; sin i and D = h cos +sin ; cos i bythismatrix? 00 10 X 2.13 Decideif1+2 x isintherangeofthemapfrom R 3 to P 2 representedwith respectto E 3 and h 1 ; 1+ x 2 ;x i bythismatrix. 130 010 101 2.14 Example2.8givesamatrixthatisnonsingular,andisthereforeassociated withmapsthatarenonsingular. a Findthesetofcolumnvectorsrepresentingthemembersofthenullspaceof anymaprepresentedbythismatrix. b Findthenullityofanysuchmap. c Findthesetofcolumnvectorsrepresentingthemembersoftherangespace ofanymaprepresentedbythismatrix. d Findtherankofanysuchmap. e Checkthatrankplusnullityequalsthedimensionofthedomain.

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210 ChapterThree.MapsBetweenSpaces X 2.15 Becausetherankofamatrixequalstherankofanymapitrepresents,if onematrixrepresentstwodierentmaps H =Rep B;D h =Rep ^ B; ^ D ^ h where h; ^ h : V W thenthedimensionoftherangespaceof h equalsthedimensionof therangespaceof ^ h .Musttheseequal-dimensionedrangespacesactuallybethe same? X 2.16 Let V bean n -dimensionalspacewithbases B and D .Consideramapthat sends,for ~v 2 V ,thecolumnvectorrepresenting ~v withrespectto B tothecolumn vectorrepresenting ~v withrespectto D .Showthatisalineartransformationof R n 2.17 Example2.2showsthatchangingthepairofbasescanchangethemapthat amatrixrepresents,eventhoughthedomainandcodomainremainthesame. Couldthemapevernotchange?Isthereamatrix H ,vectorspaces V and W andassociatedpairsofbases B 1 ;D 1 and B 2 ;D 2 with B 1 6 = B 2 or D 1 6 = D 2 or bothsuchthatthemaprepresentedby H withrespectto B 1 ;D 1 equalsthemap representedby H withrespectto B 2 ;D 2 ? X 2.18 Asquarematrixisa diagonal matrixifitisallzeroesexceptpossiblyforthe entriesonitsupper-lefttolower-rightdiagonal|its1 ; 1entry,its2 ; 2entry,etc. Showthatalinearmapisanisomorphismiftherearebasessuchthat,withrespect tothosebases,themapisrepresentedbyadiagonalmatrixwithnozeroesonthe diagonal. 2.19 Describegeometricallytheactionon R 2 ofthemaprepresentedwithrespect tothestandardbases E 2 ; E 2 bythismatrix. 30 02 Dothesameforthese. 10 00 01 10 13 01 2.20 Thefactthatforanylinearmaptherankplusthenullityequalsthedimension ofthedomainshowsthatanecessaryconditionfortheexistenceofahomomorphismbetweentwospaces,ontothesecondspace,isthattherebenogainin dimension.Thatis,where h : V W isonto,thedimensionof W mustbeless thanorequaltothedimensionof V a Showthatthisstrongconverseholds:nogainindimensionimpliesthat thereisahomomorphismand,further,anymatrixwiththecorrectsizeand correctrankrepresentssuchamap. b Aretherebasesfor R 3 suchthatthismatrix H = 100 200 010 representsamapfrom R 3 to R 3 whoserangeisthe xy planesubspaceof R 3 ? 2.21 Let V bean n -dimensionalspaceandsupposethat ~x 2 R n .Fixabasis B for V andconsiderthemap h ~x : V R given ~v 7! ~x Rep B ~v bythedot product. a Showthatthismapislinear. b Showthatforanylinearmap g : V R thereisan ~x 2 R n suchthat g = h ~x c Intheprioritemwexedthebasisandvariedthe ~x togetallpossiblelinear maps.Canwegetallpossiblelinearmapsbyxingan ~x andvaryingthebasis?

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SectionIII.ComputingLinearMaps 211 2.22 Let V;W;X bevectorspaceswithbases B;C;D a Supposethat h : V W isrepresentedwithrespectto B;C bythematrix H .Givethematrixrepresentingthescalarmultiple rh where r 2 R with respectto B;C byexpressingitintermsof H b Supposethat h;g : V W arerepresentedwithrespectto B;C by H and G .Givethematrixrepresenting h + g withrespectto B;C byexpressingitin termsof H and G c Supposethat h : V W isrepresentedwithrespectto B;C by H and g : W X isrepresentedwithrespectto C;D by G .Givethematrixrepresenting g h withrespectto B;D byexpressingitintermsof H and G .

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212 ChapterThree.MapsBetweenSpaces IVMatrixOperations Thepriorsectionshowshowmatricesrepresentlinearmaps.Agoodstrategy, onseeinganewidea,istoexplorehowitinteractswithsomealready-established ideas.Intherstsubsectionwewillaskhowtherepresentationofthesumof twomaps f + g isrelatedtotherepresentationsofthetwomaps,andhowthe representationofascalarproduct r h ofamapisrelatedtotherepresentation ofthatmap.Inlatersubsectionswewillseehowtorepresentmapcomposition andmapinverse. IV.1SumsandScalarProducts Recallthatfortwomaps f and g withthesamedomainandcodomain,the mapsum f + g hasthisdenition. ~v f + g 7)167(! f ~v + g ~v Theeasiestwaytoseehowtherepresentationsofthemapscombinetorepresent themapsumiswithanexample. 1.1Example Supposethat f;g : R 2 R 3 arerepresentedwithrespecttothe bases B and D bythesematrices. F =Rep B;D f = 0 @ 13 20 10 1 A B;D G =Rep B;D g = 0 @ 00 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 24 1 A B;D Then,forany ~v 2 V representedwithrespectto B ,computationoftherepresentationof f ~v + g ~v 0 @ 13 20 10 1 A v 1 v 2 + 0 @ 00 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 24 1 A v 1 v 2 = 0 @ 1 v 1 +3 v 2 2 v 1 +0 v 2 1 v 1 +0 v 2 1 A + 0 @ 0 v 1 +0 v 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 v 1 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 v 2 2 v 1 +4 v 2 1 A givesthisrepresentationof f + g ~v 0 @ +0 v 1 ++0 v 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 v 1 + )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 v 2 +2 v 1 ++4 v 2 1 A = 0 @ 1 v 1 +3 v 2 1 v 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 v 2 3 v 1 +4 v 2 1 A Thus,theactionof f + g isdescribedbythismatrix-vectorproduct. 0 @ 13 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 34 1 A B;D v 1 v 2 B = 0 @ 1 v 1 +3 v 2 1 v 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 v 2 3 v 1 +4 v 2 1 A D Thismatrixistheentry-by-entrysumoforiginalmatrices,e.g.,the1 ; 1entry ofRep B;D f + g isthesumofthe1 ; 1entryof F andthe1 ; 1entryof G .

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SectionIV.MatrixOperations 213 Representingascalarmultipleofamapworksthesameway. 1.2Example If t isatransformationrepresentedby Rep B;D t = 10 11 B;D sothat ~v = v 1 v 2 B 7! v 1 v 1 + v 2 D = t ~v thenthescalarmultiplemap5 t actsinthisway. ~v = v 1 v 2 B 7)167(! 5 v 1 5 v 1 +5 v 2 D =5 t ~v Therefore,thisisthematrixrepresenting5 t Rep B;D t = 50 55 B;D 1.3Denition The sum oftwosame-sizedmatricesistheirentry-by-entry sum.The scalarmultiple ofamatrixistheresultofentry-by-entryscalar multiplication. 1.4Remark Theseextendthevectoradditionandscalarmultiplicationoperationsthatwedenedintherstchapter. 1.5Theorem Let h;g : V W belinearmapsrepresentedwithrespectto bases B;D bythematrices H and G ,andlet r beascalar.Thenthemap h + g : V W isrepresentedwithrespectto B;D by H + G ,andthemap r h : V W isrepresentedwithrespectto B;D by rH Proof Exercise8;generalizetheexamplesabove. QED Anotablespecialcaseofscalarmultiplicationismultiplicationbyzero.For anymap0 h isthezerohomomorphismandforanymatrix0 H isthezero matrix. 1.6Example Thezeromapfromanythree-dimensionalspacetoanytwodimensionalspaceisrepresentedbythe2 3zeromatrix Z = 000 000 nomatterwhichdomainandcodomainbasesareused. Exercises X 1.7 Performtheindicatedoperations,ifdened. a 5 )]TJ/F29 8.9664 Tf 7.168 0 Td [(12 611 + 214 305 b 6 2 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 123

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214 ChapterThree.MapsBetweenSpaces c 21 03 + 21 03 d 4 12 3 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 +5 )]TJ/F29 8.9664 Tf 7.168 0 Td [(14 )]TJ/F29 8.9664 Tf 7.168 0 Td [(21 e 3 21 30 +2 114 305 1.8 ProveTheorem1.5. a Provethatmatrixadditionrepresentsadditionoflinearmaps. b Provethatmatrixscalarmultiplicationrepresentsscalarmultiplicationof linearmaps. X 1.9 Proveeach,wheretheoperationsaredened,where G H ,and J arematrices, where Z isthezeromatrix,andwhere r and s arescalars. a Matrixadditioniscommutative G + H = H + G b Matrixadditionisassociative G + H + J = G + H + J c Thezeromatrixisanadditiveidentity G + Z = G d 0 G = Z e r + s G = rG + sG f Matriceshaveanadditiveinverse G + )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 G = Z g r G + H = rG + rH h rs G = r sG 1.10 Fixdomainandcodomainspaces.Ingeneral,onematrixcanrepresentmany dierentmapswithrespecttodierentbases.However,provethatazeromatrix representsonlyazeromap.Arethereothersuchmatrices? X 1.11 Let V and W bevectorspacesofdimensions n and m .Showthatthespace L V;W oflinearmapsfrom V to W isisomorphicto M m n X 1.12 Showthatitfollowsfromthepriorquestionsthatforanysixtransformations t 1 ;:::;t 6 : R 2 R 2 therearescalars c 1 ;:::;c 6 2 R suchthat c 1 t 1 + + c 6 t 6 is thezeromap. Hint: thisisabitofamisleadingquestion. 1.13 The trace ofasquarematrixisthesumoftheentriesonthemaindiagonal the1 ; 1entryplusthe2 ; 2entry,etc.;wewillseethesignicanceofthetracein ChapterFive.Showthattrace H + G =trace H +trace G .Isthereasimilar resultforscalarmultiplication? 1.14 Recallthatthe transpose ofamatrix M isanothermatrix,whose i;j entryis the j;i entryof M .Veriytheseidentities. a G + H trans = G trans + H trans b r H trans = r H trans X 1.15 Asquarematrixis symmetric ifeach i;j entryequalsthe j;i entry,thatis,if thematrixequalsitstranspose. a Provethatforany H ,thematrix H + H trans issymmetric.Doesevery symmetricmatrixhavethisform? b Provethatthesetof n n symmetricmatricesisasubspaceof M n n X 1.16a Howdoesmatrixrankinteractwithscalarmultiplication|canascalar productofarank n matrixhaveranklessthan n ?Greater? b Howdoesmatrixrankinteractwithmatrixaddition|canasumofrank n matriceshaveranklessthan n ?Greater?

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SectionIV.MatrixOperations 215 IV.2MatrixMultiplication Afterrepresentingadditionandscalarmultiplicationoflinearmapsintheprior subsection,thenaturalnextmapoperationtoconsideriscomposition. 2.1Lemma Acompositionoflinearmapsislinear. Proof .Thisargumenthasappearedearlier,aspartoftheproofthatisomorphismisanequivalencerelationbetweenspaces. Let h : V W and g : W U belinear.Thecalculation g h )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(c 1 ~v 1 + c 2 ~v 2 = g )]TJ/F11 9.9626 Tf 6.227 -8.069 Td [(h c 1 ~v 1 + c 2 ~v 2 = g )]TJ/F11 9.9626 Tf 6.227 -8.069 Td [(c 1 h ~v 1 + c 2 h ~v 2 = c 1 g )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(h ~v 1 + c 2 g h ~v 2 = c 1 g h ~v 1 + c 2 g h ~v 2 showsthat g h : V U preserveslinearcombinations. QED Toseehowtherepresentationofthecompositearisesoutoftherepresentationsofthetwocompositors,consideranexample. 2.2Example Let h : R 4 R 2 and g : R 2 R 3 ,xbases B R 4 C R 2 D R 3 ,andletthesebetherepresentations. H =Rep B;C h = 4682 5793 B;C G =Rep C;D g = 0 @ 11 01 10 1 A C;D Torepresentthecomposition g h : R 4 R 3 wexa ~v ,represent h of ~v ,and thenrepresent g ofthat.Therepresentationof h ~v istheproductof h 'smatrix and ~v 'svector. Rep C h ~v = 4682 5793 B;C 0 B B @ v 1 v 2 v 3 v 4 1 C C A B = 4 v 1 +6 v 2 +8 v 3 +2 v 4 5 v 1 +7 v 2 +9 v 3 +3 v 4 C Therepresentationof g h ~v istheproductof g 'smatrixand h ~v 'svector. Rep D g h ~v = 0 @ 11 01 10 1 A C;D 4 v 1 +6 v 2 +8 v 3 +2 v 4 5 v 1 +7 v 2 +9 v 3 +3 v 4 C = 0 @ 1 v 1 +6 v 2 +8 v 3 +2 v 4 +1 v 1 +7 v 2 +9 v 3 +3 v 4 0 v 1 +6 v 2 +8 v 3 +2 v 4 +1 v 1 +7 v 2 +9 v 3 +3 v 4 1 v 1 +6 v 2 +8 v 3 +2 v 4 +0 v 1 +7 v 2 +9 v 3 +3 v 4 1 A D Distributingandregroupingonthe v 'sgives = 0 @ 4+1 5 v 1 + 6+1 7 v 2 + 8+1 9 v 3 + 2+1 3 v 4 4+1 5 v 1 + 6+1 7 v 2 + 8+1 9 v 3 + 2+1 3 v 4 4+0 5 v 1 + 6+0 7 v 2 + 8+0 9 v 3 + 2+0 3 v 4 1 A D

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216 ChapterThree.MapsBetweenSpaces whichwerecognizingastheresultofthismatrix-vectorproduct. = 0 @ 1 4+1 51 6+1 71 8+1 91 2+1 3 0 4+1 50 6+1 70 8+1 90 2+1 3 1 4+0 51 6+0 71 8+0 91 2+0 3 1 A B;D 0 B B @ v 1 v 2 v 3 v 4 1 C C A D Thus,thematrixrepresenting g h hastherowsof G combinedwiththecolumns of H 2.3Denition The matrix-multiplicativeproduct ofthe m r matrix G and the r n matrix H isthe m n matrix P ,where p i;j = g i; 1 h 1 ;j + g i; 2 h 2 ;j + + g i;r h r;j thatis,the i;j -thentryoftheproductisthedotproductofthe i -throwand the j -thcolumn. GH = 0 B B @ g i; 1 g i; 2 :::g i;r 1 C C A 0 B B B @ h 1 ;j :::h 2 ;j ::: h r;j 1 C C C A = 0 B B @ :::p i;j ::: 1 C C A 2.4Example ThematricesfromExample2.2combineinthisway. 0 @ 1 4+1 51 6+1 71 8+1 91 2+1 3 0 4+1 50 6+1 70 8+1 90 2+1 3 1 4+0 51 6+0 71 8+0 91 2+0 3 1 A = 0 @ 913175 5793 4682 1 A 2.5Example 0 @ 20 46 82 1 A 13 57 = 0 @ 2 1+0 52 3+0 7 4 1+6 54 3+6 7 8 1+2 58 3+2 7 1 A = 0 @ 26 3454 1838 1 A 2.6Theorem Acompositionoflinearmapsisrepresentedbythematrix productoftherepresentatives. Proof .ThisargumentparallelsExample2.2. Let h : V W and g : W X berepresentedby H and G withrespecttobases B V C W ,and D X ofsizes n r ,and m .Forany ~v 2 V ,the k -thcomponentofRep C h ~v is h k; 1 v 1 + + h k;n v n andsothe i -thcomponentofRep D g h ~v isthis. g i; 1 h 1 ; 1 v 1 + + h 1 ;n v n + g i; 2 h 2 ; 1 v 1 + + h 2 ;n v n + + g i;r h r; 1 v 1 + + h r;n v n

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SectionIV.MatrixOperations 217 Distributeandregrouponthe v 's. = g i; 1 h 1 ; 1 + g i; 2 h 2 ; 1 + + g i;r h r; 1 v 1 + + g i; 1 h 1 ;n + g i; 2 h 2 ;n + + g i;r h r;n v n Finishbyrecognizingthatthecoecientofeach v j g i; 1 h 1 ;j + g i; 2 h 2 ;j + + g i;r h r;j matchesthedenitionofthe i;j entryoftheproduct GH QED Thetheoremisanexampleofaresultthatsupportsadenition.Wecan picturewhatthedenitionandtheoremtogethersaywiththis arrowdiagram `wrt'abbreviates`withrespectto'. V wrtB W wrtC X wrtD h H g G g h GH Abovethearrows,themapsshowthatthetwowaysofgoingfrom V to X straightoverviathecompositionorelsebywayof W ,havethesameeect ~v g h 7)167(! g h ~v ~v h 7)167(! h ~v g 7)167(! g h ~v thisisjustthedenitionofcomposition.Belowthearrows,thematricesindicatethattheproductdoesthesamething|multiplying GH intothecolumn vectorRep B ~v hasthesameeectasmultiplyingthecolumnrstby H and thenmultiplyingtheresultby G Rep B;D g h = GH =Rep C;D g Rep B;C h Thedenitionofthematrix-matrixproductoperationdoesnotrestrictus toviewitasarepresentationofalinearmapcomposition.Wecangetinsight intothisoperationbystudyingitasamechanicalprocedure.Thestrikingthing isthewaythatrowsandcolumnscombine. Oneaspectofthatcombinationisthatthesizesofthematricesinvolvedis signicant.Briey, m r times r n equals m n 2.7Example Thisproductisnotdened )]TJ/F8 9.9626 Tf 7.749 0 Td [(120 0101 : 1 00 02 becausethenumberofcolumnsontheleftdoesnotequalthenumberofrows ontheright.

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218 ChapterThree.MapsBetweenSpaces Intermsoftheunderlyingmaps,thefactthatthesizesmustmatchupreects thefactthatmatrixmultiplicationisdenedonlywhenacorrespondingfunction composition dimension n space h )167(! dimension r space g )167(! dimension m space ispossible. 2.8Remark Theorderinwhichthesethingsarewrittencanbeconfusing.In the` m r times r n equals m n 'equation,thenumberwrittenrst m isthe dimensionof g 'scodomainandisthusthenumberthatappearslastinthemap dimensiondescriptionabove.Theexplanationisthatwhile f isdonerstand then g isapplied,thatcompositioniswritten g f ,fromthenotation` g f ~v '. Somepeopletrytolessenconfusionbyreading` g f 'aloudas g following f ".Thatorderthencarriesovertomatrices: g f isrepresentedby GF Anotheraspectofthewaythatrowsandcolumnscombineinthematrix productoperationisthatinthedenitionofthe i;j entry p i;j = g i; 1 h 1 ;j + g i; 2 h 2 ;j + + g i; r h r ;j theboxedsubscriptsonthe g 'sarecolumnindicatorswhilethoseonthe h 's indicaterows.Thatis,summationtakesplaceoverthecolumnsof G butover therowsof H ;leftistreateddierentlythanright,so GH maybeunequalto HG .Matrixmultiplicationisnotcommutative. 2.9Example Matrixmultiplicationhardlyevercommutes.Testthatbymultiplyingrandomlychosenmatricesbothways. 12 34 56 78 = 1922 4350 56 78 12 34 = 2334 3146 2.10Example Commutativitycanfailmoredramatically: 56 78 120 340 = 23340 31460 while 120 340 56 78 isn'tevendened. 2.11Remark Thefactthatmatrixmultiplicationisnotcommutativemay bepuzzlingatrstsight,perhapsjustbecausemostalgebraicoperationsin elementarymathematicsarecommutative.Butonfurtherreection,itisn't sosurprising.Afterall,matrixmultiplicationrepresentsfunctioncomposition, whichisnotcommutative|if f x =2 x and g x = x +1then g f x =2 x +1 while f g x =2 x +1=2 x +2.True,this g isnotlinearandwemight havehopedthatlinearfunctionscommute,butthisperspectiveshowsthatthe failureofcommutativityformatrixmultiplicationtsintoalargercontext.

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SectionIV.MatrixOperations 219 Exceptforthelackofcommutativity,matrixmultiplicationisalgebraically well-behaved.BelowaresomenicepropertiesandmoreareinExercise23and Exercise24. 2.12Theorem If F G ,and H arematrices,andthematrixproductsare dened,thentheproductisassociative FG H = F GH anddistributesover matrixaddition F G + H = FG + FH and G + H F = GF + HF Proof Associativityholdsbecausematrixmultiplicationrepresentsfunction composition,whichisassociative:themaps f g h and f g h areequal asbothsend ~v to f g h ~v Distributivityissimilar.Forinstance,therstonegoes f g + h ~v = f )]TJ/F8 9.9626 Tf 6.227 -8.07 Td [( g + h ~v = f )]TJ/F11 9.9626 Tf 6.226 -8.07 Td [(g ~v + h ~v = f g ~v + f h ~v = f g ~v + f h ~v the thirdequalityusesthelinearityof f QED 2.13Remark Wecouldalternativelyprovethatresultbysloggingthrough theindices.Forexample,associativitygoes:the i;j -thentryof FG H is f i; 1 g 1 ; 1 + f i; 2 g 2 ; 1 + + f i;r g r; 1 h 1 ;j + f i; 1 g 1 ; 2 + f i; 2 g 2 ; 2 + + f i;r g r; 2 h 2 ;j + f i; 1 g 1 ;s + f i; 2 g 2 ;s + + f i;r g r;s h s;j where F G ,and H are m r r s ,and s n matrices,distribute f i; 1 g 1 ; 1 h 1 ;j + f i; 2 g 2 ; 1 h 1 ;j + + f i;r g r; 1 h 1 ;j + f i; 1 g 1 ; 2 h 2 ;j + f i; 2 g 2 ; 2 h 2 ;j + + f i;r g r; 2 h 2 ;j + f i; 1 g 1 ;s h s;j + f i; 2 g 2 ;s h s;j + + f i;r g r;s h s;j andregrouparoundthe f 's f i; 1 g 1 ; 1 h 1 ;j + g 1 ; 2 h 2 ;j + + g 1 ;s h s;j + f i; 2 g 2 ; 1 h 1 ;j + g 2 ; 2 h 2 ;j + + g 2 ;s h s;j + f i;r g r; 1 h 1 ;j + g r; 2 h 2 ;j + + g r;s h s;j togetthe i;j entryof F GH Contrastthesetwowaysofverifyingassociativity,theoneintheproofand theonejustabove.Theargumentjustaboveishardtounderstandinthesense that,whilethecalculationsareeasytocheck,thearithmeticseemsunconnected toanyideaitalsoessentiallyrepeatstheproofofTheorem2.6andsoisinecient.Theargumentintheproofisshorter,clearer,andsayswhythisproperty really"holds.Thisillustratesthecommentsmadeinthepreambletothechapteronvectorspaces|atleastsomeofthetimeanargumentfromhigher-level constructsisclearer.

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220 ChapterThree.MapsBetweenSpaces Wehavenowseenhowtherepresentationofthecompositionoftwolinear mapsisderivedfromtherepresentationsofthetwomaps.Wehavecalled thecombinationtheproductofthetwomatrices.Thisoperationisextremely important.Beforewegoontostudyhowtorepresenttheinverseofalinear map,wewillexploreitsomemoreinthenextsubsection. Exercises X 2.14 Compute,orstatenotdened". a 31 )]TJ/F29 8.9664 Tf 7.168 0 Td [(42 05 00 : 5 b 11 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 403 2 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 311 311 c 2 )]TJ/F29 8.9664 Tf 7.168 0 Td [(7 74 105 )]TJ/F29 8.9664 Tf 7.168 0 Td [(111 384 d 52 31 )]TJ/F29 8.9664 Tf 7.168 0 Td [(12 3 )]TJ/F29 8.9664 Tf 7.168 0 Td [(5 X 2.15 Where A = 1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 20 B = 52 44 C = )]TJ/F29 8.9664 Tf 7.168 0 Td [(23 )]TJ/F29 8.9664 Tf 7.168 0 Td [(41 computeorstate`notdened'. a AB b AB C c BC d A BC 2.16 Whichproductsaredened? a 3 2times2 3 b 2 3times3 2 c 2 2times3 3 d 3 3times2 2 X 2.17 Givethesizeoftheproductorstatenotdened". a a2 3matrixtimesa3 1matrix b a1 12matrixtimesa12 1matrix c a2 3matrixtimesa2 1matrix d a2 2matrixtimesa2 2matrix X 2.18 Findthesystemofequationsresultingfromstartingwith h 1 ; 1 x 1 + h 1 ; 2 x 2 + h 1 ; 3 x 3 = d 1 h 2 ; 1 x 1 + h 2 ; 2 x 2 + h 2 ; 3 x 3 = d 2 andmakingthischangeofvariablei.e.,substitution. x 1 = g 1 ; 1 y 1 + g 1 ; 2 y 2 x 2 = g 2 ; 1 y 1 + g 2 ; 2 y 2 x 3 = g 3 ; 1 y 1 + g 3 ; 2 y 2 2.19 AsDenition2.3pointsout,thematrixproductoperationgeneralizesthedot product.Isthedotproductofa1 n rowvectoranda n 1columnvectorthe sameastheirmatrix-multiplicativeproduct? X 2.20 Representthederivativemapon P n withrespectto B;B where B isthe naturalbasis h 1 ;x;:::;x n i .Showthattheproductofthismatrixwithitselfis dened;whatthemapdoesitrepresent? 2.21 Showthatcompositionoflineartransformationson R 1 iscommutative.Is thistrueforanyone-dimensionalspace? 2.22 Whyismatrixmultiplicationnotdenedasentry-wisemultiplication?That wouldbeeasier,andcommutativetoo. X 2.23a Provethat H p H q = H p + q and H p q = H pq forpositiveintegers p;q b Provethat rH p = r p H p foranypositiveinteger p andscalar r 2 R X 2.24a Howdoesmatrixmultiplicationinteractwithscalarmultiplication:is r GH = rG H ?Is G rH = r GH ?

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SectionIV.MatrixOperations 221 b Howdoesmatrixmultiplicationinteractwithlinearcombinations:is F rG + sH = r FG + s FH ?Is rF + sG H = rFH + sGH ? 2.25 Wecanaskhowthematrixproductoperationinteractswiththetranspose operation. a Showthat GH trans = H trans G trans b Asquarematrixis symmetric ifeach i;j entryequalsthe j;i entry,thatis, ifthematrixequalsitsowntranspose.Showthatthematrices HH trans and H trans H aresymmetric. X 2.26 Rotationofvectorsin R 3 aboutanaxisisalinearmap.Showthatlinear mapsdonotcommutebyshowinggeometricallythatrotationsdonotcommute. 2.27 IntheproofofTheorem2.12somemapsareused.Whatarethedomainsand codomains? 2.28 Howdoesmatrixrankinteractwithmatrixmultiplication? a Cantheproductofrank n matriceshaveranklessthan n ?Greater? b Showthattherankoftheproductoftwomatricesislessthanorequalto theminimumoftherankofeachfactor. 2.29 Is`commuteswith'anequivalencerelationamong n n matrices? X 2.30 ThiswillbeusedintheMatrixInversesexercises. Hereisanotherproperty ofmatrixmultiplicationthatmightbepuzzlingatrstsight. a Provethatthecompositionoftheprojections x ; y : R 3 R 3 ontothe x and y axesisthezeromapdespitethatneitheroneisitselfthezeromap. b Provethatthecompositionofthederivatives d 2 =dx 2 ;d 3 =dx 3 : P 4 !P 4 is thezeromapdespitethatneitheristhezeromap. c Giveamatrixequationrepresentingtherstfact. d Giveamatrixequationrepresentingthesecond. Whentwothingsmultiplytogivezerodespitethatneitheriszero,eachissaidto bea zerodivisor 2.31 Showthat,forsquarematrices, S + T S )]TJ/F32 8.9664 Tf 9.215 0 Td [(T neednotequal S 2 )]TJ/F32 8.9664 Tf 9.216 0 Td [(T 2 X 2.32 Representtheidentitytransformationid: V V withrespectto B;B forany basis B .Thisisthe identitymatrix I .Showthatthismatrixplaystherolein matrixmultiplicationthatthenumber1playsinrealnumbermultiplication: HI = IH = H forallmatrices H forwhichtheproductisdened. 2.33 Inrealnumberalgebra,quadraticequationshaveatmosttwosolutions.That isnotsowithmatrixalgebra.Showthatthe2 2matrixequation T 2 = I has morethantwosolutions,where I istheidentitymatrixthismatrixhasonesin its1 ; 1and2 ; 2entriesandzeroeselsewhere;seeExercise32. 2.34a Provethatforany2 2matrix T therearescalars c 0 ;:::;c 4 thatare notall0suchthatthecombination c 4 T 4 + c 3 T 3 + c 2 T 2 + c 1 T + c 0 I isthezero matrixwhere I isthe2 2identitymatrix,with1'sinits1 ; 1and2 ; 2entries andzeroeselsewhere;seeExercise32. b Let p x beapolynomial p x = c n x n + + c 1 x + c 0 .If T isasquare matrixwedene p T tobethematrix c n T n + + c 1 T + I where I isthe appropriately-sizedidentitymatrix.Provethatforanysquarematrixthereis apolynomialsuchthat p T isthezeromatrix. c The minimalpolynomial m x ofasquarematrixisthepolynomialofleast degree,andwithleadingcoecient1,suchthat m T isthezeromatrix.Find theminimalpolynomialofthismatrix. p 3 = 2 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 = 2 1 = 2 p 3 = 2

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222 ChapterThree.MapsBetweenSpaces Thisistherepresentationwithrespectto E 2 ; E 2 ,thestandardbasis,ofarotation through = 6radianscounterclockwise. 2.35 Theinnite-dimensionalspace P ofallnite-degreepolynomialsgivesamemorableexampleofthenon-commutativityoflinearmaps.Let d=dx : P!P bethe usualderivativeandlet s : P!P bethe shift map. a 0 + a 1 x + + a n x n s 7)171(! 0+ a 0 x + a 1 x 2 + + a n x n +1 Showthatthetwomapsdon'tcommute d=dx s 6 = s d=dx ;infact,notonlyis d=dx s )]TJ/F29 8.9664 Tf 9.215 0 Td [( s d=dx notthezeromap,itistheidentitymap. 2.36 Recallthenotationforthesumofthesequenceofnumbers a 1 ;a 2 ;:::;a n n X i =1 a i = a 1 + a 2 + + a n Inthisnotation,the i;j entryoftheproductof G and H isthis. p i;j = r X k =1 g i;k h k;j Usingthisnotation, a reprovethatmatrixmultiplicationisassociative; b reproveTheorem2.6. IV.3MechanicsofMatrixMultiplication Inthissubsectionweconsidermatrixmultiplicationasamechanicalprocess, puttingasideforthemomentanyimplicationsabouttheunderlyingmaps.As describedearlier,thestrikingthingaboutmatrixmultiplicationisthewayrows andcolumnscombine.The i;j entryofthematrixproductisthedotproduct ofrow i oftheleftmatrixwithcolumn j oftherightone.Forinstance,herea secondrowandathirdcolumncombinetomakea2 ; 3entry. 0 B @ 11 01 10 1 C A 4 5 6 7 8 9 2 3 = 0 @ 913175 57 9 3 4682 1 A Wecanviewthisastheleftmatrixactingbymultiplyingitsrows,oneatatime, intothecolumnsoftherightmatrix.Ofcourse,anotherperspectiveisthatthe rightmatrixusesitscolumnstoactontheleftmatrix'srows.Below,wewill examineactionsfromtheleftandfromtherightforsomesimplematrices. Therstcase,theactionofazeromatrix,isveryeasy. 3.1Example Multiplyingbyanappropriately-sizedzeromatrixfromtheleft orfromtheright 00 00 132 )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 000 000 23 14 00 00 = 00 00 resultsinazeromatrix.

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SectionIV.MatrixOperations 223 Afterzeromatrices,thematriceswhoseactionsareeasiesttounderstand aretheoneswithasinglenonzeroentry. 3.2Denition Amatrixwithallzeroesexceptforaoneinthe i;j entryis an i;j unit matrix. 3.3Example Thisisthe1 ; 2unitmatrixwiththreerowsandtwocolumns, multiplyingfromtheleft. 0 @ 01 00 00 1 A 56 78 = 0 @ 78 00 00 1 A Actingfromtheleft,an i;j unitmatrixcopiesrow j ofthemultiplicandinto row i oftheresult.Fromtherightan i;j unitmatrixcopiescolumn i ofthe multiplicandintocolumn j oftheresult. 0 @ 123 456 789 1 A 0 @ 01 00 00 1 A = 0 @ 01 04 07 1 A 3.4Example Rescalingthesematricessimplyrescalestheresult.Thisisthe actionfromtheleftofthematrixthatistwicetheoneinthepriorexample. 0 @ 02 00 00 1 A 56 78 = 0 @ 1416 00 00 1 A Andthisistheactionofthematrixthatisminusthreetimestheonefromthe priorexample. 0 @ 123 456 789 1 A 0 @ 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 00 00 1 A = 0 @ 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(21 1 A Nextincomplicationarematriceswithtwononzeroentries.Therearetwo cases.Ifaleft-multiplierhasentriesindierentrowsthentheiractionsdon't interact. 3.5Example 0 @ 100 002 000 1 A 0 @ 123 456 789 1 A = 0 @ 100 000 000 1 A + 0 @ 000 002 000 1 A 0 @ 123 456 789 1 A = 0 @ 123 000 000 1 A + 0 @ 000 141618 000 1 A = 0 @ 123 141618 000 1 A

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224 ChapterThree.MapsBetweenSpaces Butiftheleft-multiplier'snonzeroentriesareinthesamerowthenthatrowof theresultisacombination. 3.6Example 0 @ 102 000 000 1 A 0 @ 123 456 789 1 A = 0 @ 100 000 000 1 A + 0 @ 002 000 000 1 A 0 @ 123 456 789 1 A = 0 @ 123 000 000 1 A + 0 @ 141618 000 000 1 A = 0 @ 151821 000 000 1 A Right-multiplicationactsinthesameway,withcolumns. Theseobservationsaboutmatricesthataremostlyzeroesextendtoarbitrary matrices. 3.7Lemma Inaproductoftwomatrices G and H ,thecolumnsof GH are formedbytaking G timesthecolumnsof H G 0 B B @ ~ h 1 ~ h n 1 C C A = 0 B B @ G ~ h 1 G ~ h n 1 C C A andtherowsof GH areformedbytakingtherowsof G times H 0 B B @ ~g 1 ~g r 1 C C A H = 0 B B @ ~g 1 H ~g r H 1 C C A ignoringtheextraparentheses. Proof Wewillshowthe2 2caseandleavethegeneralcaseasanexercise. GH = g 1 ; 1 g 1 ; 2 g 2 ; 1 g 2 ; 2 h 1 ; 1 h 1 ; 2 h 2 ; 1 h 2 ; 2 = g 1 ; 1 h 1 ; 1 + g 1 ; 2 h 2 ; 1 g 1 ; 1 h 1 ; 2 + g 1 ; 2 h 2 ; 2 g 2 ; 1 h 1 ; 1 + g 2 ; 2 h 2 ; 1 g 2 ; 1 h 1 ; 2 + g 2 ; 2 h 2 ; 2 Therightsideoftherstequationintheresult G h 1 ; 1 h 2 ; 1 G h 1 ; 2 h 2 ; 2 = g 1 ; 1 h 1 ; 1 + g 1 ; 2 h 2 ; 1 g 2 ; 1 h 1 ; 1 + g 2 ; 2 h 2 ; 1 g 1 ; 1 h 1 ; 2 + g 1 ; 2 h 2 ; 2 g 2 ; 1 h 1 ; 2 + g 2 ; 2 h 2 ; 2 isindeedthesameastherightsideofGH,exceptfortheextraparenthesesthe onesmarkingthecolumnsascolumnvectors.Theotherequationissimilarly easytorecognize. QED

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SectionIV.MatrixOperations 225 Anapplicationofthoseobservationsisthatthereisamatrixthatjustcopies outtherowsandcolumns. 3.8Denition The maindiagonal or principlediagonal or diagonal ofa squarematrixgoesfromtheupperlefttothelowerright. 3.9Denition An identitymatrix issquareandhaswithallentrieszero exceptforonesinthemaindiagonal. I n n = 0 B B B @ 10 ::: 0 01 ::: 0 00 ::: 1 1 C C C A 3.10Example The3 3identityleavesitsmultiplicandunchangedbothfrom theleft 0 @ 100 010 001 1 A 0 @ 236 138 )]TJ/F8 9.9626 Tf 7.749 0 Td [(710 1 A = 0 @ 236 138 )]TJ/F8 9.9626 Tf 7.749 0 Td [(710 1 A andfromtheright. 0 @ 236 138 )]TJ/F8 9.9626 Tf 7.748 0 Td [(710 1 A 0 @ 100 010 001 1 A = 0 @ 236 138 )]TJ/F8 9.9626 Tf 7.749 0 Td [(710 1 A 3.11Example Sodoesthe2 2identitymatrix. 0 B B @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 43 1 C C A 10 01 = 0 B B @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 43 1 C C A Inshort,anidentitymatrixistheidentityelementofthesetof n n matrices withrespecttotheoperationofmatrixmultiplication. Wenextseetwowaystogeneralizetheidentitymatrix. Therstisthatiftheonesarerelaxedtoarbitraryreals,theresultingmatrix willrescalewholerowsorcolumns. 3.12Denition A diagonalmatrix issquareandhaszerosothemain diagonal. 0 B B B @ a 1 ; 1 0 ::: 0 0 a 2 ; 2 ::: 0 00 :::a n;n 1 C C C A

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226 ChapterThree.MapsBetweenSpaces 3.13Example Fromtheleft,theactionofmultiplicationbyadiagonalmatrix istorescalestherows. 20 0 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 214 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1344 = 428 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 Fromtherightsuchamatrixrescalesthecolumns. 121 222 0 @ 300 020 00 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 1 A = 34 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 64 )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 Thesecondgeneralizationofidentitymatricesisthatwecanputasingleone ineachrowandcolumninwaysotherthanputtingthemdownthediagonal. 3.14Denition A permutationmatrix issquareandisallzerosexceptfora singleoneineachrowandcolumn. 3.15Example Fromtheleftthesematricespermuterows. 0 @ 001 100 010 1 A 0 @ 123 456 789 1 A = 0 @ 789 123 456 1 A Fromtherighttheypermutecolumns. 0 @ 123 456 789 1 A 0 @ 001 100 010 1 A = 0 @ 231 564 897 1 A Wenishthissubsectionbyapplyingtheseobservationstogetmatricesthat performGauss'methodandGauss-Jordanreduction. 3.16Example Wehaveseenhowtoproduceamatrixthatwillrescalerows. Multiplyingbythisdiagonalmatrixrescalesthesecondrowoftheotherbya factorofthree. 0 @ 100 030 001 1 A 0 @ 0211 01 = 31 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 1020 1 A = 0 @ 0211 013 )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 1020 1 A Wehaveseenhowtoproduceamatrixthatwillswaprows.Multiplyingbythis permutationmatrixswapstherstandthirdrows. 0 @ 001 010 100 1 A 0 @ 0211 013 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 1020 1 A = 0 @ 1020 013 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 0211 1 A

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SectionIV.MatrixOperations 227 Toseehowtoperformapivot,weobservesomethingaboutthosetwoexamples.Thematrixthatrescalesthesecondrowbyafactorofthreearisesin thiswayfromtheidentity. 0 @ 100 010 001 1 A 3 2 )167(! 0 @ 100 030 001 1 A Similarly,thematrixthatswapsrstandthirdrowsarisesinthisway. 0 @ 100 010 001 1 A 1 $ 3 )167(! 0 @ 001 010 100 1 A 3.17Example The3 3matrixthatarisesas 0 @ 100 010 001 1 A )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 2 + 3 )167(! 0 @ 100 010 0 )]TJ/F8 9.9626 Tf 7.748 0 Td [(21 1 A will,whenitactsfromtheleft,performthepivotoperation )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 2 + 3 0 @ 100 010 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(21 1 A 0 @ 1020 013 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 0211 1 A = 0 @ 1020 013 )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 00 )]TJ/F8 9.9626 Tf 7.749 0 Td [(57 1 A 3.18Denition The elementaryreductionmatrices areobtainedfromidentitymatriceswithoneGaussianoperation.Wedenotethem: I k i )167(! M i k for k 6 =0; I i $ j )167(! P i;j for i 6 = j ; I k i + j )167(! C i;j k for i 6 = j 3.19Lemma Gaussianreductioncanbedonethroughmatrixmultiplication. If H k i )167(! G then M i k H = G If H i $ j )167(! G then P i;j H = G If H k i + j )167(! G then C i;j k H = G Proof Clear. QED

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228 ChapterThree.MapsBetweenSpaces 3.20Example Thisistherstsystem,fromtherstchapter,onwhichwe performedGauss'method. 3 x 3 =9 x 1 +5 x 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 x 3 =2 = 3 x 1 +2 x 2 =3 Itcanbereducedwithmatrixmultiplication.Swaptherstandthirdrows, 0 @ 001 010 100 1 A 0 @ 003 9 15 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 2 1 = 320 3 1 A = 0 @ 1 = 320 3 15 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 2 003 9 1 A tripletherstrow, 0 @ 300 010 001 1 A 0 @ 1 = 320 3 15 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 2 003 9 1 A = 0 @ 160 9 15 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 2 003 9 1 A andthenadd )]TJ/F8 9.9626 Tf 7.749 0 Td [(1timestherstrowtothesecond. 0 @ 100 )]TJ/F8 9.9626 Tf 7.749 0 Td [(110 001 1 A 0 @ 160 9 15 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 2 003 9 1 A = 0 @ 160 9 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 003 9 1 A Nowbacksubstitutionwillgivethesolution. 3.21Example Gauss-Jordanreductionworksthesameway.Forthematrix endingthepriorexample,rstadjusttheleadingentries 0 @ 100 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 001 = 3 1 A 0 @ 160 9 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(7 003 9 1 A = 0 @ 160 9 012 7 001 3 1 A andtonish,clearthethirdcolumnandthenthesecondcolumn. 0 @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(60 010 001 1 A 0 @ 100 01 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 001 1 A 0 @ 160 9 012 7 001 3 1 A = 0 @ 100 3 010 1 001 3 1 A Wehaveobservedthefollowingresult,whichweshalluseinthenextsubsection. 3.22Corollary Foranymatrix H thereareelementaryreductionmatrices R 1 ,..., R r suchthat R r R r )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 R 1 H isinreducedechelonform. Untilnowwehavetakenthepointofviewthatourprimaryobjectsofstudy arevectorspacesandthemapsbetweenthem,andhaveadoptedmatricesonly forcomputationalconvenience.Thissubsectionshowthatthispointofview isn'tthewholestory.Matrixtheoryisafascinatingandfruitfularea. Intherestofthisbookweshallcontinuetofocusonmapsastheprimary objects,butwewillbepragmatic|ifthematrixpointofviewgivessomeclearer ideathenweshalluseit.

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SectionIV.MatrixOperations 229 Exercises X 3.23 Predicttheresultofeachmultiplicationbyanelementaryreductionmatrix, andthencheckbymultiplyingitout. a 30 00 12 34 b 40 02 12 34 c 10 )]TJ/F29 8.9664 Tf 7.168 0 Td [(21 12 34 d 12 34 1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 01 e 12 34 01 10 X 3.24 Theneedtotakelinearcombinationsofrowsandcolumnsintablesofnumbers arisesofteninpractice.Forinstance,thisisamapofpartofVermontandNew York. InpartbecauseofLakeChamplain, therearenoroadsdirectlyconnectingsomepairsoftowns.Forinstance,thereisnowaytogofrom WinooskitoGrandIslewithoutgoingthroughColchester.Ofcourse, manyotherroadsandtownshave beenleftotosimplifythegraph. Fromtoptobottomofthismapis aboutfortymiles. Burlington Colchester GrandIsle Swanton Winooski a The incidencematrix ofamapisthesquarematrixwhose i;j entryisthe numberofroadsfromcity i tocity j .Producetheincidencematrixofthismap takethecitiesinalphabeticalorder. b Amatrixis symmetric ifitequalsitstranspose.Showthatanincidence matrixissymmetric.Thesearealltwo-waystreets.Vermontdoesn'thave manyone-waystreets. c Whatisthesignicanceofthesquareoftheincidencematrix?Thecube? X 3.25 Thistablegivesthenumberofhoursofeachtypedonebyeachworker,and theassociatedpayrates.Usematricestocomputethewagesdue. regularovertime Alan 4012 Betty 356 Catherine 4018 Donald 280 wage regular $25 : 00 overtime $45 : 00 Remark. Thisillustrates,asdidthepriorproblem,thatinpracticeweoftenwant tocomputelinearcombinationsofrowsandcolumnsinacontextwherewereally aren'tinterestedinanyassociatedlinearmaps. 3.26 Findtheproductofthismatrixwithitstranspose. cos )]TJ/F29 8.9664 Tf 8.704 0 Td [(sin sin cos

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230 ChapterThree.MapsBetweenSpaces X 3.27 Provethatthediagonalmatricesformasubspaceof M n n .Whatisits dimension? 3.28 Doestheidentitymatrixrepresenttheidentitymapifthebasesareunequal? 3.29 Showthateverymultipleoftheidentitycommuteswitheverysquarematrix. Arethereothermatricesthatcommutewithallsquarematrices? 3.30 Proveordisprove:nonsingularmatricescommute. X 3.31 Showthattheproductofapermutationmatrixanditstransposeisanidentity matrix. 3.32 Showthatiftherstandsecondrowsof G areequalthensoaretherstand secondrowsof GH .Generalize. 3.33 Describetheproductoftwodiagonalmatrices. 3.34 Write 10 )]TJ/F29 8.9664 Tf 7.168 0 Td [(33 astheproductoftwoelementaryreductionmatrices. X 3.35 Showthatif G hasarowofzerosthen GH ifdenedhasarowofzeros. Doesthatworkforcolumns? 3.36 Showthatthesetofunitmatricesformsabasisfor M n m 3.37 Findtheformulaforthe n -thpowerofthismatrix. 11 10 X 3.38 The trace ofasquarematrixisthesumoftheentriesonitsdiagonalits signicanceappearsinChapterFive.Showthattrace GH =trace HG X 3.39 Asquarematrixis uppertriangular ifitsonlynonzeroentrieslieabove,or on,thediagonal.Showthattheproductoftwouppertriangularmatricesisupper triangular.Doesthisholdforlowertriangularalso? 3.40 Asquarematrixisa Markovmatrix ifeachentryisbetweenzeroandone andthesumalongeachrowisone.ProvethataproductofMarkovmatricesis Markov. X 3.41 Giveanexampleoftwomatricesofthesamerankwithsquaresofdiering rank. 3.42 Combinethetwogeneralizationsoftheidentitymatrix,theoneallowingentirestobeotherthanones,andtheoneallowingthesingleoneineachrowand columntobeothediagonal.Whatistheactionofthistypeofmatrix? 3.43 Onacomputermultiplicationsaremorecostlythanadditions,sopeopleare interestedinreducingthenumberofmultiplicationsusedtocomputeamatrix product. a Howmanyrealnumbermultiplicationsareneededinformulawegaveforthe productofa m r matrixanda r n matrix? b Matrixmultiplicationisassociative,soallassociationsyieldthesameresult. Thecostinnumberofmultiplications,however,varies.Findtheassociation requiringthefewestrealnumbermultiplicationstocomputethematrixproduct ofa5 10matrix,a10 20matrix,a20 5matrix,anda5 1matrix. c Veryhard. Findawaytomultiplytwo2 2matricesusingonlyseven multiplicationsinsteadoftheeightsuggestedbythenaiveapproach. ? 3.44 If A and B aresquarematricesofthesamesizesuchthat ABAB =0,does itfollowthat BABA =0?[Putnam,1990,A-5]

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SectionIV.MatrixOperations 231 3.45 Demonstratethesefourassertionstogetanalternateproofthatcolumnrank equalsrowrank.[Am.Math.Mon.,Dec.1966] a ~y ~y = ~ 0i ~y = ~ 0. b A~x = ~ 0i A trans A~x = ~ 0. c dim R A =dim R A trans A d colrank A =colrank A trans =rowrank A 3.46 Provewhere A isan n n matrixandsodenesatransformationofany n -dimensionalspace V withrespectto B;B where B isabasisthatdim R A N A =dim R A )]TJ/F29 8.9664 Tf 9.216 0 Td [(dim R A 2 .Conclude a N A R A idim N A =dim R A )]TJ/F29 8.9664 Tf 9.216 0 Td [(dim R A 2 ; b R A N A i A 2 =0; c R A = N A i A 2 =0anddim N A =dim R A ; d dim R A N A =0idim R A =dim R A 2 ; e RequirestheDirectSumsubsection,whichisoptional. V = R A N A idim R A =dim R A 2 [Ackerson] IV.4Inverses Wenowconsiderhowtorepresenttheinverseofalinearmap. Westartbyrecallingsomefactsaboutfunctioninverses. Somefunctions havenoinverse,orhaveaninverseontheleftsideorrightsideonly. 4.1Example Where : R 3 R 2 istheprojectionmap 0 @ x y z 1 A 7! x y and : R 2 R 3 istheembedding x y 7! 0 @ x y 0 1 A thecomposition istheidentitymapon R 2 x y 7)167(! 0 @ x y 0 1 A 7)167(! x y Wesay isa leftinversemap of or,whatisthesamething,that isa right inversemap of .However,compositionintheotherorder doesn'tgive theidentitymap|hereisavectorthatisnotsenttoitselfunder 0 @ 0 0 1 1 A 7)167(! 0 0 7)167(! 0 @ 0 0 0 1 A Moreinformationonfunctioninversesisintheappendix.

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232 ChapterThree.MapsBetweenSpaces Infact,theprojection hasnoleftinverseatall.For,if f weretobealeft inverseof thenwewouldhave 0 @ x y z 1 A 7)167(! x y f 7)167(! 0 @ x y z 1 A foralloftheinnitelymany z 's.Butnofunction f cansendasingleargument tomorethanonevalue. Anexampleofafunctionwithnoinverseoneithersideisthezerotransformationon R 2 .Somefunctionshavea two-sidedinversemap ,anotherfunction thatistheinverseoftherst,bothfromtheleftandfromtheright.Forinstance,themapgivenby ~v 7! 2 ~v hasthetwo-sidedinverse ~v 7! = 2 ~v .In thissubsectionwewillfocusontwo-sidedinverses.Theappendixshowsthat afunctionhasatwo-sidedinverseifandonlyifitisbothone-to-oneandonto. Theappendixalsoshowsthatifafunction f hasatwo-sidedinversethenitis unique,andsoitiscalled`the'inverse,andisdenoted f )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 .Soourpurpose inthissubsectionis,wherealinearmap h hasaninverse,tondtherelationshipbetweenRep B;D h andRep D;B h )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 recallthatwehaveshown,in Theorem2.21ofSectionIIofthischapter,thatifalinearmaphasaninverse thentheinverseisalinearmapalso. 4.2Denition Amatrix G isa leftinversematrix ofthematrix H if GH is theidentitymatrix.Itisa rightinversematrix if HG istheidentity.Amatrix H withatwo-sidedinverseisan invertiblematrix .Thattwo-sidedinverseis called theinversematrix andisdenoted H )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Becauseofthecorrespondencebetweenlinearmapsandmatrices,statements aboutmapinversestranslateintostatementsaboutmatrixinverses. 4.3Lemma Ifamatrixhasbothaleftinverseandarightinversethenthe twoareequal. 4.4Theorem Amatrixisinvertibleifandonlyifitisnonsingular. Proof .Forbothresults. Givenamatrix H ,xspacesofappropriatedimensionforthedomainandcodomain.Fixbasesforthesespaces.Withrespectto thesebases, H representsamap h .Thestatementsaretrueaboutthemapand thereforetheyaretrueaboutthematrix. QED 4.5Lemma Aproductofinvertiblematricesisinvertible|if G and H are invertibleandif GH isdenedthen GH isinvertibleand GH )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = H )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 G )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Proof .Thisisjustlikethepriorproofexceptthatitrequirestwomaps. Fix appropriatespacesandbasesandconsidertherepresentedmaps h and g .Note that h )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 g )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 isatwo-sidedmapinverseof gh since h )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 g )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 gh = h )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 id h = h )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 h =idand gh h )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 g )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = g id g )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = gg )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 =id.Thisequalityisreected inthematricesrepresentingthemaps,asrequired. QED

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SectionIV.MatrixOperations 233 Hereisthearrowdiagramgivingtherelationshipbetweenmapinversesand matrixinverses.Itisaspecialcaseofthediagramforfunctioncompositionand matrixmultiplication. V wrtB W wrtC V wrtB h H h )]TJ/F31 5.9776 Tf 5.757 0 Td [(1 H )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 id I Beyonditsplaceinourgeneralprogramofseeinghowtorepresentmap operations,anotherreasonforourinterestininversescomesfromsolvinglinear systems.Alinearsystemisequivalenttoamatrixequation,ashere. x 1 + x 2 =3 2 x 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 2 =2 11 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 x 1 x 2 = 3 2 Byxingspacesandbasese.g., R 2 ; R 2 and E 2 ; E 2 ,wetakethematrix H to representsomemap h .Thensolvingthesystemisthesameasasking:what domainvector ~x ismappedby h totheresult ~ d ?Ifwecouldinvert h thenwe couldsolvethesystembymultiplyingRep D;B h )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Rep D ~ d togetRep B ~x 4.6Example Wecanndaleftinverseforthematrixjustgiven mn pq 11 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 10 01 byusingGauss'methodtosolvetheresultinglinearsystem. m +2 n =1 m )]TJ/F11 9.9626 Tf 14.944 0 Td [(n =0 p +2 q =0 p )]TJ/F11 9.9626 Tf 14.943 0 Td [(q =1 Answer: m =1 = 3, n =1 = 3, p =2 = 3,and q = )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 = 3.Thismatrixisactually thetwo-sidedinverseof H ,ascaneasilybechecked.Withitwecansolvethe system abovebyapplyingtheinverse. x y = 1 = 31 = 3 2 = 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 3 3 2 = 5 = 3 4 = 3 4.7Remark Whysolvesystemsthisway,whenGauss'methodtakesless arithmeticthisassertioncanbemadeprecisebycountingthenumberofarithmeticoperations,ascomputeralgorithmdesignersdo?Beyonditsconceptual appealofttingintoourprogramofdiscoveringhowtorepresentthevarious mapoperations,solvinglinearsystemsbyusingthematrixinversehasatleast twoadvantages.

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234 ChapterThree.MapsBetweenSpaces First,oncetheworkofndinganinversehasbeendone,solvingasystem withthesamecoecientsbutdierentconstantsiseasyandfast:ifwechange theentriesontherightofthesystem thenwegetarelatedproblem 11 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 x y = 5 1 wtiharelatedsolutionmethod. x y = 1 = 31 = 3 2 = 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 3 5 1 = 2 3 Inapplications,solvingmanysystemshavingthesamematrixofcoecientsis common. Anotheradvantageofinversesisthatwecanexploreasystem'ssensitivity tochangesintheconstants.Forexample,tweakingthe3ontherightofthe system to 11 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 x 1 x 2 = 3 : 01 2 canbesolvedwiththeinverse. 1 = 31 = 3 2 = 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 3 3 : 01 2 = = 3 : 01+ = 3 = 3 : 01 )]TJ/F8 9.9626 Tf 9.962 0 Td [( = 3 toshowthat x 1 changesby1 = 3ofthetweakwhile x 2 movesby2 = 3ofthat tweak.Thissortofanalysisisused,forexample,todecidehowaccuratelydata mustbespeciedinalinearmodeltoensurethatthesolutionhasadesired accuracy. Wenishbydescribingthecomputationalprocedureusuallyusedtond theinversematrix. 4.8Lemma Amatrixisinvertibleifandonlyifitcanbewrittenasthe productofelementaryreductionmatrices.Theinversecanbecomputedby applyingtotheidentitymatrixthesamerowsteps,inthesameorder,asare usedtoGauss-Jordanreducetheinvertiblematrix. Proof Amatrix H isinvertibleifandonlyifitisnonsingularandthusGaussJordanreducestotheidentity.ByCorollary3.22thisreductioncanbedone withelementarymatrices R r R r )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 :::R 1 H = I .Thisequationgivesthetwo halvesoftheresult. First,elementarymatricesareinvertibleandtheirinversesarealsoelementary.Applying R )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 r totheleftofbothsidesofthatequation,then R )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 r )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ,etc., gives H astheproductofelementarymatrices H = R )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 1 R )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 r I the I is heretocoverthetrivial r =0case. Second,matrixinversesareuniqueandsocomparisonoftheaboveequation with H )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 H = I showsthat H )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = R r R r )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 :::R 1 I .Therefore,applying R 1 totheidentity,followedby R 2 ,etc.,yieldstheinverseof H QED

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SectionIV.MatrixOperations 235 4.9Example Tondtheinverseof 11 2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 wedoGauss-Jordanreduction,meanwhileperformingthesameoperationson theidentity.Forclericalconveniencewewritethematrixandtheidentitysideby-side,anddothereductionstepstogether. 11 10 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 01 )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 1 + 2 )167(! 11 10 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(21 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = 3 2 )167(! 11 10 01 2 = 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 3 )]TJ/F10 6.9738 Tf 6.227 0 Td [( 2 + 1 )167(! 10 1 = 31 = 3 01 2 = 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 3 Thiscalculationhasfoundtheinverse. 11 2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 1 = 31 = 3 2 = 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 3 4.10Example Thisonehappenstostartwitharowswap. 0 @ 03 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 100 101 010 1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(10 001 1 A 1 $ 2 )167(! 0 @ 101 010 03 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 100 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 001 1 A )]TJ/F10 6.9738 Tf 6.226 0 Td [( 1 + 3 )167(! 0 @ 101 010 03 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 100 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 0 )]TJ/F8 9.9626 Tf 7.748 0 Td [(11 1 A )167(! 0 @ 100 1 = 41 = 43 = 4 010 1 = 41 = 4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 4 001 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 43 = 4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 = 4 1 A 4.11Example Anon-invertiblematrixisdetectedbythefactthattheleft halfwon'treducetotheidentity. 11 10 22 01 )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 1 + 2 )167(! 11 10 00 )]TJ/F8 9.9626 Tf 7.748 0 Td [(21 Thisprocedurewillndtheinverseofageneral n n matrix.The2 2case ishandy. 4.12Corollary Theinversefora2 2matrixexistsandequals ab cd )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 1 ad )]TJ/F11 9.9626 Tf 9.962 0 Td [(bc d )]TJ/F11 9.9626 Tf 7.748 0 Td [(b )]TJ/F11 9.9626 Tf 7.749 0 Td [(ca ifandonlyif ad )]TJ/F11 9.9626 Tf 9.963 0 Td [(bc 6 =0.

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236 ChapterThree.MapsBetweenSpaces Proof ThiscomputationisExercise22. QED Wehaveseenhere,asintheMechanicsofMatrixMultiplicationsubsection, thatwecanexploitthecorrespondencebetweenlinearmapsandmatrices.So wecanfruitfullystudybothmapsandmatrices,translatingbackandforthto whicheverhelpsusthemost. Overtheentirefoursubsectionsofthissectionwehavedevelopedanalgebra systemformatrices.Wecancompareitwiththefamiliaralgebrasystemfor therealnumbers.Hereweareworkingnotwithnumbersbutwithmatrices. Wehavematrixadditionandsubtractionoperations,andtheyworkinmuch thesamewayastherealnumberoperations,exceptthattheyonlycombine same-sizedmatrices.Wealsohaveamatrixmultiplicationoperationandan operationinversetomultiplication.Thesearesomewhatlikethefamiliarreal numberoperationsassociativity,anddistributivityoveraddition,forexample, buttherearedierencesfailureofcommutativity,forexample.And,wehave scalarmultiplication,whichisinsomewaysanotherextensionofrealnumber multiplication.Thismatrixsystemprovidesanexamplethatalgebrasystems otherthantheelementaryonecanbeinterestinganduseful. Exercises 4.13 SupplytheintermediatestepsinExample4.10. X 4.14 UseCorollary4.12todecideifeachmatrixhasaninverse. a 21 )]TJ/F29 8.9664 Tf 7.168 0 Td [(11 b 04 1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(3 c 2 )]TJ/F29 8.9664 Tf 7.167 0 Td [(3 )]TJ/F29 8.9664 Tf 7.168 0 Td [(46 X 4.15 Foreachinvertiblematrixinthepriorproblem,useCorollary4.12tondits inverse. X 4.16 Findtheinverse,ifitexists,byusingtheGauss-Jordanmethod.Checkthe answersforthe2 2matriceswithCorollary4.12. a 31 02 b 21 = 2 31 c 2 )]TJ/F29 8.9664 Tf 7.168 0 Td [(4 )]TJ/F29 8.9664 Tf 7.168 0 Td [(12 d 113 024 )]TJ/F29 8.9664 Tf 7.168 0 Td [(110 e 015 0 )]TJ/F29 8.9664 Tf 7.167 0 Td [(24 23 )]TJ/F29 8.9664 Tf 7.167 0 Td [(2 f 223 1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(2 )]TJ/F29 8.9664 Tf 7.167 0 Td [(3 4 )]TJ/F29 8.9664 Tf 7.167 0 Td [(2 )]TJ/F29 8.9664 Tf 7.167 0 Td [(3 X 4.17 Whatmatrixhasthisoneforitsinverse? 13 25 4.18 Howdoestheinverseoperationinteractwithscalarmultiplicationandadditionofmatrices? a Whatistheinverseof rH ? b Is H + G )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 = H )]TJ/F31 5.9776 Tf 5.757 0 Td [(1 + G )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 ? X 4.19 Is T k )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 = T )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 k ? 4.20 Is H )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 invertible? 4.21 Foreachrealnumber let t : R 2 R 2 berepresentedwithrespecttothe standardbasesbythismatrix. cos )]TJ/F29 8.9664 Tf 8.703 0 Td [(sin sin cos Showthat t 1 + 2 = t 1 t 2 .Showalsothat t )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 = t )]TJ/F33 5.9776 Tf 5.756 0 Td [( .

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SectionIV.MatrixOperations 237 4.22 DothecalculationsfortheproofofCorollary4.12. 4.23 Showthatthismatrix H = 101 010 hasinnitelymanyrightinverses.Showalsothatithasnoleftinverse. 4.24 InExample4.1,howmanyleftinverseshas ? 4.25 Ifamatrixhasinnitelymanyright-inverses,canithaveinnitelymany left-inverses?Mustithave? X 4.26 Assumethat H isinvertibleandthat HG isthezeromatrix.Showthat G is azeromatrix. 4.27 Provethatif H isinvertiblethentheinversecommuteswithamatrix GH )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 = H )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 G ifandonlyif H itselfcommuteswiththatmatrix GH = HG X 4.28 Showthatif T issquareandif T 4 isthezeromatrixthen I )]TJ/F32 8.9664 Tf 10.153 0 Td [(T )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 = I + T + T 2 + T 3 .Generalize. X 4.29 Let D bediagonal.Describe D 2 D 3 ,...,etc.Describe D )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 D )]TJ/F31 5.9776 Tf 5.757 0 Td [(2 ,...,etc. Dene D 0 appropriately. 4.30 Provethatanymatrixrow-equivalenttoaninvertiblematrixisalsoinvertible. 4.31 TherstquestionbelowappearedasExercise28. a Showthattherankoftheproductoftwomatricesislessthanorequalto theminimumoftherankofeach. b Showthatif T and S aresquarethen TS = I ifandonlyif ST = I 4.32 Showthattheinverseofapermutationmatrixisitstranspose. 4.33 ThersttwopartsofthisquestionappearedasExercise25. a Showthat GH trans = H trans G trans b Asquarematrixis symmetric ifeach i;j entryequalsthe j;i entrythatis,if thematrixequalsitstranspose.Showthatthematrices HH trans and H trans H aresymmetric. c Showthattheinverseofthetransposeisthetransposeoftheinverse. d Showthattheinverseofasymmetricmatrixissymmetric. X 4.34 TheitemsstartingthisquestionappearedasExercise30. a Provethatthecompositionoftheprojections x ; y : R 3 R 3 isthezero mapdespitethatneitheristhezeromap. b Provethatthecompositionofthederivatives d 2 =dx 2 ;d 3 =dx 3 : P 4 !P 4 is thezeromapdespitethatneithermapisthezeromap. c Givematrixequationsrepresentingeachofthepriortwoitems. Whentwothingsmultiplytogivezerodespitethatneitheriszero,eachissaidto bea zerodivisor .Provethatnozerodivisorisinvertible. 4.35 Inrealnumberalgebra,thereareexactlytwonumbers,1and )]TJ/F29 8.9664 Tf 7.167 0 Td [(1,thatare theirownmultiplicativeinverse.Does H 2 = I haveexactlytwosolutionsfor2 2 matrices? 4.36 Istherelation`isatwo-sidedinverseof'transitive?Reexive?Symmetric? 4.37 Prove:ifthesumoftheelementsofasquarematrixis k ,thenthesumofthe elementsineachrowoftheinversematrixis1 =k .[Am.Math.Mon.,Nov.1951]

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238 ChapterThree.MapsBetweenSpaces VChangeofBasis Representations,whetherofvectorsorofmaps,varywiththebases.Forinstance,withrespecttothetwobases E 2 and B = h 1 1 ; 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 i for R 2 ,thevector ~e 1 hastwodierentrepresentations. Rep E 2 ~e 1 = 1 0 Rep B ~e 1 = 1 = 2 1 = 2 Similarly,withrespectto E 2 ; E 2 and E 2 ;B ,theidentitymaphastwodierent representations. Rep E 2 ; E 2 id= 10 01 Rep E 2 ;B id= 1 = 21 = 2 1 = 2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 = 2 Withourpointofviewthattheobjectsofourstudiesarevectorsandmaps,in xingbasesweareadoptingaschemeoftagsornamesfortheseobjects,that areconvienentforcomputation.Wewillnowseehowtotranslateamongthese names|wewillseeexactlyhowrepresentationsvaryasthebasesvary. V.1ChangingRepresentationsofVectors InconvertingRep B ~v toRep D ~v theunderlyingvector ~v doesn'tchange. Thus,thistranslationisaccomplishedbytheidentitymaponthespace,describedsothatthedomainspacevectorsarerepresentedwithrespectto B and thecodomainspacevectorsarerepresentedwithrespectto D V w.r.t. B id ? ? y V w.r.t. D Thediagramisverticaltotwiththeonesinthenextsubsection. 1.1Denition The changeofbasismatrix forbases B;D V istherepresentationoftheidentitymapid: V V withrespecttothosebases. Rep B;D id= 0 B B @ Rep D ~ 1 Rep D ~ n 1 C C A

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SectionV.ChangeofBasis 239 1.2Lemma Left-multiplicationbythechangeofbasismatrixfor B;D converts arepresentationwithrespectto B toonewithrespectto D .Conversly,ifleftmultiplicationbyamatrixchangesbases M Rep B ~v =Rep D ~v then M isa changeofbasismatrix. Proof Fortherstsentence,foreach ~v ,asmatrix-vectormultiplicationrepresentsamapapplication,Rep B;D id Rep B ~v =Rep D id ~v =Rep D ~v .For thesecondsentence,withrespectto B;D thematrix M representssomelinear map,whoseactionis ~v 7! ~v ,andisthereforetheidentitymap. QED 1.3Example Withthesebasesfor R 2 B = h 2 1 ; 1 0 i D = h )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 ; 1 1 i because Rep D id 2 1 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 3 = 2 D Rep D id 1 0 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 1 = 2 D thechangeofbasismatrixisthis. Rep B;D id= )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 = 2 3 = 21 = 2 Wecanseethismatrixatworkbyndingthetworepresentationsof ~e 2 Rep B 0 1 = 1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 Rep D 0 1 = 1 = 2 1 = 2 andcheckingthattheconversiongoesasexpected. )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 = 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 3 = 21 = 2 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 = 1 = 2 1 = 2 Wenishthissubsectionbyrecognizingthatthechangeofbasismatrices arefamiliar. 1.4Lemma Amatrixchangesbasesifandonlyifitisnonsingular. Proof Foronedirection,ifleft-multiplicationbyamatrixchangesbasesthen thematrixrepresentsaninvertiblefunction,simplybecausethefunctionis invertedbychangingthebasesback.Suchamatrixisitselfinvertible,andso nonsingular. Tonish,wewillshowthatanynonsingularmatrix M performsachangeof basisoperationfromanygivenstartingbasis B tosomeendingbasis.Because thematrixisnonsingular,itwillGauss-Jordanreducetotheidentity,sothere areelementatryreductionmatricessuchthat R r R 1 M = I .Elementary matricesareinvertibleandtheirinversesarealsoelementary,somultiplying fromtheleftrstby R r )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 ,thenby R r )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ,etc.,gives M asaproductof

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240 ChapterThree.MapsBetweenSpaces elementarymatrices M = R 1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 R r )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 .Thus,wewillbedoneifweshow thatelementarymatriceschangeagivenbasistoanotherbasis,forthen R r )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 changes B tosomeotherbasis B r ,and R r )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 changes B r tosome B r )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ...,andtheneteectisthat M changes B to B 1 .Wewillprovethisabout elementarymatricesbycoveringthethreetypesasseparatecases. Applyingarow-multiplicationmatrix M i k 0 B B B B B B @ c 1 c i c n 1 C C C C C C A = 0 B B B B B B @ c 1 kc i c n 1 C C C C C C A changesarepresentationwithrespectto h ~ 1 ;:::; ~ i ;:::; ~ n i toonewithrespect to h ~ 1 ;:::; =k ~ i ;:::; ~ n i inthisway. ~v = c 1 ~ 1 + + c i ~ i + + c n ~ n 7! c 1 ~ 1 + + kc i =k ~ i + + c n ~ n = ~v Similarly,left-multiplicationbyarow-swapmatrix P i;j changesarepresentation withrespecttothebasis h ~ 1 ;:::; ~ i ;:::; ~ j ;:::; ~ n i intoonewithrespecttothe basis h ~ 1 ;:::; ~ j ;:::; ~ i ;:::; ~ n i inthisway. ~v = c 1 ~ 1 + + c i ~ i + + c j ~ j + + c n ~ n 7! c 1 ~ 1 + + c j ~ j + + c i ~ i + + c n ~ n = ~v And,arepresentationwithrespectto h ~ 1 ;:::; ~ i ;:::; ~ j ;:::; ~ n i changesvia left-multiplicationbyarow-combinationmatrix C i;j k intoarepresentation withrespectto h ~ 1 ;:::; ~ i )]TJ/F11 9.9626 Tf 9.963 0 Td [(k ~ j ;:::; ~ j ;:::; ~ n i ~v = c 1 ~ 1 + + c i ~ i + c j ~ j + + c n ~ n 7! c 1 ~ 1 + + c i ~ i )]TJ/F11 9.9626 Tf 9.963 0 Td [(k ~ j + + kc i + c j ~ j + + c n ~ n = ~v thedenitionofreductionmatricesspeciesthat i 6 = k and k 6 =0andsothis lastoneisabasis. QED 1.5Corollary Amatrixisnonsingularifandonlyifitrepresentstheidentity mapwithrespecttosomepairofbases. Inthenextsubsectionwewillseehowtotranslateamongrepresentations ofmaps,thatis,howtochangeRep B;D h toRep ^ B; ^ D h .Theabovecorollary isaspecialcaseofthis,wherethedomainandrangearethesamespace,and wherethemapistheidentitymap.

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SectionV.ChangeofBasis 241 Exercises X 1.6 In R 2 ,where D = h 2 1 ; )]TJ/F29 8.9664 Tf 7.167 0 Td [(2 4 i ndthechangeofbasismatricesfrom D to E 2 andfrom E 2 to D .Multiplythe two. X 1.7 Findthechangeofbasismatrixfor B;D R 2 a B = E 2 D = h ~e 2 ;~e 1 i b B = E 2 D = h 1 2 ; 1 4 i c B = h 1 2 ; 1 4 i D = E 2 d B = h )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 1 ; 2 2 i D = h 0 4 ; 1 3 i 1.8 ForthebasesinExercise7,ndthechangeofbasismatrixintheotherdirection, from D to B X 1.9 Findthechangeofbasismatrixforeach B;D P 2 a B = h 1 ;x;x 2 i ;D = h x 2 ; 1 ;x i b B = h 1 ;x;x 2 i ;D = h 1 ; 1+ x; 1+ x + x 2 i c B = h 2 ; 2 x;x 2 i ;D = h 1+ x 2 ; 1 )]TJ/F32 8.9664 Tf 9.216 0 Td [(x 2 ;x + x 2 i X 1.10 Decideifeachchangesbaseson R 2 .Towhatbasisis E 2 changed? a 50 04 b 21 31 c )]TJ/F29 8.9664 Tf 7.168 0 Td [(14 2 )]TJ/F29 8.9664 Tf 7.168 0 Td [(8 d 1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 11 1.11 Findbasessuchthatthismatrixrepresentstheidentitymapwithrespectto thosebases. 314 2 )]TJ/F29 8.9664 Tf 7.167 0 Td [(11 004 1.12 Considethevectorspaceofreal-valuedfunctionswithbasis h sin x ; cos x i Showthat h 2sin x +cos x ; 3cos x i isalsoabasisforthisspace.Findthechange ofbasismatrixineachdirection. 1.13 Wheredoesthismatrix cos sin sin )]TJ/F29 8.9664 Tf 8.703 0 Td [(cos sendthestandardbasisfor R 2 ?Anyotherbases? Hint. Considertheinverse. X 1.14 Whatisthechangeofbasismatrixwithrespectto B;B ? 1.15 Provethatamatrixchangesbasesifandonlyifitisinvertible. 1.16 FinishtheproofofLemma1.4. X 1.17 Let H bea n n nonsingularmatrix.Whatbasisof R n does H changetothe standardbasis? X 1.18a In P 3 withbasis B = h 1+ x; 1 )]TJ/F32 8.9664 Tf 9.02 0 Td [(x;x 2 + x 3 ;x 2 )]TJ/F32 8.9664 Tf 9.02 0 Td [(x 3 i wehavethisrepresenatation. Rep B )]TJ/F32 8.9664 Tf 9.215 0 Td [(x +3 x 2 )]TJ/F32 8.9664 Tf 9.215 0 Td [(x 3 = 0 B @ 0 1 1 2 1 C A B Findabasis D givingthisdierentrepresentationforthesamepolynomial. Rep D )]TJ/F32 8.9664 Tf 9.215 0 Td [(x +3 x 2 )]TJ/F32 8.9664 Tf 9.215 0 Td [(x 3 = 0 B @ 1 0 2 0 1 C A D

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242 ChapterThree.MapsBetweenSpaces b Stateandprovethatanynonzerovectorrepresentationcanbechangedto anyother. Hint. TheproofofLemma1.4isconstructive|itnotonlysaysthebaseschange, itshowshowtheychange. 1.19 Let V;W bevectorspaces,andlet B; ^ B bebasesfor V and D; ^ D bebasesfor W .Where h : V W islinear,ndaformularelatingRep B;D h toRep ^ B; ^ D h X 1.20 Showthatthecolumnsofan n n changeofbasismatrixformabasisfor R n .Doallbasesappearinthatway:canthevectorsfromany R n basismakethe columnsofachangeofbasismatrix? X 1.21 Findamatrixhavingthiseect. 1 3 7! 4 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 Thatis,nda M thatleft-multipliesthestartingvectortoyieldtheendingvector. Isthereamatrixhavingthesetwoeects? a 1 3 7! 1 1 2 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 7! )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 b 1 3 7! 1 1 2 6 7! )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 Giveanecessaryandsucientconditionfortheretobeamatrixsuchthat ~v 1 7! ~w 1 and ~v 2 7! ~w 2 V.2ChangingMapRepresentations Therstsubsectionshowshowtoconverttherepresentationofavectorwith respecttoonebasistotherepresentationofthatsamevectorwithrespectto anotherbasis.Herewewillseehowtoconverttherepresentationofamapwith respecttoonepairofbasestotherepresentationofthatmapwithrespectto adierentpair.Thatis,wewanttherelationshipbetweenthematricesinthis arrowdiagram. V w.r.t. B h )333()223()222()333(! H W w.r.t. D id ? ? y id ? ? y V w.r.t. ^ B h )333()223()222()333(! ^ H W w.r.t. ^ D Tomovefromthelower-leftofthisdiagramtothelower-rightwecaneithergo straightover,orelseupto V B thenoverto W D andthendown.Restatedin termsofthematrices,wecancalculate ^ H =Rep ^ B; ^ D h eitherbysimplyusing ^ B and ^ D ,orelsebyrstchangingbaseswithRep ^ B;B idthenmultiplying by H =Rep B;D h andthenchangingbaseswithRep D; ^ D id.Thisequation summarizes. ^ H =Rep D; ^ D id H Rep ^ B;B id Tocomparethisequationwiththesentencebeforeit,rememberthattheequationisreadfromrighttoleftbecausefunctioncompositionisreadrighttoleft andmatrixmultiplicationrepresentthecomposition.

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SectionV.ChangeofBasis 243 2.1Example Thematrix T = cos = 6 )]TJ/F8 9.9626 Tf 9.409 0 Td [(sin = 6 sin = 6cos = 6 = p 3 = 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 1 = 2 p 3 = 2 represents,withrespectto E 2 ; E 2 ,thetransformation t : R 2 R 2 thatrotates vectors = 6radianscounterclockwise. 1 3 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3+ p 3 = 2 +3 p 3 = 2 t = 6 )167(! Wecantranslatethatrepresentationwithrespectto E 2 ; E 2 toonewithrespect to ^ B = h 1 1 0 2 i ^ D = h )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 0 2 3 i byusingthearrowdiagramandformula above. R 2 w.r.t. E 2 t )333()223()222()333(! T R 2 w.r.t. E 2 id ? ? y id ? ? y R 2 w.r.t. ^ B t )333()223()222()333(! ^ T R 2 w.r.t. ^ D ^ T =Rep E 2 ; ^ D id T Rep ^ B; E 2 id NotethatRep E 2 ; ^ D idcanbecalculatedasthematrixinverseofRep ^ D; E 2 id. Rep ^ B; ^ D t = )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 03 )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 p 3 = 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 1 = 2 p 3 = 2 10 12 = )]TJ 9.963 8.242 Td [(p 3 = 6+2 p 3 = 3 + p 3 = 6 p 3 = 3 Althoughthenewmatrixismessier-appearing,themapthatitrepresentsisthe same.Forinstance,toreplicatetheeectof t inthepicture,startwith ^ B Rep ^ B 1 3 = 1 1 ^ B apply ^ T )]TJ 9.963 8.241 Td [(p 3 = 6+2 p 3 = 3 + p 3 = 6 p 3 = 3 ^ B; ^ D 1 1 ^ B = +3 p 3 = 6 +3 p 3 = 6 ^ D andcheckitagainst ^ D 11+3 p 3 6 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 0 + 1+3 p 3 6 2 3 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3+ p 3 = 2 +3 p 3 = 2 toseethatitisthesameresultasabove.

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244 ChapterThree.MapsBetweenSpaces 2.2Example On R 3 themap 0 @ x y z 1 A t 7)167(! 0 @ y + z x + z x + y 1 A thatisrepresentedwithrespecttothestandardbasisinthisway Rep E 3 ; E 3 t = 0 @ 011 101 110 1 A canalsoberepresentedwithrespecttoanotherbasis if B = h 0 @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 0 1 A ; 0 @ 1 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 1 A ; 0 @ 1 1 1 1 A i thenRep B;B t = 0 @ )]TJ/F8 9.9626 Tf 7.748 0 Td [(100 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 002 1 A inawaythatissimpler,inthattheactionofadiagonalmatrixiseasyto understand. Naturally,weusuallypreferbasischangesthatmaketherepresentationeasiertounderstand.Whentherepresentationwithrespecttoequalstartingand endingbasesisadiagonalmatrixwesaythemapormatrixhasbeen diagonalized .InChaperFiveweshallseewhichmapsandmatricesarediagonalizable, andwhereoneisnot,weshallseehowtogetarepresentationthatisnearly diagonal. Wenishthissubsectionbyconsideringtheeasiercasewhererepresentationsarewithrespecttopossiblydierentstartingandendingbases.Recall thatthepriorsubsectionshowsthatamatrixchangesbasesifandonlyifit isnonsingular.Thatgivesusanotherversionoftheabovearrowdiagramand equation 2.3Denition Same-sizedmatrices H and ^ H are matrixequivalent ifthere arenonsingularmatrices P and Q suchthat ^ H = PHQ 2.4Corollary Matrixequivalentmatricesrepresentthesamemap,withrespecttoappropriatepairsofbases. Exercise19checksthatmatrixequivalenceisanequivalencerelation.Thus itpartitionsthesetofmatricesintomatrixequivalenceclasses. Allmatrices: ... H ^ H H matrixequivalent to ^ H

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SectionV.ChangeofBasis 245 Wecangetsomeinsightintotheclassesbycomparingmatrixequivalencewith rowequivalencerecallthatmatricesarerowequivalentwhentheycanbereducedtoeachotherbyrowoperations.In ^ H = PHQ ,thematrices P and Q arenonsingularandthuseachcanbewrittenasaproductofelementary reductionmatricesLemma4.8.Left-multiplicationbythereductionmatrices makingup P hastheeectofperformingrowoperations.Right-multiplication bythereductionmatricesmakingup Q performscolumnoperations.Therefore, matrixequivalenceisageneralizationofrowequivalence|twomatricesarerow equivalentifonecanbeconvertedtotheotherbyasequenceofrowreduction steps,whiletwomatricesarematrixequivalentifonecanbeconvertedtothe otherbyasequenceofrowreductionstepsfollowedbyasequenceofcolumn reductionsteps. Thus,ifmatricesarerowequivalentthentheyarealsomatrixequivalent sincewecantake Q tobetheidentitymatrixandsoperformnocolumn operations.Theconverse,however,doesnothold. 2.5Example Thesetwo 10 00 11 00 arematrixequivalentbecausethesecondcanbereducedtotherstbythe columnoperationoftaking )]TJ/F8 9.9626 Tf 7.748 0 Td [(1timestherstcolumnandaddingtothesecond. Theyarenotrowequivalentbecausetheyhavedierentreducedechelonforms infact,botharealreadyinreducedform. Wewillclosethissectionbyndingasetofrepresentativesforthematrix equivalenceclasses. 2.6Theorem Any m n matrixofrank k ismatrixequivalenttothe m n matrixthatisallzerosexceptthattherst k diagonalentriesareones. 0 B B B B B B B B B B @ 10 ::: 00 ::: 0 01 ::: 00 ::: 0 00 ::: 10 ::: 0 00 ::: 00 ::: 0 00 ::: 00 ::: 0 1 C C C C C C C C C C A Sometimesthisisdescribedasa blockpartial-identity form. I Z Z Z Moreinformationonclassrepresentativesisintheappendix.

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246 ChapterThree.MapsBetweenSpaces Proof Asdiscussedabove,Gauss-Jordanreducethegivenmatrixandcombine allthereductionmatricesusedtheretomake P .Thenusetheleadingentriesto docolumnreductionandnishbyswappingcolumnstoputtheleadingoneson thediagonal.Combinethereductionmatricesusedforthosecolumnoperations into Q QED 2.7Example Weillustratetheproofbyndingthe P and Q forthismatrix. 0 @ 121 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 001 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 242 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 1 A FirstGauss-Jordanrow-reduce. 0 @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 010 001 1 A 0 @ 100 010 )]TJ/F8 9.9626 Tf 7.749 0 Td [(201 1 A 0 @ 121 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 001 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 242 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 1 A = 0 @ 1200 001 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 0000 1 A Thencolumn-reduce,whichinvolvesright-multiplication. 0 @ 1200 001 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 0000 1 A 0 B B @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(200 0100 0010 0001 1 C C A 0 B B @ 1000 0100 0011 0001 1 C C A = 0 @ 1000 0010 0000 1 A Finishbyswappingcolumns. 0 @ 1000 0010 0000 1 A 0 B B @ 1000 0010 0100 0001 1 C C A = 0 @ 1000 0100 0000 1 A Finally,combinetheleft-multiplierstogetheras P andtheright-multipliers togetheras Q togetthe PHQ equation. 0 @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 010 )]TJ/F8 9.9626 Tf 7.749 0 Td [(201 1 A 0 @ 121 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 001 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 242 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 1 A 0 B B @ 10 )]TJ/F8 9.9626 Tf 7.749 0 Td [(20 0010 0101 0001 1 C C A = 0 @ 1000 0100 0000 1 A 2.8Corollary Twosame-sizedmatricesarematrixequivalentifandonlyif theyhavethesamerank.Thatis,thematrixequivalenceclassesarecharacterizedbyrank. Proof Twosame-sizedmatriceswiththesamerankareequivalenttothesame blockpartial-identitymatrix. QED 2.9Example The2 2matriceshaveonlythreepossibleranks:zero,one, ortwo.Thustherearethreematrix-equivalenceclasses.

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SectionV.ChangeofBasis 247 All2 2matrices: ? )]TJ/F6 4.9813 Tf 5.789 -4.98 Td [(00 00 ? )]TJ/F6 4.9813 Tf 5.788 -4.98 Td [(10 00 ? )]TJ/F6 4.9813 Tf 5.788 -4.981 Td [(10 01 Threeequivalence classes Eachclassconsistsofallofthe2 2matriceswiththesamerank.Thereisonly onerankzeromatrix,sothatclasshasonlyonemember,buttheothertwo classeseachhaveinnitelymanymembers. Inthissubsectionwehaveseenhowtochangetherepresentationofamap withrespecttoarstpairofbasestoonewithrespecttoasecondpair.That ledtoadenitiondescribingwhenmatricesareequivalentinthisway.Finally wenotedthat,withtheproperchoiceofpossiblydierentstartingandending bases,anymapcanberepresentedinblockpartial-identityform. Oneofthenicethingsaboutthisrepresentationisthat,insomesense,we cancompletelyunderstandthemapwhenitisexpressedinthisway:ifthe basesare B = h ~ 1 ;:::; ~ n i and D = h ~ 1 ;:::; ~ m i thenthemapsends c 1 ~ 1 + + c k ~ k + c k +1 ~ k +1 + + c n ~ n 7)167(! c 1 ~ 1 + + c k ~ k + ~ 0+ + ~ 0 where k isthemap'srank.Thus,wecanunderstandanylinearmapasakind ofprojection. 0 B B B B B B B B @ c 1 c k c k +1 c n 1 C C C C C C C C A B 7! 0 B B B B B B B B @ c 1 c k 0 0 1 C C C C C C C C A D Ofcourse,understanding"amapexpressedinthiswayrequiresthatweunderstandtherelationshipbetween B and D .However,despitethatdiculty, thisisagoodclassicationoflinearmaps. Exercises X 2.10 Decideifthesematricesarematrixequivalent. a 130 230 221 05 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 b 03 11 40 05 c 13 26 13 2 )]TJ/F29 8.9664 Tf 7.168 0 Td [(6 X 2.11 Findthecanonicalrepresentativeofthematrix-equivalenceclassofeachmatrix.

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248 ChapterThree.MapsBetweenSpaces a 210 420 b 0102 1104 333 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 2.12 Supposethat,withrespectto B = E 2 D = h 1 1 ; 1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 i thetransformation t : R 2 R 2 isrepresentedbythismatrix. 12 34 Usechangeofbasismatricestorepresent t withrespecttoeachpair. a ^ B = h 0 1 ; 1 1 i ^ D = h )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 0 ; 2 1 i b ^ B = h 1 2 ; 1 0 i ^ D = h 1 2 ; 2 1 i X 2.13 Whatsizesare P and Q intheequation ^ H = PHQ ? X 2.14 UseTheorem2.6toshowthatasquarematrixisnonsingularifandonlyifit isequivalenttoanidentitymatrix. X 2.15 Showthat,where A isanonsingularsquarematrix,if P and Q arenonsingular squarematricessuchthat PAQ = I then QP = A )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 X 2.16 WhydoesTheorem2.6notshowthateverymatrixisdiagonalizablesee Example2.2? 2.17 Mustmatrixequivalentmatriceshavematrixequivalenttransposes? 2.18 WhathappensinTheorem2.6if k =0? X 2.19 Showthatmatrix-equivalenceisanequivalencerelation. X 2.20 Showthatazeromatrixisaloneinitsmatrixequivalenceclass.Arethere othermatriceslikethat? 2.21 Whatarethematrixequivalenceclassesofmatricesoftransformationson R 1 ? R 3 ? 2.22 Howmanymatrixequivalenceclassesarethere? 2.23 Arematrixequivalenceclassesclosedunderscalarmultiplication?Addition? 2.24 Let t : R n R n representedby T withrespectto E n ; E n a FindRep B;B t inthisspeciccase. T = 11 3 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 B = h 1 2 ; )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 i b DescribeRep B;B t inthegeneralcasewhere B = h ~ 1 ;:::; ~ n i 2.25a Let V havebases B 1 and B 2 andsupposethat W hasthebasis D .Where h : V W ,ndtheformulathatcomputesRep B 2 ;D h fromRep B 1 ;D h b Repeatthepriorquestionwithonebasisfor V andtwobasesfor W 2.26a Iftwomatricesarematrix-equivalentandinvertible,musttheirinverses bematrix-equivalent? b Iftwomatriceshavematrix-equivalentinverses,mustthetwobematrixequivalent? c Iftwomatricesaresquareandmatrix-equivalent,musttheirsquaresbe matrix-equivalent? d Iftwomatricesaresquareandhavematrix-equivalentsquares,musttheybe matrix-equivalent?

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SectionV.ChangeofBasis 249 X 2.27 Squarematricesare similar iftheyrepresentthesametransformation,but eachwithrespecttothesameendingasstartingbasis.Thatis,Rep B 1 ;B 1 t is similartoRep B 2 ;B 2 t a GiveadenitionofmatrixsimilaritylikethatofDenition2.3. b Provethatsimilarmatricesarematrixequivalent. c Showthatsimilarityisanequivalencerelation. d Showthatif T issimilarto ^ T then T 2 issimilarto ^ T 2 ,thecubesaresimilar, etc. Contrastwiththepriorexercise. e Provethattherearematrixequivalentmatricesthatarenotsimilar.

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250 ChapterThree.MapsBetweenSpaces VIProjection Thissectionisoptional;onlythelasttwosectionsofChapterFiverequirethis material. Wehavedescribedtheprojection from R 3 intoits xy planesubspaceas a`shadowmap'.Thisshowswhy,butitalsoshowsthatsomeshadowsfall upward. 1 2 2 1 2 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Soperhapsabetterdescriptionis:theprojectionof ~v isthe ~p intheplanewith thepropertythatsomeonestandingon ~p andlookingstraightupordownsees ~v .Inthissectionwewillgeneralizethistootherprojections,bothorthogonal i.e.,`straightupanddown'andnonorthogonal. VI.1OrthogonalProjectionIntoaLine Werstconsiderorthogonalprojectionintoaline.Toorthogonallyprojecta vector ~v intoaline ` ,darkenapointonthelineifsomeoneonthatlineand lookingstraightupordownfromthatperson'spointofviewsees ~v Thepictureshowssomeonewhohaswalkedoutonthelineuntilthetipof ~v isstraightoverhead.Thatis,wherethelineisdescribedasthespanof somenonzerovector ` = f c ~s c 2 R g ,thepersonhaswalkedouttondthe coecient c ~p withthepropertythat ~v )]TJ/F11 9.9626 Tf 9.962 0 Td [(c ~p ~s isorthogonalto c ~p ~s c ~p ~s ~v ~v )]TJ/F32 8.9664 Tf 9.215 0 Td [(c ~p ~s Wecansolveforthiscoecientbynotingthatbecause ~v )]TJ/F11 9.9626 Tf 9.349 0 Td [(c ~p ~s isorthogonalto ascalarmultipleof ~s itmustbeorthogonalto ~s itself,andthentheconsequent factthatthedotproduct ~v )]TJ/F11 9.9626 Tf 9.962 0 Td [(c ~p ~s ~s iszerogivesthat c ~p = ~v ~s=~s ~s .

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SectionVI.Projection 251 1.1Denition The orthogonalprojectionof ~v intothelinespannedbya nonzero ~s isthisvector. proj [ ~s ] ~v = ~v ~s ~s ~s ~s Exercise19checksthattheoutcomeofthecalculationdependsonlyontheline andnotonwhichvector ~s happenstobeusedtodescribethatline. 1.2Remark Thewordingofthatdenitionsays`spannedby ~s 'insteadthe moreformal`thespanoftheset f ~s g '.Thiscasualrstphraseiscommon. 1.3Example Toorthogonallyprojectthevector )]TJ/F7 6.9738 Tf 4.567 -3.649 Td [(2 3 intotheline y =2 x ,we rstpickadirectionvectorfortheline.Forinstance, ~s = 1 2 willdo.Thenthecalculationisroutine. 2 3 1 2 1 2 1 2 1 2 = 8 5 1 2 = 8 = 5 16 = 5 1.4Example In R 3 ,theorthogonalprojectionofageneralvector 0 @ x y z 1 A intothe y -axisis 0 @ x y z 1 A 0 @ 0 1 0 1 A 0 @ 0 1 0 1 A 0 @ 0 1 0 1 A 0 @ 0 1 0 1 A = 0 @ 0 y 0 1 A whichmatchesourintuitiveexpectation. Thepictureabovewiththestickgurewalkingoutonthelineuntil ~v 'stip isoverheadisonewaytothinkoftheorthogonalprojectionofavectorintoa line.Wenishthissubsectionwithtwootherways. 1.5Example Arailroadcarleftonaneast-westtrackwithoutitsbrakeis pushedbyawindblowingtowardthenortheastatfteenmilesperhour;what speedwillthecarreach?

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252 ChapterThree.MapsBetweenSpaces Forthewindweuseavectoroflength15thatpointstowardthenortheast. ~v = 15 p 1 = 2 15 p 1 = 2 Thecarcanonlybeaectedbythepartofthewindblowingintheeast-west direction|thepartof ~v inthedirectionofthe x -axisisthisthepicturehas thesameperspectiveastherailroadcarpictureabove. east north ~p = 15 p 1 = 2 0 Sothecarwillreachavelocityof15 p 1 = 2milesperhourtowardtheeast. Thus,anotherwaytothinkofthepicturethatprecedesthedenitionisthat itshows ~v asdecomposedintotwoparts,thepartwiththelinehere,thepart withthetracks, ~p ,andthepartthatisorthogonaltothelineshownherelying onthenorth-southaxis.Thesetwoarenotinteracting"orindependent",in thesensethattheeast-westcarisnotatallaectedbythenorth-southpart ofthewindseeExercise11.Sotheorthogonalprojectionof ~v intotheline spannedby ~s canbethoughtofasthepartof ~v thatliesinthedirectionof ~s Finally,anotherusefulwaytothinkoftheorthogonalprojectionistohave thepersonstandnotontheline,butonthevectorthatistobeprojectedtothe line.Thispersonhasaropeoverthelineandpullsittight,naturallymaking theropeorthogonaltotheline. Thatis,wecanthinkoftheprojection ~p asbeingthevectorinthelinethatis closestto ~v seeExercise17. 1.6Example Asubmarineistrackingashipmovingalongtheline y =3 x +2. Torpedorangeisone-halfmile.Canthesubstaywhereitis,attheoriginon thechartbelow,ormustitmovetoreachaplacewheretheshipwillpasswithin range? east north

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SectionVI.Projection 253 Theformulaforprojectionintoalinedoesnotimmediatelyapplybecausethe linedoesn'tpassthroughtheorigin,andsoisn'tthespanofany ~s .Toadjust forthis,westartbyshiftingtheentiremapdowntwounits.Nowthelineis y =3 x ,whichisasubspace,andwecanprojecttogetthepoint ~p ofclosest approach,thepointonthelinethroughtheoriginclosestto ~v = 0 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 thesub'sshiftedposition. ~p = 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 1 3 1 3 1 3 1 3 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 = 5 )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 = 5 Thedistancebetween ~v and ~p isapproximately0 : 63milesandsothesubmust movetogetinrange. Thissubsectionhasdevelopedanaturalprojectionmap:orthogonalprojectionintoaline.Assuggestedbytheexamples,itisoftencalledforinapplications.Thenextsubsectionshowshowthedenitionoforthogonalprojection intoalinegivesusawaytocalculateespeciallyconvienentbasesforvector spaces,againsomethingthatiscommoninapplications.Thenalsubsection completelygeneralizesprojection,orthogonalornot,intoanysubspaceatall. Exercises X 1.7 Projecttherstvectororthogonallyintothelinespannedbythesecondvector. a 2 1 3 )]TJ/F29 8.9664 Tf 7.168 0 Td [(2 b 2 1 3 0 c 1 1 4 1 2 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 d 1 1 4 3 3 12 X 1.8 Projectthevectororthogonallyintotheline. a 2 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 4 ; f c )]TJ/F29 8.9664 Tf 7.167 0 Td [(3 1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(3 c 2 R g b )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 ,theline y =3 x 1.9 AlthoughthedevelopmentofDenition1.1isguidedbythepictures,weare notrestrictedtospacesthatwecandraw.In R 4 projectthisvectorintothisline. ~v = 0 B @ 1 2 1 3 1 C A ` = f c 0 B @ )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 1 1 C A c 2 R g X 1.10 Denition1.1usestwovectors ~s and ~v .Considerthetransformationof R 2 resultingfromxing ~s = 3 1 andprojecting ~v intothelinethatisthespanof ~s .Applyittothesevectors.

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254 ChapterThree.MapsBetweenSpaces a 1 2 b 0 4 Showthatingeneraltheprojectiontranformationisthis. x 1 x 2 7! x 1 +3 x 2 = 10 x 1 +9 x 2 = 10 Expresstheactionofthistransformationwithamatrix. 1.11 Example1.5suggeststhatprojectionbreaks ~v intotwoparts,proj [ ~s ] ~v and ~v )]TJ/F29 8.9664 Tf 9.93 0 Td [(proj [ ~s ] ~v ,thatarenotinteracting".Recallthatthetwoareorthogonal. Showthatanytwononzeroorthogonalvectorsmakeupalinearlyindependent set. 1.12a Whatistheorthogonalprojectionof ~v intoalineif ~v isamemberof thatline? b Showthatif ~v isnotamemberofthelinethentheset f ~v;~v )]TJ/F29 8.9664 Tf 9.215 0 Td [(proj [ ~s ] ~v g is linearlyindependent. 1.13 Denition1.1requiresthat ~s benonzero.Why?Whatistherightdenition oftheorthogonalprojectionofavectorintothedegeneratelinespannedbythe zerovector? 1.14 Areallvectorstheprojectionofsomeothervectorintosomeline? X 1.15 Showthattheprojectionof ~v intothelinespannedby ~s haslengthequalto theabsolutevalueofthenumber ~v ~s dividedbythelengthofthevector ~s 1.16 Findtheformulaforthedistancefromapointtoaline. 1.17 Findthescalar c suchthat cs 1 ;cs 2 isaminimumdistancefromthepoint v 1 ;v 2 byusingcalculusi.e.,considerthedistancefunction,settherstderivative equaltozero,andsolve.Generalizeto R n X 1.18 Provethattheorthogonalprojectionofavectorintoalineisshorterthanthe vector. X 1.19 Showthatthedenitionoforthogonalprojectionintoalinedoesnotdepend onthespanningvector:if ~s isanonzeromultipleof ~q then ~v ~s=~s ~s ~s equals ~v ~q=~q ~q ~q X 1.20 Considerthefunctionmappingtoplanetoitselfthattakesavectortoits projectionintotheline y = x .Thesetwoeachshowthatthemapislinear,the rstoneinawaythatisboundtothecoordinatesthatis,itxesabasisand thencomputesandthesecondinawaythatismoreconceptual. a Produceamatrixthatdescribesthefunction'saction. b Showalsothatthismapcanbeobtainedbyrstrotatingeverythinginthe plane = 4radiansclockwise,thenprojectingintothe x -axis,andthenrotating = 4radianscounterclockwise. 1.21 For ~a; ~ b 2 R n let ~v 1 betheprojectionof ~a intothelinespannedby ~ b ,let ~v 2 be theprojectionof ~v 1 intothelinespannedby ~a ,let ~v 3 betheprojectionof ~v 2 into thelinespannedby ~ b ,etc.,backandforthbetweenthespansof ~a and ~ b .Thatis, ~v i +1 istheprojectionof ~v i intothespanof ~a if i +1iseven,andintothespanof ~ b if i +1isodd.Mustthatsequenceofvectorseventuallysettledown|mustthere beasucientlylarge i suchthat ~v i +2 equals ~v i and ~v i +3 equals ~v i +1 ?Ifso,what istheearliestsuch i ?

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SectionVI.Projection 255 VI.2Gram-SchmidtOrthogonalization Thissubsectionisoptional.Itrequiresmaterialfromtheprior,alsooptional, subsection.Theworkdoneherewillonlybeneededinthenaltwosectionsof ChapterFive. Thepriorsubsectionsuggeststhatprojectingintothelinespannedby ~s decomposesavector ~v intotwoparts proj [ ~s ] ~p ~v ~v )]TJ/F7 6.9738 Tf 8.041 0 Td [(proj [ ~s ] ~p ~v =proj [ ~s ] ~v + )]TJ/F32 8.9664 Tf 4.228 -7.821 Td [(~v )]TJ/F29 8.9664 Tf 9.216 0 Td [(proj [ ~s ] ~v thatareorthogonalandsoarenotinteracting".Wewillnowdevelopthat suggestion. 2.1Denition Vectors ~v 1 ;:::;~v k 2 R n are mutuallyorthogonal whenany twoareorthogonal:if i 6 = j thenthedotproduct ~v i ~v j iszero. 2.2Theorem Ifthevectorsinaset f ~v 1 ;:::;~v k g R n aremutuallyorthogonalandnonzerothenthatsetislinearlyindependent. Proof Consideralinearrelationship c 1 ~v 1 + c 2 ~v 2 + + c k ~v k = ~ 0.If i 2 [1 ::k ] thentakingthedotproductof ~v i withbothsidesoftheequation ~v i c 1 ~v 1 + c 2 ~v 2 + + c k ~v k = ~v i ~ 0 c i ~v i ~v i =0 shows,since ~v i isnonzero,that c i iszero. QED 2.3Corollary Ifthevectorsinasize k subsetofa k dimensionalspaceare mutuallyorthogonalandnonzerothenthatsetisabasisforthespace. Proof Anylinearlyindependentsize k subsetofa k dimensionalspaceisa basis. QED Ofcourse,theconverseofCorollary2.3doesnothold|noteverybasisof everysubspaceof R n ismadeofmutuallyorthogonalvectors.However,wecan getthepartialconversethatforeverysubspaceof R n thereisatleastonebasis consistingofmutuallyorthogonalvectors. 2.4Example Themembers ~ 1 and ~ 2 ofthisbasisfor R 2 arenotorthogonal. B = h 4 2 ; 1 3 i ~ 1 ~ 2

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256 ChapterThree.MapsBetweenSpaces However,wecanderivefrom B anewbasisforthesamespacethatdoeshave mutuallyorthogonalmembers.Fortherstmemberofthenewbasiswesimply use ~ 1 ~ 1 = 4 2 Forthesecondmemberofthenewbasis,wetakeawayfrom ~ 2 itspartinthe directionof ~ 1 ~ 2 = 1 3 )]TJ/F29 8.9664 Tf 9.215 0 Td [(proj [ ~ 1 ] 1 3 = 1 3 )]TJ/F1 9.9626 Tf 9.216 13.798 Td [( 2 1 = )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 2 ~ 2 whichleavesthepart, ~ 2 picturedabove,of ~ 2 thatisorthogonalto ~ 1 itis orthogonalbythedenitionoftheprojectionintothespanof ~ 1 .Notethat, bythecorollary, f ~ 1 ;~ 2 g isabasisfor R 2 2.5Denition An orthogonalbasis foravectorspaceisabasisofmutually orthogonalvectors. 2.6Example Toturnthisbasisfor R 3 h 0 @ 1 1 1 1 A ; 0 @ 0 2 0 1 A ; 0 @ 1 0 3 1 A i intoanorthogonalbasis,wetaketherstvectorasitisgiven. ~ 1 = 0 @ 1 1 1 1 A Weget ~ 2 bystartingwiththegivensecondvector ~ 2 andsubtractingawaythe partofitinthedirectionof ~ 1 ~ 2 = 0 @ 0 2 0 1 A )]TJ/F8 9.9626 Tf 9.963 0 Td [(proj [ ~ 1 ] 0 @ 0 2 0 1 A = 0 @ 0 2 0 1 A )]TJ/F1 9.9626 Tf 9.963 20.025 Td [(0 @ 2 = 3 2 = 3 2 = 3 1 A = 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 = 3 4 = 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 = 3 1 A Finally,weget ~ 3 bytakingthethirdgivenvectorandsubtractingthepartof itinthedirectionof ~ 1 ,andalsothepartofitinthedirectionof ~ 2 ~ 3 = 0 @ 1 0 3 1 A )]TJ/F8 9.9626 Tf 9.962 0 Td [(proj [ ~ 1 ] 0 @ 1 0 3 1 A )]TJ/F8 9.9626 Tf 9.963 0 Td [(proj [ ~ 2 ] 0 @ 1 0 3 1 A = 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 0 1 1 A

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SectionVI.Projection 257 Againthecorollarygivesthat h 0 @ 1 1 1 1 A ; 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 = 3 4 = 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 = 3 1 A ; 0 @ )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 0 1 1 A i isabasisforthespace. Thenextresultveriesthattheprocessusedinthoseexamplesworkswith anybasisforanysubspaceofan R n wearerestrictedto R n onlybecausewe havenotgivenadenitionoforthogonalityforothervectorspaces. 2.7TheoremGram-Schmidtorthogonalization If h ~ 1 ;::: ~ k i isabasis forasubspaceof R n then,where ~ 1 = ~ 1 ~ 2 = ~ 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(proj [ ~ 1 ] ~ 2 ~ 3 = ~ 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(proj [ ~ 1 ] ~ 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(proj [ ~ 2 ] ~ 3 ~ k = ~ k )]TJ/F8 9.9626 Tf 9.963 0 Td [(proj [ ~ 1 ] ~ k )-222()]TJ/F8 9.9626 Tf 33.762 0 Td [(proj [ ~ k )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 ] ~ k the ~ 'sformanorthogonalbasisforthesamesubspace. Proof Wewilluseinductiontocheckthateach ~ i isnonzero,isinthespanof h ~ 1 ;::: ~ i i andisorthogonaltoallprecedingvectors: ~ 1 ~ i = = ~ i )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ~ i =0. Withthose,andwithCorollary2.3,wewillhavethat h ~ 1 ;:::~ k i isabasisfor thesamespaceas h ~ 1 ;::: ~ k i Weshallcoverthecasesupto i =3,whichgivethesenseoftheargument. CompletingthedetailsisExercise23. The i =1caseistrivial|setting ~ 1 equalto ~ 1 makesitanonzerovector since ~ 1 isamemberofabasis,itisobviouslyinthedesiredspan,andthe `orthogonaltoallprecedingvectors'conditionisvacuouslymet. Forthe i =2case,expandthedenitionof ~ 2 ~ 2 = ~ 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(proj [ ~ 1 ] ~ 2 = ~ 2 )]TJ/F11 9.9626 Tf 11.865 9.369 Td [(~ 2 ~ 1 ~ 1 ~ 1 ~ 1 = ~ 2 )]TJ/F11 9.9626 Tf 11.865 9.369 Td [(~ 2 ~ 1 ~ 1 ~ 1 ~ 1 Thisexpansionshowsthat ~ 2 isnonzeroorelsethiswouldbeanon-triviallinear dependenceamongthe ~ 'sitisnontrivialbecausethecoecientof ~ 2 is1and alsoshowsthat ~ 2 isinthedesiredspan.Finally, ~ 2 isorthogonaltotheonly precedingvector ~ 1 ~ 2 = ~ 1 ~ 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(proj [ ~ 1 ] ~ 2 =0 becausethisprojectionisorthogonal.

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258 ChapterThree.MapsBetweenSpaces The i =3caseisthesameasthe i =2caseexceptforonedetail.Asinthe i =2case,expandingthedenition ~ 3 = ~ 3 )]TJ/F11 9.9626 Tf 11.864 9.369 Td [(~ 3 ~ 1 ~ 1 ~ 1 ~ 1 )]TJ/F11 9.9626 Tf 11.864 9.369 Td [(~ 3 ~ 2 ~ 2 ~ 2 ~ 2 = ~ 3 )]TJ/F11 9.9626 Tf 11.864 9.369 Td [(~ 3 ~ 1 ~ 1 ~ 1 ~ 1 )]TJ/F11 9.9626 Tf 11.864 9.369 Td [(~ 3 ~ 2 ~ 2 ~ 2 )]TJ/F11 9.9626 Tf 5.22 -5.441 Td [(~ 2 )]TJ/F11 9.9626 Tf 11.865 9.369 Td [(~ 2 ~ 1 ~ 1 ~ 1 ~ 1 showsthat ~ 3 isnonzeroandisinthespan.Acalculationshowsthat ~ 3 is orthogonaltotheprecedingvector ~ 1 ~ 1 ~ 3 = ~ 1 )]TJ/F11 9.9626 Tf 5.22 -5.441 Td [(~ 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(proj [ ~ 1 ] ~ 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(proj [ ~ 2 ] ~ 3 = ~ 1 )]TJ/F11 9.9626 Tf 5.22 -5.44 Td [(~ 3 )]TJ/F8 9.9626 Tf 9.962 0 Td [(proj [ ~ 1 ] ~ 3 )]TJ/F11 9.9626 Tf 9.576 0 Td [(~ 1 proj [ ~ 2 ] ~ 3 =0 Here'sthedierencefromthe i =2case|thesecondlinehastwokindsof terms.Thersttermiszerobecausethisprojectionisorthogonal,asinthe i =2case.Thesecondtermiszerobecause ~ 1 isorthogonalto ~ 2 andsois orthogonaltoanyvectorinthelinespannedby ~ 2 .Thecheckthat ~ 3 isalso orthogonaltotheotherprecedingvector ~ 2 issimilar. QED Beyondhavingthevectorsinthebasisbeorthogonal,wecandomore;we canarrangeforeachvectortohavelengthonebydividingeachbyitsownlength wecan normalize thelengths. 2.8Example Normalizingthelengthofeachvectorintheorthogonalbasisof Example2.6producesthis orthonormalbasis h 0 @ 1 = p 3 1 = p 3 1 = p 3 1 A ; 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = p 6 2 = p 6 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = p 6 1 A ; 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = p 2 0 1 = p 2 1 A i Besidesitsintuitiveappeal,anditsanalogywiththestandardbasis E n for R n anorthonormalbasisalsosimpliessomecomputations.SeeExercise17,for example. Exercises 2.9 PerformtheGram-Schmidtprocessoneachofthesebasesfor R 2 a h 1 1 ; 2 1 i b h 0 1 ; )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 3 i c h 0 1 ; )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 0 i Thenturnthoseorthogonalbasesintoorthonormalbases. X 2.10 PerformtheGram-Schmidtprocessoneachofthesebasesfor R 3 .

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SectionVI.Projection 259 a h 2 2 2 ; 1 0 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 ; 0 3 1 i b h 1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 0 ; 0 1 0 ; 2 3 1 i Thenturnthoseorthogonalbasesintoorthonormalbases. X 2.11 Findanorthonormalbasisforthissubspaceof R 3 :theplane x )]TJ/F32 8.9664 Tf 9.215 0 Td [(y + z =0. 2.12 Findanorthonormalbasisforthissubspaceof R 4 f 0 B @ x y z w 1 C A x )]TJ/F32 8.9664 Tf 9.216 0 Td [(y )]TJ/F32 8.9664 Tf 9.215 0 Td [(z + w =0and x + z =0 g 2.13 Showthatanylinearlyindependentsubsetof R n canbeorthogonalizedwithoutchangingitsspan. X 2.14 WhathappensifweapplytheGram-Schmidtprocesstoabasisthatisalready orthogonal? 2.15 Let h ~ 1 ;:::;~ k i beasetofmutuallyorthogonalvectorsin R n a Provethatforany ~v inthespace,thevector ~v )]TJ/F29 8.9664 Tf 7.461 0 Td [(proj [ ~ 1 ] ~v + +proj [ ~v k ] ~v isorthogonaltoeachof ~ 1 ,..., ~ k b Illustratetheprioritemin R 3 byusing ~e 1 as ~ 1 ,using ~e 2 as ~ 2 ,andtaking ~v tohavecomponents1,2,and3. c Showthatproj [ ~ 1 ] ~v + +proj [ ~v k ] ~v isthevectorinthespanoftheset of ~ 'sthatisclosestto ~v Hint .Totheillustrationdoneforthepriorpart, addavector d 1 ~ 1 + d 2 ~ 2 andapplythePythagoreanTheoremtotheresulting triangle. 2.16 Findavectorin R 3 thatisorthogonaltobothofthese. 1 5 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 2 2 0 X 2.17 Oneadvantageoforthogonalbasesisthattheysimplifyndingtherepresentationofavectorwithrespecttothatbasis. a Forthisvectorandthisnon-orthogonalbasisfor R 2 ~v = 2 3 B = h 1 1 ; 1 0 i rstrepresentthevectorwithrespecttothebasis.Thenprojectthevectorinto thespanofeachbasisvector[ ~ 1 ]and[ ~ 2 ]. b Withthisorthogonalbasisfor R 2 K = h 1 1 ; 1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 i representthesamevector ~v withrespecttothebasis.Thenprojectthevector intothespanofeachbasisvector.Notethatthecoecientsintherepresentation andtheprojectionarethesame. c Let K = h ~ 1 ;:::;~ k i beanorthogonalbasisforsomesubspaceof R n .Prove thatforany ~v inthesubspace,the i -thcomponentoftherepresentationRep K ~v isthescalarcoecient ~v ~ i = ~ i ~ i fromproj [ ~ i ] ~v d Provethat ~v =proj [ ~ 1 ] ~v + +proj [ ~ k ] ~v 2.18 Bessel'sInequality .Considertheseorthonormalsets B 1 = f ~e 1 g B 2 = f ~e 1 ;~e 2 g B 3 = f ~e 1 ;~e 2 ;~e 3 g B 4 = f ~e 1 ;~e 2 ;~e 3 ;~e 4 g alongwiththevector ~v 2 R 4 whosecomponentsare4,3,2,and1. a Findthecoecient c 1 fortheprojectionof ~v intothespanofthevectorin B 1 .Checkthat k ~v k 2 j c 1 j 2 .

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260 ChapterThree.MapsBetweenSpaces b Findthecoecients c 1 and c 2 fortheprojectionof ~v intothespansofthe twovectorsin B 2 .Checkthat k ~v k 2 j c 1 j 2 + j c 2 j 2 c Find c 1 c 2 ,and c 3 associatedwiththevectorsin B 3 ,and c 1 c 2 c 3 ,and c 4 forthevectorsin B 4 .Checkthat k ~v k 2 j c 1 j 2 + + j c 3 j 2 andthat k ~v k 2 j c 1 j 2 + + j c 4 j 2 Showthatthisholdsingeneral:where f ~ 1 ;:::;~ k g isanorthonormalsetand c i is coecientoftheprojectionofavector ~v fromthespacethen k ~v k 2 j c 1 j 2 + + j c k j 2 Hint .Onewayistolookattheinequality0 k ~v )]TJ/F29 8.9664 Tf 9.423 0 Td [( c 1 ~ 1 + + c k ~ k k 2 andexpandthe c 's. 2.19 Proveordisprove:everyvectorin R n isinsomeorthogonalbasis. 2.20 Showthatthecolumnsofan n n matrixformanorthonormalsetifandonly iftheinverseofthematrixisitstranspose.Producesuchamatrix. 2.21 DoestheproofofTheorem2.2failtoconsiderthepossibilitythatthesetof vectorsisemptyi.e.,that k =0? 2.22 Theorem2.7describesachangeofbasisfromanybasis B = h ~ 1 ;:::; ~ k i to onethatisorthogonal K = h ~ 1 ;:::;~ k i .Considerthechangeofbasismatrix Rep B;K id. a ProvethatthematrixRep K;B idchangingbasesinthedirectionopposite tothatofthetheoremhasanuppertriangularshape|allofitsentriesbelow themaindiagonalarezeros. b Provethattheinverseofanuppertriangularmatrixisalsouppertriangular ifthematrixisinvertible,thatis.ThisshowsthatthematrixRep B;K id changingbasesinthedirectiondescribedinthetheoremisuppertriangular. 2.23 CompletetheinductionargumentintheproofofTheorem2.7. VI.3ProjectionIntoaSubspace Thissubsection,liketheothersinthissection,isoptional.Italsorequires materialfromtheoptionalearliersubsectiononCombiningSubspaces. Thepriorsubsectionsprojectavectorintoalinebydecomposingitintotwo parts:thepartinthelineproj [ ~s ] ~v andtherest ~v )]TJ/F8 9.9626 Tf 10.066 0 Td [(proj [ ~s ] ~v .Togeneralize projectiontoarbitrarysubspaces,wefollowthisidea. 3.1Denition Foranydirectsum V = M N andany ~v 2 V ,the projection of ~v into M along N is proj M;N ~v = ~m where ~v = ~m + ~n with ~m 2 M;~n 2 N Thisdenitiondoesn'tinvolveasenseof`orthogonal'sowecanapplyitto spacesotherthansubspacesofan R n .Denitionsoforthogonalityforother spacesareperfectlypossible,butwehaven'tseenanyinthisbook. 3.2Example Thespace M 2 2 of2 2matricesisthedirectsumofthesetwo. M = f ab 00 a;b 2 R g N = f 00 cd c;d 2 R g

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SectionVI.Projection 261 Toproject A = 31 04 into M along N ,werstxbasesforthetwosubspaces. B M = h 10 00 ; 01 00 i B N = h 00 10 ; 00 01 i Theconcatenationofthese B = B M B N = h 10 00 ; 01 00 ; 00 10 ; 00 01 i isabasisfortheentirespace,becausethespaceisthedirectsum,sowecan useittorepresent A 31 04 =3 10 00 +1 01 00 +0 00 10 +4 00 01 Nowtheprojectionof A into M along N isfoundbykeepingthe M partofthis sumanddroppingthe N part. proj M;N 31 04 =3 10 00 +1 01 00 = 31 00 3.3Example Bothsubscriptsonproj M;N ~v aresignicant.Therstsubscript M mattersbecausetheresultoftheprojectionisan ~m 2 M ,andchanging thissubspacewouldchangethepossibleresults.Foranexampleshowingthat thesecondsubscriptmatters,xthisplanesubspaceof R 3 anditsbasis M = f 0 @ x y z 1 A y )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 z =0 g B M = h 0 @ 1 0 0 1 A ; 0 @ 0 2 1 1 A i andcomparetheprojectionsalongtwodierentsubspaces. N = f k 0 @ 0 0 1 1 A k 2 R g ^ N = f k 0 @ 0 1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 1 A k 2 R g Vericationthat R 3 = M N and R 3 = M ^ N isroutine.Wewillcheck thattheseprojectionsaredierentbycheckingthattheyhavedierenteects onthisvector. ~v = 0 @ 2 2 5 1 A Fortherstonewendabasisfor N B N = h 0 @ 0 0 1 1 A i

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262 ChapterThree.MapsBetweenSpaces andrepresent ~v withrespecttotheconcatenation B M B N 0 @ 2 2 5 1 A =2 0 @ 1 0 0 1 A +1 0 @ 0 2 1 1 A +4 0 @ 0 0 1 1 A Theprojectionof ~v into M along N isfoundbykeepingthe M partanddropping the N part. proj M;N ~v =2 0 @ 1 0 0 1 A +1 0 @ 0 2 1 1 A = 0 @ 2 2 1 1 A Fortheothersubspace ^ N ,thisbasisisnatural. B ^ N = h 0 @ 0 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 1 A i Representing ~v withrespecttotheconcatenation 0 @ 2 2 5 1 A =2 0 @ 1 0 0 1 A + = 5 0 @ 0 2 1 1 A )]TJ/F8 9.9626 Tf 9.962 0 Td [( = 5 0 @ 0 1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 1 A andthenkeepingonlythe M partgivesthis. proj M; ^ N ~v =2 0 @ 1 0 0 1 A + = 5 0 @ 0 2 1 1 A = 0 @ 2 18 = 5 9 = 5 1 A Thereforeprojectionalongdierentsubspacesmayyielddierentresults. Thesepicturescomparethetwomaps.Bothshowthattheprojectionis indeed`into'theplaneand`along'theline. M N M N Noticethattheprojectionalong N isnotorthogonal|therearemembersof theplane M thatarenotorthogonaltothedottedline.Buttheprojection along ^ N isorthogonal. Anaturalquestionis:whatistherelationshipbetweentheprojectionoperationdenedabove,andtheoperationoforthogonalprojectionintoaline? Thesecondpictureabovesuggeststheanswer|orthogonalprojectionintoa lineisaspecialcaseoftheprojectiondenedabove;itisjustprojectionalong asubspaceperpendiculartotheline.

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SectionVI.Projection 263 N M Inadditiontopointingoutthatprojectionalongasubspaceisageneralization, thisschemeshowshowtodeneorthogonalprojectionintoanysubspaceof R n ofanydimension. 3.4Denition The orthogonalcomplement ofasubspace M of R n is M ? = f ~v 2 R n ~v isperpendiculartoallvectorsin M g read M perp".The orthogonalprojection proj M ~v ofavectorisitsprojectioninto M along M ? 3.5Example In R 3 ,tondtheorthogonalcomplementoftheplane P = f 0 @ x y z 1 A 3 x +2 y )]TJ/F11 9.9626 Tf 9.962 0 Td [(z =0 g westartwithabasisfor P B = h 0 @ 1 0 3 1 A ; 0 @ 0 1 2 1 A i Any ~v perpendiculartoeveryvectorin B isperpendiculartoeveryvectorinthe spanof B theproofofthisassertionisExercise19.Therefore,thesubspace P ? consistsofthevectorsthatsatisfythesetwoconditions. 0 @ 1 0 3 1 A 0 @ v 1 v 2 v 3 1 A =0 0 @ 0 1 2 1 A 0 @ v 1 v 2 v 3 1 A =0 Wecanexpressthoseconditionsmorecompactlyasalinearsystem. P ? = f 0 @ v 1 v 2 v 3 1 A 103 012 0 @ v 1 v 2 v 3 1 A = 0 0 g Wearethusleftwithndingthenullspaceofthemaprepresentedbythematrix, thatis,withcalculatingthesolutionsetofahomogeneouslinearsystem. P ? = f 0 @ v 1 v 2 v 3 1 A v 1 +3 v 3 =0 v 2 +2 v 3 =0 g = f k 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 1 1 A k 2 R g

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264 ChapterThree.MapsBetweenSpaces 3.6Example Where M isthe xy -planesubspaceof R 3 ,whatis M ? ?A commonrstreactionisthat M ? isthe yz -plane,butthat'snotright.Some vectorsfromthe yz -planearenotperpendiculartoeveryvectorinthe xy -plane. 1 1 0 6? 0 3 2 =arccos 1 0+1 3+0 2 p 2 p 13 0 : 94rad Instead M ? isthe z -axis,sinceproceedingasinthepriorexampleandtaking thenaturalbasisforthe xy -planegivesthis. M ? = f 0 @ x y z 1 A 100 010 0 @ x y z 1 A = 0 0 g = f 0 @ x y z 1 A x =0and y =0 g Thetwoexamplesthatwe'veseensinceDenition3.4illustratetherst sentenceinthatdenition.Thenextresultjustiesthesecondsentence. 3.7Lemma Let M beasubspaceof R n .Theorthogonalcomplementof M is alsoasubspace.Thespaceisthedirectsumofthetwo R n = M M ? .And, forany ~v 2 R n ,thevector ~v )]TJ/F8 9.9626 Tf 9.664 0 Td [(proj M ~v isperpendiculartoeveryvectorin M Proof First,theorthogonalcomplement M ? isasubspaceof R n because,as notedinthepriortwoexamples,itisanullspace. Next,wecanstartwithanybasis B M = h ~ 1 ;:::;~ k i for M andexpanditto abasisfortheentirespace.ApplytheGram-Schmidtprocesstogetanorthogonalbasis K = h ~ 1 ;:::;~ n i for R n .This K istheconcatenationoftwobases h ~ 1 ;:::;~ k i withthesamenumberofmembersas B M and h ~ k +1 ;:::;~ n i Therstisabasisfor M ,soifweshowthatthesecondisabasisfor M ? then wewillhavethattheentirespaceisthedirectsumofthetwosubspaces. Exercise17fromthepriorsubsectionprovesthisaboutanyorthogonalbasis:eachvector ~v inthespaceisthesumofitsorthogonalprojectionsontothe linesspannedbythebasisvectors. ~v =proj [ ~ 1 ] ~v + +proj [ ~ n ] ~v Tocheckthis,representthevector ~v = r 1 ~ 1 + + r n ~ n ,apply ~ i tobothsides ~v ~ i = r 1 ~ 1 + + r n ~ n ~ i = r 1 0+ + r i ~ i ~ i + + r n 0,and solvetoget r i = ~v ~ i = ~ i ~ i ,asdesired. Sinceobviouslyanymemberofthespanof h ~ k +1 ;:::;~ n i isorthogonalto anyvectorin M ,toshowthatthisisabasisfor M ? weneedonlyshowthe othercontainment|thatany ~w 2 M ? isinthespanofthisbasis.Theprior paragraphdoesthis.Onprojectionsintobasisvectorsfrom M ,any ~w 2 M ? givesproj [ ~ 1 ] ~w = ~ 0 ;:::; proj [ ~ k ] ~w = ~ 0andtherefore givesthat ~w isa linearcombinationof ~ k +1 ;:::;~ n .Thusthisisabasisfor M ? and R n isthe directsumofthetwo.

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SectionVI.Projection 265 Thenalsentenceisprovedinmuchthesameway.Write ~v =proj [ ~ 1 ] ~v + +proj [ ~ n ] ~v .Thenproj M ~v isgottenbykeepingonlythe M partand droppingthe M ? partproj M ~v =proj [ ~ k +1 ] ~v + +proj [ ~ k ] ~v .Therefore ~v )]TJ/F8 9.9626 Tf 10.876 0 Td [(proj M ~v consistsofalinearcombinationofelementsof M ? andsois perpendiculartoeveryvectorin M QED Wecanndtheorthogonalprojectionintoasubspacebyfollowingthesteps oftheproof,butthenextresultgivesaconvienentformula. 3.8Theorem Let ~v beavectorin R n andlet M beasubspaceof R n withbasis h ~ 1 ;:::; ~ k i .If A isthematrixwhosecolumnsarethe ~ 'sthen proj M ~v = c 1 ~ 1 + + c k ~ k wherethecoecients c i aretheentriesofthe vector A trans A A trans ~v .Thatis,proj M ~v = A A trans A )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 A trans ~v Proof Thevectorproj M ~v isamemberof M andsoitisalinearcombination ofbasisvectors c 1 ~ 1 + + c k ~ k .Since A 'scolumnsarethe ~ 's,thatcan beexpressedas:thereisa ~c 2 R k suchthatproj M ~v = A~c thisisexpressed compactlywithmatrixmultiplicationasinExample3.5and3.6.Because ~v )]TJ/F8 9.9626 Tf 9.064 0 Td [(proj M ~v isperpendiculartoeachmemberofthebasis,wehavethisagain, expressedcompactly. ~ 0= A trans )]TJ/F11 9.9626 Tf 4.179 -8.07 Td [(~v )]TJ/F11 9.9626 Tf 9.963 0 Td [(A~c = A trans ~v )]TJ/F11 9.9626 Tf 9.962 0 Td [(A trans A~c Solvingfor ~c showingthat A trans A isinvertibleisanexercise ~c = )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(A trans A )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 A trans ~v givestheformulafortheprojectionmatrixasproj M ~v = A ~c QED 3.9Example Toorthogonallyprojectthisvectorintothissubspace ~v = 0 @ 1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 1 1 A P = f 0 @ x y z 1 A x + z =0 g rstmakeamatrixwhosecolumnsareabasisforthesubspace A = 0 @ 01 10 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 A andthencompute. A )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(A trans A )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 A trans = 0 @ 01 10 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 A 01 1 = 20 10 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 010 = 0 @ 1 = 20 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 010 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 201 = 2 1 A

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266 ChapterThree.MapsBetweenSpaces Withthematrix,calculatingtheorthogonalprojectionofanyvectorinto P is easy. proj P ~v = 0 @ 1 = 20 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 010 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 201 = 2 1 A 0 @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 1 A = 0 @ 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 0 1 A Exercises X 3.10 Projectthevectorsinto M along N a 3 )]TJ/F29 8.9664 Tf 7.168 0 Td [(2 ;M = f x y x + y =0 g ;N = f x y )]TJ/F32 8.9664 Tf 7.167 0 Td [(x )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 y =0 g b 1 2 ;M = f x y x )]TJ/F32 8.9664 Tf 9.216 0 Td [(y =0 g ;N = f x y 2 x + y =0 g c 3 0 1 ;M = f x y z x + y =0 g ;N = f c 1 0 1 c 2 R g X 3.11 Find M ? a M = f x y x + y =0 g b M = f x y )]TJ/F29 8.9664 Tf 7.168 0 Td [(2 x +3 y =0 g c M = f x y x )]TJ/F32 8.9664 Tf 9.215 0 Td [(y =0 g d M = f ~ 0 g e M = f x y x =0 g f M = f x y z )]TJ/F32 8.9664 Tf 7.168 0 Td [(x +3 y + z =0 g g M = f x y z x =0and y + z =0 g 3.12 Thissubsectionshowshowtoprojectorthogonallyintwoways,themethodof Example3.2and3.3,andthemethodofTheorem3.8.Tocomparethem,consider theplane P speciedby3 x +2 y )]TJ/F32 8.9664 Tf 9.216 0 Td [(z =0in R 3 a Findabasisfor P b Find P ? andabasisfor P ? c Representthisvectorwithrespecttotheconcatenationofthetwobasesfrom theprioritem. ~v = 1 1 2 d Findtheorthogonalprojectionof ~v into P bykeepingonlythe P partfrom theprioritem. e CheckthatagainsttheresultfromapplyingTheorem3.8. X 3.13 Wehavethreewaystondtheorthogonalprojectionofavectorintoaline, theDenition1.1wayfromtherstsubsectionofthissection,theExample3.2 and3.3wayofrepresentingthevectorwithrespecttoabasisforthespaceand thenkeepingthe M part,andthewayofTheorem3.8.Forthesecases,doall threeways. a ~v = 1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(3 ;M = f x y x + y =0 g b ~v = 0 1 2 ;M = f x y z x + z =0and y =0 g 3.14 CheckthattheoperationofDenition3.1iswell-dened.Thatis,inExample3.2and3.3,doesn'ttheanswerdependonthechoiceofbases?

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SectionVI.Projection 267 3.15 Whatistheorthogonalprojectionintothetrivialsubspace? 3.16 Whatistheprojectionof ~v into M along N if ~v 2 M ? 3.17 Showthatif M R n isasubspacewithorthonormalbasis h ~ 1 ;:::;~ n i then theorthogonalprojectionof ~v into M isthis. ~v ~ 1 ~ 1 + + ~v ~ n ~ n X 3.18 Provethatthemap p : V V istheprojectioninto M along N ifandonly ifthemapid )]TJ/F32 8.9664 Tf 9.422 0 Td [(p istheprojectioninto N along M .Recallthedenitionofthe dierenceoftwomaps:id )]TJ/F32 8.9664 Tf 9.216 0 Td [(p ~v =id ~v )]TJ/F32 8.9664 Tf 9.215 0 Td [(p ~v = ~v )]TJ/F32 8.9664 Tf 9.215 0 Td [(p ~v X 3.19 Showthatifavectorisperpendiculartoeveryvectorinasetthenitis perpendiculartoeveryvectorinthespanofthatset. 3.20 Trueorfalse:theintersectionofasubspaceanditsorthogonalcomplementis trivial. 3.21 Showthatthedimensionsoforthogonalcomplementsaddtothedimension oftheentirespace. X 3.22 Supposethat ~v 1 ;~v 2 2 R n aresuchthatforallcomplements M;N R n ,the projectionsof ~v 1 and ~v 2 into M along N areequal.Must ~v 1 equal ~v 2 ?Ifso,what ifwerelaxtheconditionto:allorthogonalprojectionsofthetwoareequal? X 3.23 Let M;N besubspacesof R n .Theperpoperatoractsonsubspaces;wecan askhowitinteractswithothersuchoperations. a Showthattwoperpscancel: M ? ? = M b Provethat M N impliesthat N ? M ? c Showthat M + N ? = M ? N ? X 3.24 Thematerialinthissubsectionallowsustoexpressageometricrelationship thatwehavenotyetseenbetweentherangespaceandthenullspaceofalinear map. a Represent f : R 3 R givenby v 1 v 2 v 3 7! 1 v 1 +2 v 2 +3 v 3 withrespecttothestandardbasesandshowthat 1 2 3 isamemberoftheperpofthenullspace.Provethat N f ? isequaltothe spanofthisvector. b Generalizethattoapplytoany f : R n R c Represent f : R 3 R 2 v 1 v 2 v 3 7! 1 v 1 +2 v 2 +3 v 3 4 v 1 +5 v 2 +6 v 3 withrespecttothestandardbasesandshowthat 1 2 3 ; 4 5 6 arebothmembersoftheperpofthenullspace.Provethat N f ? isthespan ofthesetwo. Hint .SeethethirditemofExercise23.

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268 ChapterThree.MapsBetweenSpaces d Generalizethattoapplytoany f : R n R m This,andrelatedresults,iscalledthe FundamentalTheoremofLinearAlgebra in [Strang93]. 3.25 Denea projection tobealineartransformation t : V V withtheproperty thatrepeatingtheprojectiondoesnothingmorethandoestheprojectionalone: t t ~v = t ~v forall ~v 2 V a Showthatorthogonalprojectionintoalinehasthatproperty. b Showthatprojectionalongasubspacehasthatproperty. c Showthatforanysuch t thereisabasis B = h ~ 1 ;:::; ~ n i for V suchthat t ~ i = ~ i i =1 ; 2 ;:::;r ~ 0 i = r +1 ;r +2 ;:::;n where r istherankof t d Concludethateveryprojectionisaprojectionalongasubspace. e Alsoconcludethateveryprojectionhasarepresentation Rep B;B t = I Z Z Z inblockpartial-identityform. 3.26 Asquarematrixis symmetric ifeach i;j entryequalsthe j;i entryi.e.,ifthe matrixequalsitstranspose.Showthattheprojectionmatrix A A trans A )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 A trans issymmetric.[Strang80] Hint .Findpropertiesoftransposesbylookinginthe indexunder`transpose'.

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Topic:LineofBestFit 269 Topic:LineofBestFit ThisTopicrequirestheformulasfromthesubsectionsonOrthogonalProjection IntoaLine,andProjectionIntoaSubspace. Scientistsareoftenpresentedwithasystemthathasnosolutionandthey mustndanansweranyway.Thatis,theymustndavaluethatisascloseas possibletobeingananswer. Forinstance,supposethatwehaveacointouseinipping.Thiscoinhas someproportion m ofheadstototalips,determinedbyhowitisphysically constructed,andwewanttoknowif m isnear1 = 2.Wecangetexperimental databyippingitmanytimes.Thisistheresultapennyexperiment,including someintermediatenumbers. numberofips 306090 numberofheads 163451 Becauseofrandomness,wedonotndtheexactproportionwiththissample| thereisnosolutiontothissystem. 30 m =16 60 m =34 90 m =51 Thatis,thevectorofexperimentaldataisnotinthesubspaceofsolutions. 0 @ 16 34 51 1 A 62f m 0 @ 30 60 90 1 A m 2 R g However,asdescribedabove,wewanttondthe m thatmostnearlyworks.An orthogonalprojectionofthedatavectorintothelinesubspacegivesourbest guess. 0 @ 16 34 51 1 A 0 @ 30 60 90 1 A 0 @ 30 60 90 1 A 0 @ 30 60 90 1 A 0 @ 30 60 90 1 A = 7110 12600 0 @ 30 60 90 1 A Theestimate m =7110 = 12600 0 : 56isabithighbutnotmuch,soprobably thepennyisfairenough. Thelinewiththeslope m 0 : 56iscalledthe lineofbestt forthisdata. ips 306090 heads 30 60

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270 ChapterThree.MapsBetweenSpaces Minimizingthedistancebetweenthegivenvectorandthevectorusedasthe right-handsideminimizesthetotaloftheseverticallengths,andconsequently wesaythatthelinehasbeenobtainedthrough ttingbyleast-squares theverticalscaleherehasbeenexaggeratedtentimestomakethelengths visible. Wearrangedtheequationabovesothatthelinemustpassthrough ; 0 becausewetaketakeittobeourbestguessatthelinewhoseslopeisthis coin'strueproportionofheadstoips.Wecanalsohandlecaseswheretheline neednotpassthroughtheorigin. Forexample,thedierentdenominationsofU.S.moneyhavedierentaveragetimesincirculationthe$2billisleftoasaspecialcase.Howlongshould weexpecta$25billtolast? denomination 15102050100 averagelifeyears 1 : 5235920 Theplotseebelowlooksroughlylinear.Itisn'taperfectline,i.e.,thelinear systemwithequations b +1 m =1 : 5,..., b +100 m =20hasnosolution,but wecanagainuseorthogonalprojectiontondabestapproximation.Consider thematrixofcoecientsofthatlinearsystemandalsoitsvectorofconstants, theexperimentally-determinedvalues. A = 0 B B B B B B @ 11 15 110 120 150 1100 1 C C C C C C A ~v = 0 B B B B B B @ 1 : 5 2 3 5 9 20 1 C C C C C C A TheendingresultinthesubsectiononProjectionintoaSubspacesaysthat coecients b and m sothatthelinearcombinationofthecolumnsof A isas closeaspossibletothevector ~v aretheentriesof A trans A )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 A trans ~v .Some calculationgivesaninterceptof b =1 : 05andaslopeof m =0 : 18. denom 1030507090 avglife 5 15 Plugging x =25intotheequationofthelineshowsthatsuchabillshouldlast betweenveandsixyears.

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Topic:LineofBestFit 271 Weclosebyconsideringthetimesforthemen'smilerace[Oakley&Baker]. ThesearetheworldrecordsthatwereinforceonJanuaryrstofthegiven years.Wewanttoprojectwhena3:40milewillberun. year 1870188018901900191019201930 seconds 268 : 8264 : 5258 : 4255 : 6255 : 6252 : 6250 : 4 1940195019601970198019902000 246 : 4241 : 4234 : 5231 : 1229 : 0226 : 3223 : 1 Wecanseebelowthatthedataissurprisinglylinear.Withthisinput A = 0 B B B B B @ 11860 11870 11990 12000 1 C C C C C A ~v = 0 B B B B B @ 280 : 0 268 : 8 226 : 3 223 : 1 1 C C C C C A thePythonprogramatthisTopic'sendgivesslope= )]TJ/F8 9.9626 Tf 7.748 0 Td [(0 : 35andintercept= 925 : 53roundedtotwoplaces;theoriginaldataisgoodtoonlyaboutaquarter ofasecondsincemuchofitwashand-timed. year 1870189019101930195019701990 seconds 220 240 260 Whenwilla220secondmileberun?Solvingtheequationofthelineofbestt givesanestimateoftheyear2008. Thisexampleisamusing,butservesasacaution|obviouslythelinearity ofthedatawillbreakdownsomedayasindeeditdoespriorto1860. Exercises Thecalculationsherearebestdoneonacomputer.Inaddition,someoftheproblemsrequiremoredata,availableinyourlibrary,onthenet,orintheAnswersto theExercises. 1 Useleast-squarestojudgeifthecoininthisexperimentisfair. ips 816243240 heads 49131720 2 Forthemen'smilerecord,ratherthangiveeachofthemanyrecordsandits exactdate,we'vesmoothed"thedatasomewhatbytakingaperiodicsample.Do thelongercalculationandcomparetheconclusions. 3 Findthelineofbesttforthemen's1500meterrun.Howdoestheslope comparewiththatforthemen'smile?Thedistancesareclose;amileisabout 1609meters.

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272 ChapterThree.MapsBetweenSpaces 4 Findthelineofbesttfortherecordsforwomen'smile. 5 Dothelinesofbesttforthemen'sandwomen'smilescross? 6 WhenthespaceshuttleChallengerexplodedin1986,oneofthecriticismsmadeof NASA'sdecisiontolaunchwasinthewaytheanalysisofnumberofO-ringfailures versustemperaturewasmadeofcourse,O-ringfailurecausedtheexplosion.Four O-ringfailureswillcausetherockettoexplode.NASAhaddatafrom24previous ights. temp F 5375575863707066676767 failures 32111110000 68697070727375767678798081 0000000000000 Thetemperaturethatdaywasforecasttobe31 F. a NASAbasedthedecisiontolaunchpartiallyonachartshowingonlythe ightsthathadatleastoneO-ringfailure.Findthelinethatbesttsthese sevenights.Onthebasisofthisdata,predictthenumberofO-ringfailures whenthetemperatureis31,andwhenthenumberoffailureswillexceedfour. b Findthelinethatbesttsall24ights.Onthebasisofthisextradata, predictthenumberofO-ringfailureswhenthetemperatureis31,andwhenthe numberoffailureswillexceedfour. Whichdoyouthinkisthemoreaccuratemethodofpredicting?Anexcellent discussionappearsin[Dalal,et.al.]. 7 Thistableliststheaveragedistancefromthesuntoeachoftherstsevenplanets, usingearth'saverageasaunit. MercuryVenusEarthMarsJupiterSaturnUranus 0 : 390 : 721 : 001 : 525 : 209 : 5419 : 2 a PlotthenumberoftheplanetMercuryis1,etc.versusthedistance.Note thatitdoesnotlooklikealine,andsondingthelineofbesttisnotfruitful. b Itdoes,howeverlooklikeanexponentialcurve.Therefore,plotthenumber oftheplanetversusthelogarithmofthedistance.Doesthislooklikealine? c TheasteroidbeltbetweenMarsandJupiteristhoughttobewhatisleftofa planetthatbrokeapart.RenumbersothatJupiteris6,Saturnis7,andUranus is8,andplotagainstthelogagain.Doesthislookbetter? d UseleastsquaresonthatdatatopredictthelocationofNeptune. e RepeattopredictwherePlutois. f IstheformulaaccurateforNeptuneandPluto? ThismethodwasusedtohelpdiscoverNeptunealthoughtheseconditemismisleadingaboutthehistory;actually,thediscoveryofNeptuneinposition9prompted peopletolookforthemissingplanet"inposition5.See[Gardner,1970] 8 WilliamBennetthasproposedanIndexofLeadingCulturalIndicatorsforthe US[Bennett],in1993.Amongthestatisticscitedaretheaveragedailyhours spentwatchingTV,andtheaveragecombinedSATscores. 19601965197019751980198519901992 TV 5:065:295:566:076:367:076:557:04 SAT 975969948910890906900899 Supposethatacauseandeectrelationshipisproposedbetweenthetimespent watchingTVandthedeclineinSATscoresinthisarticle,Mr.Bennettdoesnot arguethatthereisadirectconnection. a Findthelineofbesttrelatingtheindependentvariableofaveragedaily TVhourstothedependentvariableofSATscores.

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Topic:LineofBestFit 273 b FindthemostrecentestimateoftheaveragedailyTVhoursBennett'scites NeilsenMediaResearchasthesourceoftheseestimates.EstimatetheassociatedSATscore.Howcloseisyourestimatetotheactualaverage?Warning:a changehasbeenmaderecentlyintheSAT,soyoushouldinvestigatewhether someadjustmentneedstobemadetothereportedaveragetomakeavalid comparison. ComputerCode #!/usr/bin/python #least_squares.pycalculatethelineofbestfitforadataset #datafileformat:eachlineistwonumbers,xandy importmath,string n=0 sum__x=0 sum_y=0 sum_x_squared=0 sum_xy=0 fn=raw_input"Nameofthedatafile?" datafile=openfn,"r" while1: ln=datafile.readline ifln: data=string.splitln x=string.atofdata[0] y=string.atofdata[1] n=n+1 sum_x=sum_x+x sum_y=sum_y+y sum_x_squared=sum_x_squared+x*x sum_xy=sum_xy+x*y else: break datafile.close slope=sum_of_xy-powsum_x,2/n/sum_x_squared-powsum_x,2/n intercept=sum_y-slope*sum_x/n print"lineofbestfit:slope=",slope,"intercept=",intercept

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274 ChapterThree.MapsBetweenSpaces Topic:GeometryofLinearMaps Thepicturesbelowcontrast f 1 x = e x and f 2 x = x 2 ,whicharenonlinear, with h 1 x =2 x and h 2 x = )]TJ/F11 9.9626 Tf 7.749 0 Td [(x ,whicharelinear.Eachofthefourpictures showsthedomain R 1 ontheleftmappedtothecodomain R 1 ontheright. Arrowstraceoutwhereeachmapsends x =0, x =1, x =2, x = )]TJ/F8 9.9626 Tf 7.748 0 Td [(1,and x = )]TJ/F8 9.9626 Tf 7.749 0 Td [(2.Notehowthenonlinearmapsdistortthedomainintransformingit intotherange.Forinstance, f 1 isfurtherfrom f 1 thanitisfrom f 1 | themapisspreadingthedomainoutunevenlysothatanintervalnear x =2is spreadapartmorethanisanintervalnear x =0whentheyarecarriedoverto therange. -5 0 5 -5 0 5 -5 0 5 -5 0 5 Thelinearmapsarenicer,moreregular,inthatforeachmapallofthedomain isspreadbythesamefactor. -5 0 5 -5 0 5 -5 0 5 -5 0 5 Theonlylinearmapsfrom R 1 to R 1 aremultiplicationsbyascalar.In higherdimensionsmorecanhappen.Forinstance,thislineartransformationof R 2 ,rotatesvectorscounterclockwise,andisnotjustascalarmultiplication.

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Topic:GeometryofLinearMaps 275 )]TJ/F9 4.9813 Tf 4.566 -3.786 Td [(x y 7! )]TJ/F9 4.9813 Tf 4.566 -3.786 Td [(x cos )]TJ/F9 4.9813 Tf 7.026 0 Td [(y sin x sin + y cos 7)245(! Thetransformationof R 3 whichprojectsvectorsintothe xz -planeisalsonot justarescaling. x y z 7! x 0 z 7)245(! Nonetheless,eveninhigherdimensionsthesituationisn'ttoocomplicated. Below,weusethestandardbasestorepresenteachlinearmap h : R n R m byamatrix H .Recallthatany H canbefactored H = PBQ ,where P and Q are nonsingularand B isapartial-identitymatrix.Further,recallthatnonsingular matricesfactorintoelementarymatrices PBQ = T n T n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 T j BT j )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 T 1 whicharematricesthatareobtainedfromtheidentity I withoneGaussian step I k i )167(! M i k I i $ j )167(! P i;j I k i + j )167(! C i;j k i 6 = j k 6 =0.Soifweunderstandtheeectofalinearmapdescribed byapartial-identitymatrix,andtheeectoflinearmapssdescribedbythe elementarymatrices,thenwewillinsomesenseunderstandtheeectofany linearmap.Thepicturesbelowsticktotransformationsof R 2 foreaseof drawing,butthestatementsholdformapsfromany R n toany R m Thegeometriceectofthelineartransformationrepresentedbyapartialidentitymatrixisprojection. 0 @ x y z 1 A 100 010 000 E 3 ; E 3 )167(! 0 @ x y 0 1 A Forthe M i k matrices,thegeometricactionofatransformationrepresented bysuchamatrixwithrespecttothestandardbasisistostretchvectorsby afactorof k alongthe i -thaxis.Thismapstretchesbyafactorof3alongthe x -axis. )]TJ/F9 4.9813 Tf 4.566 -3.786 Td [(x y 7! )]TJ/F6 4.9813 Tf 4.567 -3.786 Td [(3 x y 7)245(!

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276 ChapterThree.MapsBetweenSpaces Notethatif0 k< 1orif k< 0thenthe i -thcomponentgoestheotherway; here,towardtheleft. )]TJ/F9 4.9813 Tf 4.566 -3.786 Td [(x y 7! )]TJ/F12 4.9813 Tf 4.566 -3.786 Td [()]TJ/F6 4.9813 Tf 5.397 0 Td [(2 x y 7)245(! Eitheroftheseisa dilation Theactionofatransformationrepresentedbya P i;j permutationmatrixis tointerchangethe i -thand j -thaxes;thisisaparticularkindofreection. )]TJ/F9 4.9813 Tf 4.567 -3.786 Td [(x y 7! )]TJ/F9 4.9813 Tf 4.629 -3.786 Td [(y x 7)245(! Inhigherdimensions,permutationsinvolvingmanyaxescanbedecomposed intoacombinationofswapsofpairsofaxes|seeExercise5. Theremainingcaseisthatofmatricesoftheform C i;j k .Recallthat,for instance,that C 1 ; 2 performs2 1 + 2 x y 10 21 E 2 ; E 2 )167(! x 2 x + y Inthepicturebelow,thevector ~u withtherstcomponentof1isaectedless thanthevector ~v withtherstcomponentof2| h ~u isonly2higherthan ~u while h ~v is4higherthan ~v )]TJ/F9 4.9813 Tf 4.566 -3.786 Td [(x y 7! )]TJ/F9 4.9813 Tf 12.36 -3.786 Td [(x 2 x + y 7)245(! ~u ~v h ~u h ~v Anyvectorwitharstcomponentof1wouldbeaectedasis ~u ;itwouldbeslid upby2.Andanyvectorwitharstcomponentof2wouldbeslidup4,aswas ~v .Thatis,thetransformationrepresentedby C i;j k aectsvectorsdepending ontheir i -thcomponent. Anotherwaytoseethissamepointistoconsidertheactionofthismap ontheunitsquare.Inthenextpicture,vectorswitharstcomponentof0, liketheorigin,arenotpushedverticallyatallbutvectorswithapositiverst componentareslidup.Here,allvectorswitharstcomponentof1|theentire rightsideofthesquare|isaectedtothesameextent.Moregenerally,vectors onthesameverticallineareslidupthesameamount,namely,theyareslidup bytwicetheirrstcomponent.Theresultingshape,arhombus,hasthesame baseandheightasthesquareandthusthesameareabuttherightanglesare gone.

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Topic:GeometryofLinearMaps 277 )]TJ/F9 4.9813 Tf 4.566 -3.786 Td [(x y 7! )]TJ/F9 4.9813 Tf 12.36 -3.786 Td [(x 2 x + y 7)245(! Forcontrastthenextpictureshowstheeectofthemaprepresentedby C 2 ; 1 Inthiscase,vectorsareaectedaccordingtotheirsecondcomponent.The vector )]TJ/F10 6.9738 Tf 4.566 -3.649 Td [(x y isslidhorozontallybytwice y )]TJ/F9 4.9813 Tf 4.566 -3.786 Td [(x y 7! )]TJ/F9 4.9813 Tf 4.566 -3.786 Td [(x +2 y y 7)245(! Becauseofthisaction,thiskindofmapiscalleda skew Withthat,wehavecoveredthegeometriceectofthefourtypesofcomponentsintheexpansion H = T n T n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 T j BT j )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 T 1 ,thepartial-identity projection B andtheelementary T i 's.Sinceweunderstanditscomponents, weinsomesenseunderstandtheactionofany H .Asanillustrationofthis assertion,recallthatunderalinearmap,theimageofasubspaceisasubspace andthusthelineartransformation h representedby H mapslinesthroughthe origintolinesthroughtheorigin.Thedimensionoftheimagespacecannot begreaterthanthedimensionofthedomainspace,soalinecan'tmaponto, say,aplane.Wewillextendthattoshowthatanyline,notjustthosethrough theorigin,ismappedby h toaline.Theproofissimplythatthepartialidentityprojection B andtheelementary T i 'seachturnalineinputintoaline outputverifyingthefourcasesisExercise6,andthereforetheircomposition alsopreserveslines.Thus,byunderstandingitscomponentswecanunderstand arbitrarysquarematrices H ,inthesensethatwecanprovethingsaboutthem. Anunderstandingofthegeometriceectoflineartransformationson R n is veryimportantinmathematics.Hereisafamiliarapplicationfromcalculus. Ontheleftisapictureoftheactionofthenonlinearfunction y x = x 2 + x .As atthatstartofthisTopic,overallthegeometriceectofthismapisirregular inthatatdierentdomainpointsithasdierenteectse.g.,asthedomain point x goesfrom2to )]TJ/F8 9.9626 Tf 7.748 0 Td [(2,theassociatedrangepoint f x atrstdecreases, thenpausesinstantaneously,andthenincreases. 0 5 0 5

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278 ChapterThree.MapsBetweenSpaces Butincalculuswedon'tfocusonthemapoverall,wefocusinsteadonthelocal eectofthemap.At x =1thederivativeis y 0 =3,sothatnear x =1 wehave y 3 x .Thatis,inaneighborhoodof x =1,incarryingthe domaintothecodomainthismapcausesittogrowbyafactorof3|itis, locally,approximately,adilation.Thepicturebelowshowsasmallinterval inthedomain x )]TJ/F8 9.9626 Tf 10.386 0 Td [( x::x + x carriedovertoanintervalinthecodomain y )]TJ/F8 9.9626 Tf 9.962 0 Td [( y::y + y thatisthreetimesaswide: y 3 x x =1 y =2 Whentheabovepictureisdrawninthetraditionalcartesianwaythenthe priorsentenceabouttherateofgrowthof y x isusuallystated:thederivative y 0 =3givestheslopeofthelinetangenttothegraphatthepoint ; 2. Inhigherdimensions,theideaisthesamebuttheapproximationisnotjust the R 1 -toR 1 scalarmultiplicationcase.Instead,forafunction y : R n R m andapoint ~x 2 R n ,thederivativeisdenedtobethelinearmap h : R n R m bestapproximatinghow y changesnear y ~x .Sothegeometrystudiedabove applies. WewillclosethisTopicbyremarkinghowthispointofviewmakesclearan often-misunderstood,butveryimportant,resultaboutderivatives:thederivativeofthecompositionoftwofunctionsiscomputedbyusingtheChainRule forcombiningtheirderivatives.Recallthatwithsuitableconditionsonthetwo functions d g f dx x = dg dx f x df dx x sothat,forinstance,thederivativeofsin x 2 +3 x iscos x 2 +3 x x +3.How doesthiscombinationarise?Fromthispictureoftheactionofthecomposition. x f x g f x

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Topic:GeometryofLinearMaps 279 Therstmap f dilatestheneighborhoodof x byafactorof df dx x andthesecondmap g dilatessomemore,thistimedilatinganeighborhoodof f x byafactorof dg dx f x andasaresult,thecompositiondilatesbytheproductofthesetwo. Inhigherdimensionsthemapexpressinghowafunctionchangesneara pointisalinearmap,andisexpressedasamatrix.Soweunderstandthe basicgeometryofhigher-dimensionalderivatives;theyarecompositionsofdilations,interchangesofaxes,shears,andaprojection.And,theChainRulejust multipliesthematrices. Thus,thegeometryoflinearmaps h : R n R m isappealingbothforits simplicityandforitsusefulness. Exercises 1 Let h : R 2 R 2 bethetransformationthatrotatesvectorsclockwiseby = 4radians. a Findthematrix H representing h withrespecttothestandardbases.Use Gauss'methodtoreduce H totheidentity. b Translatetherowreductiontotoamatrixequation T j T j )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 T 1 H = I the prioritemshowsboththat H issimilarto I ,andthatnocolumnoperationsare neededtoderive I from H c Solvethismatrixequationfor H d Sketchthegeometriceectmatrix,thatis,sketchhow H isexpressedasa combinationofdilations,ips,skews,andprojectionstheidentityisatrivial projection. 2 Whatcombinationofdilations,ips,skews,andprojectionsproducesarotation counterclockwiseby2 = 3radians? 3 Whatcombinationofdilations,ips,skews,andprojectionsproducesthemap h : R 3 R 3 representedwithrespecttothestandardbasesbythismatrix? 121 360 122 4 Showthatanylineartransformationof R 1 isthemapthatmultipliesbyascalar x 7! kx 5 Showthatforanypermutationthatis,reordering p ofthenumbers1,..., n themap 0 B B @ x 1 x 2 x n 1 C C A 7! 0 B B @ x p x p x p n 1 C C A canbeaccomplishedwithacompositionofmaps,eachofwhichonlyswapsasingle pairofcoordinates. Hint: itcanbedonebyinductionon n Remark: inthefourth

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280 ChapterThree.MapsBetweenSpaces chapterwewillshowthisandwewillalsoshowthattheparityofthenumberof swapsusedisdeterminedby p .Thatis,althoughaparticularpermutationcould beaccomplishedintwodierentwayswithtwodierentnumbersofswaps,either bothwaysuseanevennumberofswaps,orbothuseanoddnumber. 6 Showthatlinearmapspreservethelinearstructuresofaspace. a Showthatforanylinearmapfrom R n to R m ,theimageofanylineisaline. Theimagemaybeadegenerateline,thatis,asinglepoint. b Showthattheimageofanylinearsurfaceisalinearsurface.Thisgeneralizes theresultthatunderalinearmaptheimageofasubspaceisasubspace. c Linearmapspreserveotherlinearideas.Showthatlinearmapspreserve betweeness":ifthepoint B isbetween A and C thentheimageof B isbetween theimageof A andtheimageof C 7 UseapictureliketheonethatappearsinthediscussionoftheChainRuleto answer:ifafunction f : R R hasaninverse,what'stherelationshipbetweenhow thefunction|locally,approximately|dilatesspace,andhowitsinversedilates spaceassuming,ofcourse,thatithasaninverse?

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Topic:MarkovChains 281 Topic:MarkovChains Hereisasimplegame:aplayerbetsoncointosses,adollareachtime,andthe gameendseitherwhentheplayerhasnomoneyleftorisuptovedollars.If theplayerstartswiththreedollars,whatisthechancethatthegametakesat leastveips?Twenty-veips? Atanypoint,thisplayerhaseither$0,or$1,...,or$5.Wesaythatthe playerisinthe state s 0 s 1 ,...,or s 5 .Agameconsistsofmovingfromstateto state.Forinstance,aplayernowinstate s 3 hasonthenextipa : 5chanceof movingtostate s 2 anda : 5chanceofmovingto s 4 .Theboundarystatesarea bitdierent;onceinstate s 0 orstate s 5 ,theplayerneverleaves. Let p i n betheprobabilitythattheplayerisinstate s i after n ips.Then, forinstance,wehavethattheprobabilityofbeinginstate s 0 afterip n +1is p 0 n +1= p 0 n +0 : 5 p 1 n .Thismatrixequationsumarizes. 0 B B B B B B @ 1 : 50000 00 : 5000 0 : 50 : 500 00 : 50 : 50 000 : 500 0000 : 51 1 C C C C C C A 0 B B B B B B @ p 0 n p 1 n p 2 n p 3 n p 4 n p 5 n 1 C C C C C C A = 0 B B B B B B @ p 0 n +1 p 1 n +1 p 2 n +1 p 3 n +1 p 4 n +1 p 5 n +1 1 C C C C C C A Withtheinitialconditionthattheplayerstartswiththreedollars,calculation givesthis. n =0 n =1 n =2 n =3 n =4 n =24 0 B B B B B B @ 0 0 0 1 0 0 1 C C C C C C A 0 B B B B B B @ 0 0 : 5 0 : 5 0 1 C C C C C C A 0 B B B B B B @ 0 : 25 0 : 5 0 : 25 1 C C C C C C A 0 B B B B B B @ : 125 0 : 375 0 : 25 : 25 1 C C C C C C A 0 B B B B B B @ : 125 : 1875 0 : 3125 0 : 375 1 C C C C C C A 0 B B B B B B @ : 39600 : 00276 0 : 00447 0 : 59676 1 C C C C C C A Asthiscomputationalexplorationsuggests,thegameisnotlikelytogoonfor long,withtheplayerquicklyendingineitherstate s 0 orstate s 5 .Forinstance, afterthefourthipthereisaprobabilityof0 : 50thatthegameisalreadyover. Becauseaplayerwhoenterseitheroftheboundarystatesneverleaves,they aresaidtobe absorbtive Thisgameisanexampleofa Markovchain ,namedforA.A.Markov,who workedinthersthalfofthe1900's.Eachvectorof p 'sisa probabilityvector andthematrixisa transitionmatrix .ThenotablefeatureofaMarkovchain modelisthatitis historyless inthatwithaxedtransitionmatrix,thenext statedependsonlyonthecurrentstate,notonanypriorstates.Thusaplayer, say,whoarrivesat s 2 bystartinginstate s 3 ,thengoingtostate s 2 ,thento s 1 ,andthento s 2 hasatthispointexactlythesamechanceofmovingnextto state s 3 asdoesaplayerwhosehistorywastostartin s 3 ,thengoto s 4 ,andto s 3 ,andthento s 2 .

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282 ChapterThree.MapsBetweenSpaces HereisaMarkovchainfromsociology.Astudy[Macdonald&Ridge], p.202dividedoccupationsintheUnitedKingdomintoupperlevelexecutives andprofessionals,middlelevelsupervisorsandskilledmanualworkers,and lowerlevelunskilled.Todeterminethemobilityacrosstheselevelsinageneration,abouttwothousandmenwereasked,Atwhichlevelareyou,andat whichlevelwasyourfatherwhenyouwerefourteenyearsold?"Thisequation summarizestheresults. 0 @ : 60 : 29 : 16 : 26 : 37 : 27 : 14 : 34 : 57 1 A 0 @ p U n p M n p L n 1 A = 0 @ p U n +1 p M n +1 p L n +1 1 A Forinstance,achildofalowerclassworkerhasa : 27probabilityofgrowingupto bemiddleclass.NoticethattheMarkovmodelassumptionabouthistoryseems reasonable|weexpectthatwhileaparent'soccupationhasadirectinuence ontheoccupationofthechild,thegrandparent'soccupationhasnosuchdirect inuence.Withtheinitialdistributionoftherespondents'sfathersgivenbelow, thistableliststhedistributionsforthenextvegenerations. n =0 n =1 n =2 n =3 n =4 n =5 0 @ : 12 : 32 : 56 1 A 0 @ : 23 : 34 : 42 1 A 0 @ : 29 : 34 : 37 1 A 0 @ : 31 : 34 : 35 1 A 0 @ : 32 : 33 : 34 1 A 0 @ : 33 : 33 : 34 1 A Onemoreexample,fromaveryimportantsubject,indeed.TheWorldSeries ofAmericanbaseballisplayedbetweentheteamwinningtheAmericanLeague andtheteamwinningtheNationalLeaguewefollow[Brunner]butseealso [Woodside].Theseriesiswonbytherstteamtowinfourgames.Thatmeans thataseriesisinoneoftwenty-fourstates:0-0nogameswonyetbyeither team,1-0onegamewonfortheAmericanLeagueteamandnogamesforthe NationalLeagueteam,etc.Ifweassumethatthereisaprobability p thatthe AmericanLeagueteamwinseachgamethenwehavethefollowingtransition matrix. 0 B B B B B B B B B @ 0000 ::: p 000 ::: 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(p 000 ::: 0 p 00 ::: 01 )]TJ/F11 9.9626 Tf 9.962 0 Td [(pp 0 ::: 001 )]TJ/F11 9.9626 Tf 9.962 0 Td [(p 0 ::: . 1 C C C C C C C C C A 0 B B B B B B B B B @ p 0-0 n p 1-0 n p 0-1 n p 2-0 n p 1-1 n p 0-2 n 1 C C C C C C C C C A = 0 B B B B B B B B B @ p 0-0 n +1 p 1-0 n +1 p 0-1 n +1 p 2-0 n +1 p 1-1 n +1 p 0-2 n +1 1 C C C C C C C C C A Anespeciallyinterestingspecialcaseis p =0 : 50;thistableliststheresulting componentsofthe n =0through n =7vectors.Thecodetogeneratethis tableinthecomputeralgebrasystemOctavefollowstheexercises.

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Topic:MarkovChains 283 n =0 n =1 n =2 n =3 n =4 n =5 n =6 n =7 0 )]TJ/F29 8.9664 Tf 9.215 0 Td [(0 1 )]TJ/F29 8.9664 Tf 9.215 0 Td [(0 0 )]TJ/F29 8.9664 Tf 9.215 0 Td [(1 2 )]TJ/F29 8.9664 Tf 9.215 0 Td [(0 1 )]TJ/F29 8.9664 Tf 9.215 0 Td [(1 0 )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 3 )]TJ/F29 8.9664 Tf 9.215 0 Td [(0 2 )]TJ/F29 8.9664 Tf 9.215 0 Td [(1 1 )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 0 )]TJ/F29 8.9664 Tf 9.215 0 Td [(3 4 )]TJ/F29 8.9664 Tf 9.215 0 Td [(0 3 )]TJ/F29 8.9664 Tf 9.215 0 Td [(1 2 )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 1 )]TJ/F29 8.9664 Tf 9.215 0 Td [(3 0 )]TJ/F29 8.9664 Tf 9.215 0 Td [(4 4 )]TJ/F29 8.9664 Tf 9.215 0 Td [(1 3 )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 2 )]TJ/F29 8.9664 Tf 9.215 0 Td [(3 1 )]TJ/F29 8.9664 Tf 9.215 0 Td [(4 4 )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 3 )]TJ/F29 8.9664 Tf 9.215 0 Td [(3 2 )]TJ/F29 8.9664 Tf 9.215 0 Td [(4 4 )]TJ/F29 8.9664 Tf 9.215 0 Td [(3 3 )]TJ/F29 8.9664 Tf 9.215 0 Td [(4 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 : 5 0 : 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 : 25 0 : 5 0 : 25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 : 125 0 : 375 0 : 375 0 : 125 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 : 0625 0 : 25 0 : 375 0 : 25 0 : 0625 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 : 0625 0 0 0 0 : 0625 0 : 125 0 : 3125 0 : 3125 0 : 125 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 : 0625 0 0 0 0 : 0625 0 : 125 0 0 0 : 125 0 : 15625 0 : 3125 0 : 15625 0 0 0 0 0 0 0 0 0 0 0 0 0 : 0625 0 0 0 0 : 0625 0 : 125 0 0 0 : 125 0 : 15625 0 0 : 15625 0 : 15625 0 : 15625 Notethatevenly-matchedteamsarelikelytohavealongseries|thereisa probabilityof0 : 625thattheseriesgoesatleastsixgames. OnereasonfortheinclusionofthisTopicisthatMarkovchainsareone ofthemostwidely-usedapplicationsofmatrixoperations.Anotherreasonis thatitprovidesanexampleoftheuseofmatriceswherewedonotconsider thesignicanceofthemapsrepresentedbythematrices.FormoreonMarkov chains,therearemanysourcessuchas[Kemeny&Snell]and[Iosifescu]. Exercises Useacomputerfortheseproblems.Youcan,forinstance,adapttheOctavescript givenbelow. 1 Thesequestionsrefertothecoin-ippinggame. a Checkthecomputationsinthetableattheendoftherstparagraph. b Considerthesecondrowofthevectortable.Notethatthisrowhasalternating0's.Must p 1 j be0when j isodd?Provethatitmustbe,orproducea counterexample. c Performacomputationalexperimenttoestimatethechancethattheplayer endsatvedollars,startingwithonedollar,twodollars,andfourdollars. 2 Weconsiderthrowsofadie,andsaythesystemisinstate s i ifthelargestnumber yetappearingonthediewas i a Givethetransitionmatrix. b Startthesysteminstate s 1 ,andrunitforvethrows.Whatisthevector attheend?

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284 ChapterThree.MapsBetweenSpaces [Feller],p.424 3 TherehasbeenmuchinterestinwhetherindustriesintheUnitedStatesare movingfromtheNortheastandNorthCentralregionstotheSouthandWest, motivatedbythewarmerclimate,bylowerwages,andbylessunionization.Hereis thetransitionmatrixforlargermsinElectricandElectronicEquipment[Kelton], p.43 NENCSWZ NE NC S W Z 0 : 787 0 0 0 0 : 021 0 0 : 966 0 : 063 0 0 : 009 0 0 : 034 0 : 937 0 : 074 0 : 005 0 : 111 0 0 0 : 612 0 : 010 0 : 102 0 0 0 : 314 0 : 954 Forexample,armintheNortheastregionwillbeintheWestregionnextyear withprobability0 : 111.The Z entryisabirth-death"state.Forinstance,with probability0 : 102alargeElectricandElectronicEquipmentrmfromtheNortheastwillmoveoutofthissystemnextyear:gooutofbusiness,moveabroad,or movetoanothercategoryofrm.Thereisa0 : 021probabilitythatarminthe NationalCensusofManufacturers willmoveintoElectronics,orbecreated,or moveinfromabroad,intotheNortheast.Finally,withprobability0 : 954arm outofthecategorieswillstayout,accordingtothisresearch. a DoestheMarkovmodelassumptionoflackofhistoryseemjustied? b Assumethattheinitialdistributioniseven,exceptthatthevalueat Z is 0 : 9.Computethevectorsfor n =1through n =4. c Supposethattheinitialdistributionisthis. NENCSWZ 0 : 00000 : 65220 : 34780 : 00000 : 0000 Calculatethedistributionsfor n =1through n =4. d Findthedistributionfor n =50and n =51.Hasthesystemsettleddown toanequilibrium? 4 Thismodelhasbeensuggestedforsomekindsoflearning[Wickens],p.41.The learnerstartsinanundecidedstate s U .Eventuallythelearnerhastodecidetodo eitherresponse A thatis,endinstate s A orresponse B endingin s B .However, thelearnerdoesn'tjumprightfrombeingundecidedtobeingsure A isthecorrect thingtodoor B .Instead,thelearnerspendssometimeinatentativeA state,oratentativeB "state,tryingtheresponseoutdenotedhere t A and t B Imaginethatoncethelearnerhasdecided,itisnal,soonce s A or s B isentered itisneverleft.Fortheotherstatechanges,imagineatransitionismadewith probability p ineitherdirection. a Constructthetransitionmatrix. b Take p =0 : 25andtaketheinitialvectortobe1at s U .Runthisforve steps.Whatisthechanceofendingupat s A ? c Dothesamefor p =0 : 20. d Graph p versusthechanceofendingat s A .Isthereathresholdvaluefor p abovewhichthelearnerisalmostsurenottotakelongerthanvesteps? 5 Acertaintownisinacertaincountrythisisahypotheticalproblem.Eachyear tenpercentofthetowndwellersmovetootherpartsofthecountry.Eachyear onepercentofthepeoplefromelsewheremovetothetown.Assumethatthere aretwostates s T ,livingintown,and s C ,livingelsewhere. a Constructthetransistionmatrix.

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Topic:MarkovChains 285 b Startingwithaninitialdistribution s T =0 : 3and s C =0 : 7,gettheresults forthersttenyears. c Dothesamefor s T =0 : 2. d Arethetwooutcomesalikeordierent? 6 FortheWorldSeriesapplication,useacomputertogeneratethesevenvectors for p =0 : 55and p =0 : 6. a WhatisthechanceoftheNationalLeagueteamwinningitall,eventhough theyhaveonlyaprobabilityof0 : 45or0 : 40ofwinninganyonegame? b Graphtheprobability p againstthechancethattheAmericanLeagueteam winsitall.Isthereathresholdvalue|a p abovewhichthebetterteamis essentiallyensuredofwinning? Somesamplecodeisincludedbelow. 7 A Markovmatrix haseachentrypositiveandeachcolumnsumsto1. a CheckthatthethreetransistionmatricesshowninthisTopicmeetthesetwo conditions.Mustanytransitionmatrixdoso? b Observethatif A~v 0 = ~v 1 and A~v 1 = ~v 2 then A 2 isatransitionmatrixfrom ~v 0 to ~v 2 .ShowthatapowerofaMarkovmatrixisalsoaMarkovmatrix. c GeneralizetheprioritembyprovingthattheproductoftwoappropriatelysizedMarkovmatricesisaMarkovmatrix. ComputerCode Thisscript markov.m forthecomputeralgebrasystemOctavewasusedto generatethetableofWorldSeriesoutcomes.Thesharpcharacter # marksthe restofalineasacomment. #OctavescriptfiletocomputechanceofWorldSeriesoutcomes. functionw=markovp,v q=1-p; A=[0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;#0-0 p,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;#1-0 q,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;#0-1_ 0,p,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;#2-0 0,q,p,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;#1-1 0,0,q,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;#0-2__ 0,0,0,p,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;#3-0 0,0,0,q,p,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;#2-1 0,0,0,0,q,p,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;#1-2_ 0,0,0,0,0,q,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;#0-3 0,0,0,0,0,0,p,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0;#4-0 0,0,0,0,0,0,q,p,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;#3-1__ 0,0,0,0,0,0,0,q,p,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0;#2-2 0,0,0,0,0,0,0,0,q,p,0,0,0,0,0,0,0,0,0,0,0,0,0,0;#1-3 0,0,0,0,0,0,0,0,0,q,0,0,0,0,1,0,0,0,0,0,0,0,0,0;#0-4_ 0,0,0,0,0,0,0,0,0,0,0,p,0,0,0,1,0,0,0,0,0,0,0,0;#4-1 0,0,0,0,0,0,0,0,0,0,0,q,p,0,0,0,0,0,0,0,0,0,0,0;#3-2 0,0,0,0,0,0,0,0,0,0,0,0,q,p,0,0,0,0,0,0,0,0,0,0;#2-3__ 0,0,0,0,0,0,0,0,0,0,0,0,0,q,0,0,0,0,1,0,0,0,0,0;#1-4 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,p,0,0,1,0,0,0,0;#4-2 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q,p,0,0,0,0,0,0;#3-3_ 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q,0,0,0,1,0,0;#2-4

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286 ChapterThree.MapsBetweenSpaces 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,p,0,1,0;#4-3 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,q,0,0,1];#3-4 w=A*v; endfunction ThentheOctavesessionwasthis. >v0=[1;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0;0] >p=.5 >v1=markovp,v0 >v2=markovp,v1 ... Translatingtoanothercomputeralgebrasystemshouldbeeasy|allhavecommandssimilartothese.

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Topic:OrthonormalMatrices 287 Topic:OrthonormalMatrices In TheElements ,Euclidconsiderstwogurestobethesameiftheyhavethe samesizeandshape.Thatis,thetrianglesbelowarenotequalbecausethey arenotthesamesetofpoints.Buttheyare congruent |essentiallyindistinguishableforEuclid'spurposes|becausewecanimaginepickingtheplaneup, slidingitoverandrotatingitabit,althoughnotwarpingorstretchingit,and thenputtingitbackdown,tosuperimposetherstgureonthesecond.Euclid neverexplicitlystatesthisprinciplebutheusesitoften[Casey]. P 1 P 2 P 3 Q 1 Q 2 Q 3 Inmodernterminology,pickingtheplaneup..."meansconsideringamap fromtheplanetoitself.Euclidhaslimitedconsiderationtoonlycertaintransformationsoftheplane,onesthatmaypossiblyslideorturntheplanebutnot bendorstretchit.Accordingly,wedeneamap f : R 2 R 2 tobe distancepreserving ora rigidmotion oran isometry ,ifforallpoints P 1 ;P 2 2 R 2 ,the distancefrom f P 1 to f P 2 equalsthedistancefrom P 1 to P 2 .Wealsodene aplane gure tobeasetofpointsintheplaneandwesaythattwogures are congruent ifthereisadistance-preservingmapfromtheplanetoitselfthat carriesonegureontotheother. ManystatementsfromEuclideangeometryfolloweasilyfromthesedenitions.Someare:icollinearityisinvariantunderanydistance-preservingmap thatis,if P 1 P 2 ,and P 3 arecollinearthensoare f P 1 f P 2 ,and f P 3 iibetweenessisinvariantunderanydistance-preservingmapif P 2 isbetween P 1 and P 3 thensois f P 2 between f P 1 and f P 3 ,iiithepropertyof beingatriangleisinvariantunderanydistance-preservingmapifagureisa trianglethentheimageofthatgureisalsoatriangle,ivandthepropertyof beingacircleisinvariantunderanydistance-preservingmap.In1872,F.Klein suggestedthatEuclideangeometrycanbecharacterizedasthestudyofpropertiesthatareinvariantunderthesemaps.ThisformspartofKlein'sErlanger Program,whichproposestheorganizingprinciplethateachkindofgeometry| Euclidean,projective,etc.|canbedescribedasthestudyoftheproperties thatareinvariantundersomegroupoftransformations.Theword`group'here meansmorethanjust`collection',butthatliesoutsideofourscope. Wecanuselinearalgebratocharacterizethedistance-preservingmapsof theplane. First,therearedistance-preservingtransformationsoftheplanethatarenot linear.Theobviousexampleisthis translation x y 7! x y + 1 0 = x +1 y

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288 ChapterThree.MapsBetweenSpaces However,thisexampleturnsouttobetheonlyexample,inthesensethatif f is distance-preservingandsends ~ 0to ~v 0 thenthemap ~v 7! f ~v )]TJ/F11 9.9626 Tf 8.043 0 Td [(~v 0 islinear.That willfollowimmediatelyfromthisstatement:amap t thatisdistance-preserving andsends ~ 0toitselfislinear.Toprovethisequivalentstatement,let t ~e 1 = a b t ~e 2 = c d forsome a;b;c;d 2 R .Thentoshowthat t islinear,wecanshowthatitcan berepresentedbyamatrix,thatis,that t actsinthiswayforall x;y 2 R ~v = x y t 7)167(! ax + cy bx + dy Recallthatifwexthreenon-collinearpointsthenanypointintheplanecan bedescribedbygivingitsdistancefromthosethree.Soanypoint ~v inthe domainisdeterminedbyitsdistancefromthethreexedpoints ~ 0, ~e 1 ,and ~e 2 Similarly,anypoint t ~v inthecodomainisdeterminedbyitsdistancefromthe threexedpoints t ~ 0, t ~e 1 ,and t ~e 2 thesethreearenotcollinearbecause,as mentionedabove,collinearityisinvariantand ~ 0, ~e 1 ,and ~e 2 arenotcollinear. Infact,because t isdistance-preserving,wecansaymore:forthepoint ~v inthe planethatisdeterminedbybeingthedistance d 0 from ~ 0,thedistance d 1 from ~e 1 ,andthedistance d 2 from ~e 2 ,itsimage t ~v mustbetheuniquepointinthe codomainthatisdeterminedbybeing d 0 from t ~ 0, d 1 from t ~e 1 ,and d 2 from t ~e 2 .Becauseoftheuniqueness,checkingthattheactionin worksinthe d 0 d 1 ,and d 2 cases dist x y ; ~ 0=dist t x y ;t ~ 0=dist ax + cy bx + dy ; ~ 0 t isassumedtosend ~ 0toitself dist x y ;~e 1 =dist t x y ;t ~e 1 =dist ax + cy bx + dy ; a b and dist x y ;~e 2 =dist t x y ;t ~e 2 =dist ax + cy bx + dy ; c d sucestoshowthat describes t .Thosechecksareroutine. Thus,anydistance-preserving f : R 2 R 2 canbewritten f ~v = t ~v + ~v 0 forsomeconstantvector ~v 0 andlinearmap t thatisdistance-preserving. Noteverylinearmapisdistance-preserving,forexample, ~v 7! 2 ~v doesnot preservedistances.Butthereisaneatcharacterization:alineartransformation t oftheplaneisdistance-preservingifandonlyifboth k t ~e 1 k = k t ~e 2 k =1and t ~e 1 isorthogonalto t ~e 2 .The`onlyif'halfofthatstatementiseasy|because t isdistance-preservingitmustpreservethelengthsofvectors,andbecause t isdistance-preservingthePythagoreantheoremshowsthatitmustpreserve

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Topic:OrthonormalMatrices 289 orthogonality.Forthe`if'half,itsucestocheckthatthemappreserves lengthsofvectors,becausethenforall ~p and ~q thedistancebetweenthetwois preserved k t ~p )]TJ/F11 9.9626 Tf 9.938 0 Td [(~q k = k t ~p )]TJ/F11 9.9626 Tf 9.963 0 Td [(t ~q k = k ~p )]TJ/F11 9.9626 Tf 9.938 0 Td [(~q k .Forthatcheck,let ~v = x y t ~e 1 = a b t ~e 2 = c d and,withthe`if'assumptionsthat a 2 + b 2 = c 2 + d 2 =1and ac + bd =0we havethis. k t ~v k 2 = ax + cy 2 + bx + dy 2 = a 2 x 2 +2 acxy + c 2 y 2 + b 2 x 2 +2 bdxy + d 2 y 2 = x 2 a 2 + b 2 + y 2 c 2 + d 2 +2 xy ac + bd = x 2 + y 2 = k ~v k 2 Onethingthatisneataboutthischaracterizationisthatwecaneasily recognizematricesthatrepresentsuchamapwithrespecttothestandardbases. Thosematriceshavethatwhenthecolumnsarewrittenasvectorsthenthey areoflengthoneandaremutuallyorthogonal.Suchamatrixiscalledan orthonormalmatrix or orthogonalmatrix thesecondtermiscommonlyused tomeannotjustthatthecolumnsareorthogonal,butalsothattheyhavelength one. Wecanusethisinsighttodelimitthegeometricactionspossibleindistancepreservingmaps.Because k t ~v k = k ~v k ,any ~v ismappedby t toliesomewhere onthecircleabouttheoriginthathasradiusequaltothelengthof ~v .In particular, ~e 1 and ~e 2 aremappedtotheunitcircle.What'smore,oncewex theunitvector ~e 1 asmappedtothevectorwithcomponents a and b thenthere areonlytwoplaceswhere ~e 2 canbemappedifthatimageistobeperpendicular totherstvector:onewhere ~e 2 maintainsitspositionaquartercircleclockwise from ~e 1 a b )]TJ/F10 6.9738 Tf 6.227 0 Td [(b a Rep E 2 ; E 2 t = a )]TJ/F11 9.9626 Tf 7.749 0 Td [(b ba andonewhereisismappedaquartercirclecounterclockwise. a b b )]TJ/F10 6.9738 Tf 6.226 0 Td [(a Rep E 2 ; E 2 t = ab b )]TJ/F11 9.9626 Tf 7.748 0 Td [(a

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290 ChapterThree.MapsBetweenSpaces Wecangeometricallydescribethesetwocases.Let betheanglebetween the x -axisandtheimageof ~e 1 ,measuredcounterclockwise.Therstmatrix aboverepresents,withrespecttothestandardbases,a rotation oftheplaneby radians. a b )]TJ/F10 6.9738 Tf 6.227 0 Td [(b a x y t 7)167(! x cos )]TJ/F11 9.9626 Tf 9.962 0 Td [(y sin x sin + y cos Thesecondmatrixaboverepresentsa reection oftheplanethroughtheline bisectingtheanglebetween ~e 1 and t ~e 1 a b b )]TJ/F10 6.9738 Tf 6.227 0 Td [(a x y t 7)167(! x cos + y sin x sin )]TJ/F11 9.9626 Tf 9.962 0 Td [(y cos Thispictureshows ~e 1 reectedupintotherstquadrantand ~e 2 reecteddown intothefourthquadrant. Noteagain:theanglebetween ~e 1 and ~e 2 runscounterclockwise,andinthe rstmapabovetheanglefrom t ~e 1 to t ~e 2 isalsocounterclockwise,sothe orientationoftheangleispreserved.Butinthesecondmaptheorientationis reversed.Adistance-preservingmapis direct ifitpreservesorientationsand opposite ifitreversesorientation. So,wehavecharacterizedtheEuclideanstudyofcongruence:itconsiders, forplanegures,thepropertiesthatareinvariantundercombinationsofia rotationfollowedbyatranslation,oriiareectionfollowedbyatranslation areectionfollowedbyanon-trivialtranslationisa glidereection Anotheridea,besidescongruenceofgures,encounteredinelementarygeometryisthatguresare similar iftheyarecongruentafterachangeofscale. Thesetwotrianglesaresimilarsincethesecondisthesameshapeastherst, but3 = 2-thsthesize. P 1 P 2 P 3 Q 1 Q 2 Q 3 Fromtheabovework,wehavethatguresaresimilarifthereisanorthonormal matrix T suchthatthepoints ~q ononearederivedfromthepoints ~p by ~q = kT ~v + ~p 0 forsomenonzerorealnumber k andconstantvector ~p 0 .

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Topic:OrthonormalMatrices 291 AlthoughmanyoftheseideaswererstexploredbyEuclid,mathematics istimelessandtheyareverymuchinusetoday.Oneapplicationofthemaps studiedaboveisincomputergraphics.Wecan,forexample,animatethistop viewofacubebyputtingtogetherlmframesofitrotating;that'sarigid motion. Frame1Frame2Frame3 Wecouldalsomakethecubeappeartobemovingawayfromusbyproducing lmframesofitshrinking,whichgivesusguresthataresimilar. Frame1:Frame2:Frame3: Computergraphicsincorporatestechniquesfromlinearalgebrainmanyother waysseeExercise4. Sotheanalysisaboveofdistance-preservingmapsisusefulaswellasinteresting.Abeautifulbookthatexploressomeofthisareais[Weyl].Moreon groups,oftransformationsandotherwise,canbefoundinanybookonModern Algebra,forinstance[Birkho&MacLane].MoreonKleinandtheErlanger Programisin[Yaglom]. Exercises 1 Decideifeachoftheseisanorthonormalmatrix. a 1 = p 2 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 = p 2 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 = p 2 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 = p 2 b 1 = p 3 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 = p 3 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 = p 3 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 = p 3 c 1 = p 3 )]TJ 7.168 7.432 Td [(p 2 = p 3 )]TJ 7.168 7.432 Td [(p 2 = p 3 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 = p 3 2 Writedowntheformulaforeachofthesedistance-preservingmaps. a themapthatrotates = 6radians,andthentranslatesby ~e 2 b themapthatreectsabouttheline y =2 x c themapthatreectsabout y = )]TJ/F29 8.9664 Tf 7.168 0 Td [(2 x andtranslatesover1andup1 3a Theproofthatamapthatisdistance-preservingandsendsthezerovector toitselfincidentallyshowsthatsuchamapisone-to-oneandontothepoint inthedomaindeterminedby d 0 d 1 ,and d 2 correspondstothepointinthe codomaindeterminedbythosethree.Thereforeanydistance-preservingmap hasaninverse.Showthattheinverseisalsodistance-preserving. b Provethatcongruenceisanequivalencerelationbetweenplanegures. 4 Inpracticethematrixforthedistance-preservinglineartransformationandthe translationareoftencombinedintoone.Checkthatthesetwocomputationsyield thesamersttwocomponents. ac bd x y + e f ace bdf 001 x y 1 !

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292 ChapterThree.MapsBetweenSpaces Theseare homogeneouscoordinates ;seetheTopiconProjectiveGeometry. 5a VerifythatthepropertiesdescribedinthesecondparagraphofthisTopic asinvariantunderdistance-preservingmapsareindeedso. b GivetwomorepropertiesthatareofinterestinEuclideangeometryfrom yourexperienceinstudyingthatsubjectthatarealsoinvariantunderdistancepreservingmaps. c GiveapropertythatisnotofinterestinEuclideangeometryandisnot invariantunderdistance-preservingmaps.

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ChapterFour Determinants Intherstchapterofthisbookweconsideredlinearsystemsandwepickedout thespecialcaseofsystemswiththesamenumberofequationsasunknowns, thoseoftheform T~x = ~ b where T isasquarematrix.Wenotedadistinction betweentwoclassesof T 's.Whilesuchsystemsmayhaveauniquesolutionor nosolutionsorinnitelymanysolutions,ifaparticular T isassociatedwitha uniquesolutioninanysystem,suchasthehomogeneoussystem ~ b = ~ 0,then T isassociatedwithauniquesolutionforevery ~ b .Wecallsuchamatrixof coecients`nonsingular'.Theotherkindof T ,whereeverylinearsystemfor whichitisthematrixofcoecientshaseithernosolutionorinnitelymany solutions,wecall`singular'. Throughthesecondandthirdchaptersthevalueofthisdistinctionhasbeen atheme.Forinstance,wenowknowthatnonsingularityofan n n matrix T isequivalenttoeachofthese: asystem T~x = ~ b hasasolution,andthatsolutionisunique; Gauss-Jordanreductionof T yieldsanidentitymatrix; therowsof T formalinearlyindependentset; thecolumnsof T formabasisfor R n ; anymapthat T representsisanisomorphism; aninversematrix T )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 exists. Sowhenwelookataparticularsquarematrix,thequestionofwhetherit isnonsingularisoneoftherstthingsthatweask.Thischapterdevelops aformulatodeterminethis.Sincewewillrestrictthediscussiontosquare matrices,inthischapterwewillusuallysimplysay`matrix'inplaceof`square matrix'. Moreprecisely,wewilldevelopinnitelymanyformulas,onefor1 1matrices,onefor2 2matrices,etc.Ofcourse,theseformulasarerelated|that is,wewilldevelopafamilyofformulas,aschemethatdescribestheformulafor eachsize. 293

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294 ChapterFour.Determinants IDefinition For1 1matrices,determiningnonsingularityistrivial. )]TJ/F11 9.9626 Tf 4.566 -7.97 Td [(a isnonsingulari a 6 =0 The2 2formulacameoutinthecourseofdevelopingtheinverse. ab cd isnonsingulari ad )]TJ/F11 9.9626 Tf 9.963 0 Td [(bc 6 =0 The3 3formulacanbeproducedsimilarlyseeExercise9. 0 @ abc def ghi 1 A isnonsingulari aei + bfg + cdh )]TJ/F11 9.9626 Tf 9.963 0 Td [(hfa )]TJ/F11 9.9626 Tf 9.962 0 Td [(idb )]TJ/F11 9.9626 Tf 9.963 0 Td [(gec 6 =0 Withthesecasesinmind,wepositafamilyofformulas, a ad )]TJ/F11 9.9626 Tf 8.747 0 Td [(bc ,etc.Foreach n theformulagivesrisetoa determinant functiondet n n : M n n R suchthat an n n matrix T isnonsingularifandonlyifdet n n T 6 =0.Weusuallyomit thesubscriptbecauseif T is n n then`det T 'couldonlymean`det n n T '. I.1Exploration Thissubsectionisoptional.Itbrieydescribeshowaninvestigatormightcome toagoodgeneraldenition,whichisgiveninthenextsubsection. Thethreecasesabovedon'tshowanevidentpatterntouseforthegeneral n n formula.Wemayspotthatthe1 1term a hasoneletter,thatthe2 2 terms ad and bc havetwoletters,andthatthe3 3terms aei ,etc.,havethree letters.Wemayalsoobservethatinthosetermsthereisaletterfromeachrow andcolumnofthematrix,e.g.,thelettersinthe cdh term 0 @ c d h 1 A comeonefromeachrowandonefromeachcolumn.Buttheseobservations perhapsseemmorepuzzlingthanenlightening.Forinstance,wemightwonder whysomeofthetermsareaddedwhileothersaresubtracted. Agoodproblemsolvingstrategyistoseewhatpropertiesasolutionmust haveandthensearchforsomethingwiththoseproperties.Soweshallstartby askingwhatpropertieswerequireoftheformulas. Atthispoint,ourprimarywaytodecidewhetheramatrixissingularis todoGaussianreductionandthencheckwhetherthediagonalofresulting echelonformmatrixhasanyzeroesthatis,tocheckwhethertheproduct downthediagonaliszero.So,wemayexpectthattheproofthataformula

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SectionI.Definition 295 determinessingularitywillinvolveapplyingGauss'methodtothematrix,to showthatintheendtheproductdownthediagonaliszeroifandonlyifthe determinantformulagiveszero.Thissuggestsourinitialplan:wewilllookfor afamilyoffunctionswiththepropertyofbeingunaectedbyrowoperations andwiththepropertythatadeterminantofanechelonformmatrixisthe productofitsdiagonalentries.Underthisplan,aproofthatthefunctions determinesingularitywouldgo,Where T !! ^ T istheGaussianreduction, thedeterminantof T equalsthedeterminantof ^ T becausethedeterminantis unchangedbyrowoperations,whichistheproductdownthediagonal,which iszeroifandonlyifthematrixissingular".Intherestofthissubsectionwe willtestthisplanonthe2 2and3 3determinantsthatweknow.Wewillend upmodifyingtheunaectedbyrowoperations"part,butnotbymuch. Therststepincheckingtheplanistotestwhetherthe2 2and3 3 formulasareunaectedbytherowoperationofpivoting:if T k i + j )167(! ^ T thenisdet ^ T =det T ?Thischeckofthe2 2determinantafterthe k 1 + 2 operation det ab ka + ckb + d = a kb + d )]TJ/F8 9.9626 Tf 9.962 0 Td [( ka + c b = ad )]TJ/F11 9.9626 Tf 9.962 0 Td [(bc showsthatitisindeedunchanged,andtheother2 2pivot k 2 + 1 givesthe sameresult.The3 3pivot k 3 + 2 leavesthedeterminantunchanged det 0 @ abc kg + dkh + eki + f ghi 1 A = a kh + e i + b ki + f g + c kg + d h )]TJ/F11 9.9626 Tf 9.963 0 Td [(h ki + f a )]TJ/F11 9.9626 Tf 9.962 0 Td [(i kg + d b )]TJ/F11 9.9626 Tf 9.963 0 Td [(g kh + e c = aei + bfg + cdh )]TJ/F11 9.9626 Tf 9.963 0 Td [(hfa )]TJ/F11 9.9626 Tf 9.963 0 Td [(idb )]TJ/F11 9.9626 Tf 9.962 0 Td [(gec asdotheother3 3pivotoperations. Sothereseemstobepromiseintheplan.Ofcourse,perhapsthe4 4 determinantformulaisaectedbypivoting.Weareexploringapossibilityhere andwedonotyethaveallthefacts.Nonetheless,sofar,sogood. Thenextstepistocomparedet ^ T withdet T fortheoperation T i $ j )167(! ^ T ofswappingtworows.The2 2rowswap 1 $ 2 det cd ab = cb )]TJ/F11 9.9626 Tf 9.963 0 Td [(ad doesnotyield ad )]TJ/F11 9.9626 Tf 9.962 0 Td [(bc .This 1 $ 3 swapinsideofa3 3matrix det 0 @ ghi def abc 1 A = gec + hfa + idb )]TJ/F11 9.9626 Tf 9.963 0 Td [(bfg )]TJ/F11 9.9626 Tf 9.963 0 Td [(cdh )]TJ/F11 9.9626 Tf 9.962 0 Td [(aei

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296 ChapterFour.Determinants alsodoesnotgivethesamedeterminantasbeforetheswap|againthereisa signchange.Tryingadierent3 3swap 1 $ 2 det 0 @ def abc ghi 1 A = dbi + ecg + fah )]TJ/F11 9.9626 Tf 9.963 0 Td [(hcd )]TJ/F11 9.9626 Tf 9.962 0 Td [(iae )]TJ/F11 9.9626 Tf 9.963 0 Td [(gbf alsogivesachangeofsign. Thus,rowswapsappeartochangethesignofadeterminant.Thismodiesourplan,butdoesnotwreckit.Weintendtodecidenonsingularityby consideringonlywhetherthedeterminantiszero,notbyconsideringitssign. Therefore,insteadofexpectingdeterminantstobeentirelyunaectedbyrow operations,willlookforthemtochangesignonaswap. Tonish,wecomparedet ^ T todet T fortheoperation T k i )167(! ^ T ofmultiplyingarowbyascalar k 6 =0.Oneofthe2 2casesis det ab kckd = a kd )]TJ/F8 9.9626 Tf 9.962 0 Td [( kc b = k ad )]TJ/F11 9.9626 Tf 9.963 0 Td [(bc andtheothercasehasthesameresult.Hereisone3 3case det 0 @ abc def kgkhki 1 A = ae ki + bf kg + cd kh )]TJ/F8 9.9626 Tf 7.749 0 Td [( kh fa )]TJ/F8 9.9626 Tf 9.963 0 Td [( ki db )]TJ/F8 9.9626 Tf 9.963 0 Td [( kg ec = k aei + bfg + cdh )]TJ/F11 9.9626 Tf 9.963 0 Td [(hfa )]TJ/F11 9.9626 Tf 9.962 0 Td [(idb )]TJ/F11 9.9626 Tf 9.962 0 Td [(gec andtheothertwoaresimilar.Theseleadustosuspectthatmultiplyingarow by k multipliesthedeterminantby k .Thistswithourmodiedplanbecause weareaskingonlythatthezeronessofthedeterminantbeunchangedandwe arenotfocusingonthedeterminant'ssignormagnitude. Insummary,todeveloptheschemefortheformulastocomputedeterminants,welookfordeterminantfunctionsthatremainunchangedunderthe pivotingoperation,thatchangesignonarowswap,andthatrescaleonthe rescalingofarow.Inthenexttwosubsectionswewillndthatforeach n such afunctionexistsandisunique. Forthenextsubsection,notethat,asabove,scalarscomeoutofeachrow withoutaectingotherrows.Forinstance,inthisequality det 0 @ 339 211 510 )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 1 A =3 det 0 @ 113 211 510 )]TJ/F8 9.9626 Tf 7.749 0 Td [(5 1 A the3isn'tfactoredoutofallthreerows,onlyoutofthetoprow.Thedeterminantactsoneachrowofindependentlyoftheotherrows.Whenwewanttouse thispropertyofdeterminants,weshallwritethedeterminantasafunctionof therows:`det ~ 1 ;~ 2 ;:::~ n ',insteadofas`det T 'or`det t 1 ; 1 ;:::;t n;n '.The denitionofthedeterminantthatstartsthenextsubsectioniswritteninthis way.

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SectionI.Definition 297 Exercises X 1.1 Evaluatethedeterminantofeach. a 31 )]TJ/F29 8.9664 Tf 7.167 0 Td [(11 b 201 311 )]TJ/F29 8.9664 Tf 7.168 0 Td [(101 c 401 001 13 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 1.2 Evaluatethedeterminantofeach. a 20 )]TJ/F29 8.9664 Tf 7.167 0 Td [(13 b 211 05 )]TJ/F29 8.9664 Tf 7.168 0 Td [(2 1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(34 c 234 567 891 X 1.3 Verifythatthedeterminantofanupper-triangular3 3matrixistheproduct downthediagonal. det abc 0 ef 00 i = aei Dolower-triangularmatricesworkthesameway? X 1.4 Usethedeterminanttodecideifeachissingularornonsingular. a 21 31 b 01 1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 c 42 21 1.5 Singularornonsingular?Usethedeterminanttodecide. a 211 322 014 b 101 211 413 c 210 3 )]TJ/F29 8.9664 Tf 7.168 0 Td [(20 100 X 1.6 Eachpairofmatricesdierbyonerowoperation.Usethisoperationtocompare det A withdet B a A = 12 23 B = 12 0 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 b A = 310 001 012 B = 310 012 001 c A = 1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(13 22 )]TJ/F29 8.9664 Tf 7.167 0 Td [(6 104 B = 1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(13 11 )]TJ/F29 8.9664 Tf 7.168 0 Td [(3 104 1.7 Showthis. det 111 abc a 2 b 2 c 2 = b )]TJ/F32 8.9664 Tf 9.215 0 Td [(a c )]TJ/F32 8.9664 Tf 9.216 0 Td [(a c )]TJ/F32 8.9664 Tf 9.216 0 Td [(b X 1.8 Whichrealnumbers x makethismatrixsingular? 12 )]TJ/F32 8.9664 Tf 9.216 0 Td [(x 4 88 )]TJ/F32 8.9664 Tf 9.216 0 Td [(x 1.9 DotheGaussianreductiontochecktheformulafor3 3matricesstatedinthe preambletothissection. abc def ghi isnonsingulari aei + bfg + cdh )]TJ/F32 8.9664 Tf 9.216 0 Td [(hfa )]TJ/F32 8.9664 Tf 9.215 0 Td [(idb )]TJ/F32 8.9664 Tf 9.216 0 Td [(gec 6 =0 1.10 Showthattheequationofalinein R 2 thru x 1 ;y 1 and x 2 ;y 2 isexpressed bythisdeterminant. det xy 1 x 1 y 1 1 x 2 y 2 1 =0 x 1 6 = x 2

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298 ChapterFour.Determinants X 1.11 Manypeopleknowthismnemonicforthedeterminantofa3 3matrix:rst repeatthersttwocolumnsandthensumtheproductsontheforwarddiagonals andsubtracttheproductsonthebackwarddiagonals.Thatis,rstwrite h 1 ; 1 h 1 ; 2 h 1 ; 3 h 1 ; 1 h 1 ; 2 h 2 ; 1 h 2 ; 2 h 2 ; 3 h 2 ; 1 h 2 ; 2 h 3 ; 1 h 3 ; 2 h 3 ; 3 h 3 ; 1 h 3 ; 2 andthencalculatethis. h 1 ; 1 h 2 ; 2 h 3 ; 3 + h 1 ; 2 h 2 ; 3 h 3 ; 1 + h 1 ; 3 h 2 ; 1 h 3 ; 2 )]TJ/F32 8.9664 Tf 7.167 0 Td [(h 3 ; 1 h 2 ; 2 h 1 ; 3 )]TJ/F32 8.9664 Tf 9.216 0 Td [(h 3 ; 2 h 2 ; 3 h 1 ; 1 )]TJ/F32 8.9664 Tf 9.215 0 Td [(h 3 ; 3 h 2 ; 1 h 1 ; 2 a Checkthatthisagreeswiththeformulagiveninthepreambletothissection. b Doesitextendtoother-sizeddeterminants? 1.12 The crossproduct ofthevectors ~x = x 1 x 2 x 3 ~y = y 1 y 2 y 3 isthevectorcomputedasthisdeterminant. ~x ~y =det ~e 1 ~e 2 ~e 3 x 1 x 2 x 3 y 1 y 2 y 3 Notethattherstrowiscomposedofvectors,thevectorsfromthestandardbasis for R 3 .Showthatthecrossproductoftwovectorsisperpendiculartoeachvector. 1.13 Provethateachstatementholdsfor2 2matrices. a Thedeterminantofaproductistheproductofthedeterminantsdet ST = det S det T b If T isinvertiblethenthedeterminantoftheinverseistheinverseofthe determinantdet T )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 =det T )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 Matrices T and T 0 are similar ifthereisanonsingularmatrix P suchthat T 0 = PTP )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 .ThisdenitionisinChapterFive.Showthatsimilar2 2matriceshave thesamedeterminant. X 1.14 Provethattheareaofthisregionintheplane x 1 y 1 x 2 y 2 isequaltothevalueofthisdeterminant. det x 1 x 2 y 1 y 2 Comparewiththis. det x 2 x 1 y 2 y 1 1.15 Provethatfor2 2matrices,thedeterminantofamatrixequalsthedeterminantofitstranspose.Doesthatalsoholdfor3 3matrices? X 1.16 Isthedeterminantfunctionlinear|isdet x T + y S = x det T + y det S ? 1.17 Showthatif A is3 3thendet c A = c 3 det A foranyscalar c .

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SectionI.Definition 299 1.18 Whichrealnumbers make cos )]TJ/F29 8.9664 Tf 8.704 0 Td [(sin sin cos singular?Explaingeometrically. ? 1.19 Ifathirdorderdeterminanthaselements1,2,...,9,whatisthemaximum valueitmayhave?[Am.Math.Mon.,Apr.1955] I.2PropertiesofDeterminants Asdescribedabove,wewantaformulatodeterminewhetheran n n matrix isnonsingular.Wewillnotbeginbystatingsuchaformula.Instead,wewill beginbyconsideringthefunctionthatsuchaformulacalculates.Wewilldene thefunctionbyitsproperties,thenprovethatthefunctionwiththesepropertiesexistandisuniqueandalsodescribeformulasthatcomputethisfunction. Becausewewillshowthatthefunctionexistsandisunique,fromthestartwe willsay`det T 'insteadof`ifthereisadeterminantfunctionthendet T 'and `thedeterminant'insteadof`anydeterminant'. 2.1Denition A n n determinant isafunctiondet: M n n R suchthat det ~ 1 ;:::;k ~ i + ~ j ;:::;~ n =det ~ 1 ;:::;~ j ;:::;~ n for i 6 = j det ~ 1 ;:::;~ j ;:::;~ i ;:::;~ n = )]TJ/F8 9.9626 Tf 9.409 0 Td [(det ~ 1 ;:::;~ i ;:::;~ j ;:::;~ n for i 6 = j det ~ 1 ;:::;k~ i ;:::;~ n = k det ~ 1 ;:::;~ i ;:::;~ n for k 6 =0 det I =1where I isanidentitymatrix the ~ 'saretherowsofthematrix.Weoftenwrite j T j fordet T 2.2Remark Propertyisredundantsince T i + j )167(! )]TJ/F10 6.9738 Tf 6.226 0 Td [( j + i )167(! i + j )167(! )]TJ/F10 6.9738 Tf 6.227 0 Td [( i )167(! ^ T swapsrows i and j .Itislistedonlyforconvenience. Therstresultshowsthatafunctionsatisfyingtheseconditionsgivesa criteriafornonsingularity.Itslastsentenceisthat,inthecontextoftherst threeconditions,isequivalenttotheconditionthatthedeterminantofan echelonformmatrixistheproductdownthediagonal. 2.3Lemma Amatrixwithtwoidenticalrowshasadeterminantofzero.A matrixwithazerorowhasadeterminantofzero.Amatrixisnonsingularif andonlyifitsdeterminantisnonzero.Thedeterminantofanechelonform matrixistheproductdownitsdiagonal.

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300 ChapterFour.Determinants Proof Toverifytherstsentence,swapthetwoequalrows.Thesignofthe determinantchanges,butthematrixisunchangedandsoitsdeterminantis unchanged.Thusthedeterminantiszero. Thesecondsentenceisclearlytrueifthematrixis1 1.Ifithasatleast tworowsthenapplypropertyofthedenitionwiththezerorowasrow j andwith k =1. det :::;~ i ;:::; ~ 0 ;::: =det :::;~ i ;:::;~ i + ~ 0 ;::: Therstsentenceofthislemmagivesthatthedeterminantiszero. Forthethirdsentence,where T !! ^ T istheGauss-Jordanreduction, bythedenitionthedeterminantof T iszeroifandonlyifthedeterminantof ^ T iszeroalthoughtheycoulddierinsignormagnitude.Anonsingular T Gauss-Jordanreducestoanidentitymatrixandsohasanonzerodeterminant. Asingular T reducestoa ^ T withazerorow;bythesecondsentenceofthis lemmaitsdeterminantiszero. Finally,forthefourthsentence,ifanechelonformmatrixissingularthenit hasazeroonitsdiagonal,thatis,theproductdownitsdiagonaliszero.The thirdsentencesaysthatifamatrixissingularthenitsdeterminantiszero.So iftheechelonformmatrixissingularthenitsdeterminantequalstheproduct downitsdiagonal. Ifanechelonformmatrixisnonsingularthennoneofitsdiagonalentriesis zerosowecanusepropertyofthedenitiontofactorthemoutagain,the verticalbars jj indicatethedeterminantoperation. t 1 ; 1 t 1 ; 2 t 1 ;n 0 t 2 ; 2 t 2 ;n 0 t n;n = t 1 ; 1 t 2 ; 2 t n;n 1 t 1 ; 2 =t 1 ; 1 t 1 ;n =t 1 ; 1 01 t 2 ;n =t 2 ; 2 01 Next,theJordanhalfofGauss-Jordanelimination,usingpropertyofthe denition,leavestheidentitymatrix. = t 1 ; 1 t 2 ; 2 t n;n 100 010 01 = t 1 ; 1 t 2 ; 2 t n;n 1 Therefore,ifanechelonformmatrixisnonsingularthenitsdeterminantisthe productdownitsdiagonal. QED Thatresultgivesusawaytocomputethevalueofadeterminantfunctionon amatrix.DoGaussianreduction,keepingtrackofanychangesofsigncausedby rowswapsandanyscalarsthatarefactoredout,andthennishbymultiplying downthediagonaloftheechelonformresult.Thisproceduretakesthesame timeasGauss'methodandsoissucientlyfasttobepracticalonthesize matricesthatweseeinthisbook.

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SectionI.Definition 301 2.4Example Doing2 2determinants 24 )]TJ/F8 9.9626 Tf 7.749 0 Td [(13 = 24 05 =10 withGauss'methodwon'tgiveabigsavingsbecausethe2 2determinant formulaissoeasy.However,a3 3determinantisusuallyeasiertocalculate withGauss'methodthanwiththeformulagivenearlier. 226 443 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(35 = 226 00 )]TJ/F8 9.9626 Tf 7.748 0 Td [(9 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(35 = )]TJ/F1 9.9626 Tf 9.409 20.423 Td [( 226 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(35 00 )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(54 2.5Example Determinantsofmatricesanybiggerthan3 3arealmostalways mostquicklydonewiththisGauss'methodprocedure. 1013 0114 0005 0101 = 1013 0114 0005 00 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 = )]TJ/F1 9.9626 Tf 9.409 26.401 Td [( 1013 0114 00 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 0005 = )]TJ/F8 9.9626 Tf 7.749 0 Td [( )]TJ/F8 9.9626 Tf 7.749 0 Td [(5=5 Thepriorexampleillustratesanimportantpoint.Althoughwehavenotyet founda4 4determinantformula,ifoneexiststhenweknowwhatvalueitgives tothematrix|ifthereisafunctionwithproperties-thenontheabove matrixthefunctionmustreturn5. 2.6Lemma Foreach n ,ifthereisan n n determinantfunctionthenitis unique. Proof Forany n n matrixwecanperformGauss'methodonthematrix, keepingtrackofhowthesignalternatesonrowswaps,andthenmultiplydown thediagonaloftheechelonformresult.Bythedenitionandthelemma,all n n determinantfunctionsmustreturnthisvalueonthismatrix.Thusall n n determinantfunctionsareequal,thatis,thereisonlyoneinputargument/output valuerelationshipsatisfyingthefourconditions. QED The`ifthereisan n n determinantfunction'emphasizesthat,althoughwe canuseGauss'methodtocomputetheonlyvaluethatadeterminantfunction couldpossiblyreturn,wehaven'tyetshownthatsuchadeterminantfunction existsforall n .Intherestofthesectionwewillproducedeterminantfunctions. Exercises Forthese,assumethatan n n determinantfunctionexistsforall n X 2.7 UseGauss'methodtondeachdeterminant. a 312 310 014 b 1001 2110 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1010 1110 2.8 UseGauss'methodtondeach.

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302 ChapterFour.Determinants a 2 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 b 110 302 522 2.9 Forwhichvaluesof k doesthissystemhaveauniquesolution? x + z )]TJ/F32 8.9664 Tf 9.215 0 Td [(w =2 y )]TJ/F29 8.9664 Tf 9.668 0 Td [(2 z =3 x + kz =4 z )]TJ/F32 8.9664 Tf 9.215 0 Td [(w =2 X 2.10 Expresseachoftheseintermsof j H j a h 3 ; 1 h 3 ; 2 h 3 ; 3 h 2 ; 1 h 2 ; 2 h 2 ; 3 h 1 ; 1 h 1 ; 2 h 1 ; 3 b )]TJ/F32 8.9664 Tf 7.168 0 Td [(h 1 ; 1 )]TJ/F32 8.9664 Tf 7.168 0 Td [(h 1 ; 2 )]TJ/F32 8.9664 Tf 7.168 0 Td [(h 1 ; 3 )]TJ/F29 8.9664 Tf 7.167 0 Td [(2 h 2 ; 1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(2 h 2 ; 2 )]TJ/F29 8.9664 Tf 7.167 0 Td [(2 h 2 ; 3 )]TJ/F29 8.9664 Tf 7.167 0 Td [(3 h 3 ; 1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(3 h 3 ; 2 )]TJ/F29 8.9664 Tf 7.167 0 Td [(3 h 3 ; 3 c h 1 ; 1 + h 3 ; 1 h 1 ; 2 + h 3 ; 2 h 1 ; 3 + h 3 ; 3 h 2 ; 1 h 2 ; 2 h 2 ; 3 5 h 3 ; 1 5 h 3 ; 2 5 h 3 ; 3 X 2.11 Findthedeterminantofadiagonalmatrix. 2.12 Describethesolutionsetofahomogeneouslinearsystemifthedeterminant ofthematrixofcoecientsisnonzero. X 2.13 Showthatthisdeterminantiszero. y + zx + zx + y xyz 111 2.14a Findthe1 1,2 2,and3 3matriceswith i;j entrygivenby )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 i + j b Findthedeterminantofthesquarematrixwith i;j entry )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 i + j 2.15a Findthe1 1,2 2,and3 3matriceswith i;j entrygivenby i + j b Findthedeterminantofthesquarematrixwith i;j entry i + j X 2.16 Showthatdeterminantfunctionsarenotlinearbygivingacasewhere j A + B j6 = j A j + j B j 2.17 Thesecondconditioninthedenition,thatrowswapschangethesignofa determinant,issomewhatannoying.Itmeanswehavetokeeptrackofthenumber ofswaps,tocomputehowthesignalternates.Canwegetridofit?Canwereplace itwiththeconditionthatrowswapsleavethedeterminantunchanged?Ifsothen wewouldneednew1 1,2 2,and3 3formulas,butthatwouldbeaminor matter. 2.18 Provethatthedeterminantofanytriangularmatrix,upperorlower,isthe productdownitsdiagonal. 2.19 RefertothedenitionofelementarymatricesintheMechanicsofMatrix Multiplicationsubsection. a Whatisthedeterminantofeachkindofelementarymatrix? b Provethatif E isanyelementarymatrixthen j ES j = j E jj S j foranyappropriatelysized S c Thisquestiondoesn'tinvolvedeterminants. Provethatif T issingularthen aproduct TS isalsosingular. d Showthat j TS j = j T jj S j e Showthatif T isnonsingularthen j T )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 j = j T j )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 .

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SectionI.Definition 303 2.20 Provethatthedeterminantofaproductistheproductofthedeterminants j TS j = j T jj S j inthisway.Fixthe n n matrix S andconsiderthefunction d : M n n R givenby T 7!j TS j = j S j a Checkthat d satisespropertyinthedenitionofadeterminantfunction. b Checkproperty. c Checkproperty. d Checkproperty. e Concludethedeterminantofaproductistheproductofthedeterminants. 2.21 A submatrix ofagivenmatrix A isonethatcanbeobtainedbydeletingsome oftherowsandcolumnsof A .Thus,therstmatrixhereisasubmatrixofthe second. 31 25 341 09 )]TJ/F29 8.9664 Tf 7.168 0 Td [(2 2 )]TJ/F29 8.9664 Tf 7.168 0 Td [(15 Provethatforanysquarematrix,therankofthematrixis r ifandonlyif r isthe largestintegersuchthatthereisan r r submatrixwithanonzerodeterminant. X 2.22 Provethatamatrixwithrationalentrieshasarationaldeterminant. ? 2.23 Findtheelementoflikenessinasimplifyingafraction,bpowderingthe nose,cbuildingnewstepsonthechurch,dkeepingemeritusprofessorson campus,eputting B C D inthedeterminant 1 aa 2 a 3 a 3 1 aa 2 Ba 3 1 a CDa 3 1 : [Am.Math.Mon.,Feb.1953] I.3ThePermutationExpansion Thepriorsubsectiondenesafunctiontobeadeterminantifitsatisesfour conditionsandshowsthatthereisatmostone n n determinantfunctionfor each n .Whatisleftistoshowthatforeach n suchafunctionexists. Howcouldsuchafunctionnotexist?Afterall,wehavedonecomputations thatstartwithasquarematrix,followtheconditions,andendwithanumber. Thedicultyisthat,asfarasweknow,thecomputationmightnotgivea well-denedresult.Toillustratethispossibility,supposethatweweretochange thesecondconditioninthedenitionofdeterminanttobethatthevalueofa determinantdoesnotchangeonarowswap.ByRemark2.2weknowthat thisconictswiththerstandthirdconditions.Hereisaninstanceofthe conict:herearetwoGauss'methodreductionsofthesamematrix,therst withoutanyrowswap 12 34 )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 1 + 2 )167(! 12 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2

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304 ChapterFour.Determinants andthesecondwithaswap. 12 34 1 $ 2 )167(! 34 12 )]TJ/F7 6.9738 Tf 6.226 0 Td [( = 3 1 + 2 )167(! 34 02 = 3 FollowingDenition2.1givesthatbothcalculationsyieldthedeterminant )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 sinceinthesecondonewekeeptrackofthefactthattherowswapchanges thesignoftheresultofmultiplyingdownthediagonal.Butifwefollowthe suppositionandchangethesecondconditionthenthetwocalculationsyield dierentvalues, )]TJ/F8 9.9626 Tf 7.748 0 Td [(2and2.Thatis,underthesuppositiontheoutcomewouldnot bewell-dened|nofunctionexiststhatsatisesthechangedsecondcondition alongwiththeotherthree. Ofcourse,observingthatDenition2.1doestherightthinginthisone instanceisnotenough;whatwewilldointherestofthissectionistoshow thatthereisneveraconict.Thenaturalwaytotrythiswouldbetodene thedeterminantfunctionwith:Thevalueofthefunctionistheresultofdoing Gauss'method,keepingtrackofrowswaps,andnishingbymultiplyingdown thediagonal".SinceGauss'methodallowsforsomevariation,suchasachoice ofwhichrowtousewhenswapping,wewouldhavetoxanexplicitalgorithm. Thenwewouldbedoneifweveriedthatthiswayofcomputingthedeterminant satisesthefourproperties.Forinstance,if T and ^ T arerelatedbyarowswap thenwewouldneedtoshowthatthisalgorithmreturnsdeterminantsthatare negativesofeachother.However,howtoverifythisisnotevident.Sothe developmentbelowwillnotproceedinthisway.Instead,inthissubsectionwe willdeneadierentwaytocomputethevalueofadeterminant,aformula, andwewillusethiswaytoprovethattheconditionsaresatised. Theformulathatweshalluseisbasedonaninsightgottenfromproperty ofthedenitionofdeterminants.Thispropertyshowsthatdeterminantsare notlinear. 3.1Example Forthismatrixdet A 6 =2 det A A = 21 )]TJ/F8 9.9626 Tf 7.749 0 Td [(13 Instead,thescalarcomesoutofeachofthetworows. 42 )]TJ/F8 9.9626 Tf 7.749 0 Td [(26 =2 21 )]TJ/F8 9.9626 Tf 7.749 0 Td [(26 =4 21 )]TJ/F8 9.9626 Tf 7.749 0 Td [(13 Sincescalarscomeoutarowatatime,wemightguessthatdeterminants arelineararowatatime. 3.2Denition Let V beavectorspace.Amap f : V n R is multilinear if f ~ 1 ;:::;~v + ~w;:::;~ n = f ~ 1 ;:::;~v;:::;~ n + f ~ 1 ;:::;~w;:::;~ n f ~ 1 ;:::;k~v;:::;~ n = k f ~ 1 ;:::;~v;:::;~ n for ~v;~w 2 V and k 2 R

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SectionI.Definition 305 3.3Lemma Determinantsaremultilinear. Proof ThedenitionofdeterminantsgivespropertyLemma2.3following thatdenitioncoversthe k =0casesoweneedonlycheckproperty. det ~ 1 ;:::;~v + ~w;:::;~ n =det ~ 1 ;:::;~v;:::;~ n +det ~ 1 ;:::;~w;:::;~ n Iftheset f ~ 1 ;:::;~ i )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ;~ i +1 ;:::;~ n g islinearlydependentthenallthreematrices aresingularandsoallthreedeterminantsarezeroandtheequalityistrivial. Thereforeassumethatthesetislinearlyindependent.Thissetof n -widerow vectorshas n )]TJ/F8 9.9626 Tf 9.984 0 Td [(1members,sowecanmakeabasisbyaddingonemorevector h ~ 1 ;:::;~ i )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ; ~ ;~ i +1 ;:::;~ n i .Express ~v and ~w withrespecttothisbasis ~v = v 1 ~ 1 + + v i )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ~ i )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 + v i ~ + v i +1 ~ i +1 + + v n ~ n ~w = w 1 ~ 1 + + w i )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 ~ i )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 + w i ~ + w i +1 ~ i +1 + + w n ~ n givingthis. ~v + ~w = v 1 + w 1 ~ 1 + + v i + w i ~ + + v n + w n ~ n Bythedenitionofdeterminant,thevalueofdet ~ 1 ;:::;~v + ~w;:::;~ n isunchangedbythepivotoperationofadding )]TJ/F8 9.9626 Tf 7.749 0 Td [( v 1 + w 1 ~ 1 to ~v + ~w ~v + ~w )]TJ/F8 9.9626 Tf 9.962 0 Td [( v 1 + w 1 ~ 1 = v 2 + w 2 ~ 2 + + v i + w i ~ + + v n + w n ~ n Then,totheresult,wecanadd )]TJ/F8 9.9626 Tf 7.749 0 Td [( v 2 + w 2 ~ 2 ,etc.Thus det ~ 1 ;:::;~v + ~w;:::;~ n =det ~ 1 ;:::; v i + w i ~ ;:::;~ n = v i + w i det ~ 1 ;:::; ~ ;:::;~ n = v i det ~ 1 ;:::; ~ ;:::;~ n + w i det ~ 1 ;:::; ~ ;:::;~ n usingforthesecondequality.Tonish,bring v i and w i backinsidein frontof ~ andusepivotingagain,thistimetoreconstructtheexpressionsof ~v and ~w intermsofthebasis,e.g.,startwiththepivotoperationsofadding v 1 ~ 1 to v i ~ and w 1 ~ 1 to w i ~ 1 ,etc. QED Multilinearityallowsustoexpandadeterminantintoasumofdeterminants, eachofwhichinvolvesasimplematrix. 3.4Example Wecanusemultilinearitytosplitthisdeterminantintotwo, rstbreakinguptherstrow 21 43 = 20 43 + 01 43 andthenseparatingeachofthosetwo,breakingalongthesecondrows. = 20 40 + 20 03 + 01 40 + 01 03

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306 ChapterFour.Determinants Weareleftwithfourdeterminants,suchthatineachrowofeachmatrixthere isasingleentryfromtheoriginalmatrix. 3.5Example Inthesameway,a3 3determinantseparatesintoasumof manysimplerdeterminants.Westartbysplittingalongtherstrow,producing threedeterminantsthezerointhe1 ; 3positionisunderlinedtosetitovisually fromthezeroesthatappearinthesplitting. 21 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 430 215 = 200 430 215 + 010 430 215 + 00 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 430 215 Eachofthesethreewillitselfsplitinthreealongthesecondrow.Eachof theresultingninesplitsinthreealongthethirdrow,resultingintwentyseven determinants = 200 400 200 + 200 400 010 + 200 400 005 + 200 030 200 + + 00 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 000 005 suchthateachrowcontainsasingleentryfromthestartingmatrix. Soan n n determinantexpandsintoasumof n n determinantswhereeach rowofeachsummandscontainsasingleentryfromthestartingmatrix.However,manyofthesesummanddeterminantsarezero. 3.6Example Ineachofthesethreematricesfromtheaboveexpansion,two oftherowshavetheirentryfromthestartingmatrixinthesamecolumn,e.g., intherstmatrix,the2andthe4bothcomefromtherstcolumn. 200 400 010 00 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 030 005 010 000 005 Anysuchmatrixissingular,becauseineach,onerowisamultipleoftheother orisazerorow.Thus,anysuchdeterminantiszero,byLemma2.3. Therefore,theaboveexpansionofthe3 3determinantintothesumofthe twentysevendeterminantssimpliestothesumofthesesix. 21 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 430 215 = 200 030 005 + 200 000 010 + 010 400 005 + 010 000 200 + 00 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 400 010 + 00 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 030 200

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SectionI.Definition 307 Wecanbringoutthescalars. = 100 010 001 + 100 001 010 + 010 100 001 + 010 001 100 + )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 001 100 010 + )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 001 010 100 Tonish,weevaluatethosesixdeterminantsbyrow-swappingthemtothe identitymatrix,keepingtrackoftheresultingsignchanges. =30 +1+0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 +20 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1+0 +1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 +1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(6 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1=12 Thatexampleillustratesthekeyidea.We'veappliedmultilinearitytoa3 3 determinanttoget3 3 separatedeterminants,eachwithonedistinguishedentry perrow.Wecandropmostofthesenewdeterminantsbecausethematrices aresingular,withonerowamultipleofanother.Weareleftwiththeoneentry-per-rowdeterminantsalsohavingonlyoneentrypercolumnoneentry fromtheoriginaldeterminant,thatis.And,sincewecanfactorscalarsout,we canfurtherreducetoonlyconsideringdeterminantsofone-entry-per-row-andcolumnmatriceswheretheentriesareones. Thesearepermutationmatrices.Thus,thedeterminantcanbecomputed inthisthree-stepway Step1 foreachpermutationmatrix,multiplytogether theentriesfromtheoriginalmatrixwherethatpermutationmatrixhasones, Step2 multiplythatbythedeterminantofthepermutationmatrixand Step3 dothatforallpermutationmatricesandsumtheresultstogether. Tostatethisasaformula,weintroduceanotationforpermutationmatrices. Let j betherowvectorthatisallzeroesexceptforaoneinits j -thentry,so thatthefour-wide 2 is )]TJ/F8 9.9626 Tf 4.566 -7.97 Td [(0100 .Wecanconstructpermutationmatrices bypermuting|thatis,scrambling|thenumbers1,2,..., n ,andusingthem asindicesonthe 's.Forinstance,togeta4 4permutationmatrixmatrix,we canscramblethenumbersfrom1to4intothissequence h 3 ; 2 ; 1 ; 4 i andtakethe correspondingrowvector 's. 0 B B @ 3 2 1 4 1 C C A = 0 B B @ 0010 0100 1000 0001 1 C C A 3.7Denition An n -permutation isasequenceconsistingofanarrangement ofthenumbers1,2,..., n

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308 ChapterFour.Determinants 3.8Example The2-permutationsare 1 = h 1 ; 2 i and 2 = h 2 ; 1 i .Theseare theassociatedpermutationmatrices. P 1 = 1 2 = 10 01 P 2 = 2 1 = 01 10 Wesometimeswritepermutationsasfunctions,e.g., 2 =2,and 2 =1. Thentherowsof P 2 are 2 = 2 and 2 = 1 The3-permutationsare 1 = h 1 ; 2 ; 3 i 2 = h 1 ; 3 ; 2 i 3 = h 2 ; 1 ; 3 i 4 = h 2 ; 3 ; 1 i 5 = h 3 ; 1 ; 2 i ,and 6 = h 3 ; 2 ; 1 i .Herearetwooftheassociatedpermutationmatrices. P 2 = 0 @ 1 3 2 1 A = 0 @ 100 001 010 1 A P 5 = 0 @ 3 1 2 1 A = 0 @ 001 100 010 1 A Forinstance,therowsof P 5 are 5 = 3 5 = 1 ,and 5 = 2 3.9Denition The permutationexpansion fordeterminantsis t 1 ; 1 t 1 ; 2 :::t 1 ;n t 2 ; 1 t 2 ; 2 :::t 2 ;n t n; 1 t n; 2 :::t n;n = t 1 ; 1 t 2 ; 1 t n; 1 n j P 1 j + t 1 ; 2 t 2 ; 2 t n; 2 n j P 2 j + t 1 ; k t 2 ; k t n; k n j P k j where 1 ;:::; k areallofthe n -permutations. Thisformulaisoftenwrittenin summationnotation j T j = X permutations t 1 ; t 2 ; t n; n j P j readaloudasthesum,overallpermutations ,oftermshavingtheform t 1 ; t 2 ; t n; n j P j ".Thisphraseisjustarestatingofthethree-step process Step1 foreachpermutationmatrix,compute t 1 ; t 2 ; t n; n Step2 multiplythatby j P j and Step3 sumallsuchtermstogether. 3.10Example Thefamiliarformulaforthedeterminantofa2 2matrixcan bederivedinthisway. t 1 ; 1 t 1 ; 2 t 2 ; 1 t 2 ; 2 = t 1 ; 1 t 2 ; 2 j P 1 j + t 1 ; 2 t 2 ; 1 j P 2 j = t 1 ; 1 t 2 ; 2 10 01 + t 1 ; 2 t 2 ; 1 01 10 = t 1 ; 1 t 2 ; 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(t 1 ; 2 t 2 ; 1

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SectionI.Definition 309 thesecondpermutationmatrixtakesonerowswaptopasstotheidentity. Similarly,theformulaforthedeterminantofa3 3matrixisthis. t 1 ; 1 t 1 ; 2 t 1 ; 3 t 2 ; 1 t 2 ; 2 t 2 ; 3 t 3 ; 1 t 3 ; 2 t 3 ; 3 = t 1 ; 1 t 2 ; 2 t 3 ; 3 j P 1 j + t 1 ; 1 t 2 ; 3 t 3 ; 2 j P 2 j + t 1 ; 2 t 2 ; 1 t 3 ; 3 j P 3 j + t 1 ; 2 t 2 ; 3 t 3 ; 1 j P 4 j + t 1 ; 3 t 2 ; 1 t 3 ; 2 j P 5 j + t 1 ; 3 t 2 ; 2 t 3 ; 1 j P 6 j = t 1 ; 1 t 2 ; 2 t 3 ; 3 )]TJ/F11 9.9626 Tf 9.963 0 Td [(t 1 ; 1 t 2 ; 3 t 3 ; 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(t 1 ; 2 t 2 ; 1 t 3 ; 3 + t 1 ; 2 t 2 ; 3 t 3 ; 1 + t 1 ; 3 t 2 ; 1 t 3 ; 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(t 1 ; 3 t 2 ; 2 t 3 ; 1 Computingadeterminantbypermutationexpansionusuallytakeslonger thanGauss'method.However,herewearenottryingtodothecomputation eciently,weareinsteadtryingtogiveadeterminantformulathatwecan provetobewell-dened.Whilethepermutationexpansionisimpracticalfor computations,itisusefulinproofs.Inparticular,wecanuseitfortheresult thatweareafter. 3.11Theorem Foreach n thereisa n n determinantfunction. Theproofisdeferredtothefollowingsubsection.Alsothereistheproofof thenextresulttheysharesomefeatures. 3.12Theorem Thedeterminantofamatrixequalsthedeterminantofits transpose. Theconsequenceofthistheoremisthat,whilewehavesofarstatedresults intermsofrowse.g.,determinantsaremultilinearintheirrows,rowswaps changethesignum,etc.,alloftheresultsalsoholdintermsofcolumns.The nalresultgivesexamples. 3.13Corollary Amatrixwithtwoequalcolumnsissingular.Columnswaps changethesignofadeterminant.Determinantsaremultilinearintheircolumns. Proof Fortherststatement,transposingthematrixresultsinamatrixwith thesamedeterminant,andwithtwoequalrows,andhenceadeterminantof zero.Theothertwoareprovedinthesameway. QED Wenishwithasummaryalthoughthenalsubsectioncontainstheunnishedbusinessofprovingthetwotheorems.Determinantfunctionsexist, areunique,andweknowhowtocomputethem.Asforwhatdeterminantsare about,perhapstheselines[Kemp]helpmakeitmemorable. Determinantnone, Solution:lotsornone. Determinantsome, Solution:justone.

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310 ChapterFour.Determinants Exercises Thesesummarizethenotationusedinthisbookforthe 2 -and 3 -permutations. i 12 1 i 12 2 i 21 i 123 1 i 123 2 i 132 3 i 213 4 i 231 5 i 312 6 i 321 X 3.14 Computethedeterminantbyusingthepermutationexpansion. a 123 456 789 b 221 3 )]TJ/F29 8.9664 Tf 7.168 0 Td [(10 )]TJ/F29 8.9664 Tf 7.167 0 Td [(205 X 3.15 ComputethesebothwithGauss'methodandwiththepermutationexpansion formula. a 21 31 b 014 023 151 X 3.16 Usethepermutationexpansionformulatoderivetheformulafor3 3determinants. 3.17 Listallofthe4-permutations. 3.18 Apermutation,regardedasafunctionfromtheset f 1 ;::;n g toitself,isoneto-oneandonto.Therefore,eachpermutationhasaninverse. a Findtheinverseofeach2-permutation. b Findtheinverseofeach3-permutation. 3.19 Provethat f ismultilinearifandonlyifforall ~v;~w 2 V and k 1 ;k 2 2 R ,this holds. f ~ 1 ;:::;k 1 ~v 1 + k 2 ~v 2 ;:::;~ n = k 1 f ~ 1 ;:::;~v 1 ;:::;~ n + k 2 f ~ 1 ;:::;~v 2 ;:::;~ n 3.20 Findtheonlynonzeroterminthepermutationexpansionofthismatrix. 0100 1010 0101 0010 Computethatdeterminantbyndingthesignumoftheassociatedpermutation. 3.21 Howwoulddeterminantschangeifwechangedpropertyofthedenition toreadthat j I j =2? 3.22 VerifythesecondandthirdstatementsinCorollary3.13. X 3.23 Showthatifan n n matrixhasanonzerodeterminantthenanycolumnvector ~v 2 R n canbeexpressedasalinearcombinationofthecolumnsofthematrix. 3.24 Trueorfalse:amatrixwhoseentriesareonlyzerosoroneshasadeterminant equaltozero,one,ornegativeone.[Strang80] 3.25a Showthatthereare120termsinthepermutationexpansionformulaof a5 5matrix. b Howmanyaresuretobezeroifthe1 ; 2entryiszero? 3.26 Howmany n -permutationsarethere?

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SectionI.Definition 311 3.27 Amatrix A is skew-symmetric if A trans = )]TJ/F32 8.9664 Tf 7.168 0 Td [(A ,asinthismatrix. A = 03 )]TJ/F29 8.9664 Tf 7.167 0 Td [(30 Showthat n n skew-symmetricmatriceswithnonzerodeterminantsexistonlyfor even n X 3.28 Whatisthesmallestnumberofzeros,andtheplacementofthosezeros,needed toensurethata4 4matrixhasadeterminantofzero? X 3.29 Ifwehave n datapoints x 1 ;y 1 ; x 2 ;y 2 ;:::; x n ;y n andwanttonda polynomial p x = a n )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 x n )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 + a n )]TJ/F31 5.9776 Tf 5.756 0 Td [(2 x n )]TJ/F31 5.9776 Tf 5.756 0 Td [(2 + + a 1 x + a 0 passingthroughthose pointsthenwecanpluginthepointstogetan n equation/ n unknownlinear system.Thematrixofcoecientsforthatsystemiscalledthe Vandermonde matrix .Provethatthedeterminantofthetransposeofthatmatrixofcoecients 11 ::: 1 x 1 x 2 :::x n x 1 2 x 2 2 :::x n 2 x 1 n )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 x 2 n )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 :::x n n )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 equalstheproduct,overallindices i;j 2f 1 ;:::;n g with i
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312 ChapterFour.Determinants ? 3.35 Showthatthedeterminantofthe n 2 elementsintheupperleftcornerofthe Pascaltriangle 1111 :: 123 :: 13 :: 1 :: : : hasthevalueunity.[Am.Math.Mon.,Jun.1931] I.4DeterminantsExist Thissubsectionisoptional.Itconsistsofproofsoftworesultsfromtheprior subsection.Theseproofsinvolvethepropertiesofpermutations,whichwillnot beusedlater,exceptintheoptionalJordanCanonicalFormsubsection. Thepriorsubsectionattackstheproblemofshowingthatforanysizethere isadeterminantfunctiononthesetofsquarematricesofthatsizebyusing multilinearitytodevelopthepermutationexpansion. t 1 ; 1 t 1 ; 2 :::t 1 ;n t 2 ; 1 t 2 ; 2 :::t 2 ;n t n; 1 t n; 2 :::t n;n = t 1 ; 1 t 2 ; 1 t n; 1 n j P 1 j + t 1 ; 2 t 2 ; 2 t n; 2 n j P 2 j + t 1 ; k t 2 ; k t n; k n j P k j = X permutations t 1 ; t 2 ; t n; n j P j Thisreducestheproblemtoshowingthatthereisadeterminantfunctionon thesetofpermutationmatricesofthatsize. Ofcourse,apermutationmatrixcanberow-swappedtotheidentitymatrix andtocalculateitsdeterminantwecankeeptrackofthenumberofrowswaps. However,theproblemisstillnotsolved.Westillhavenotshownthattheresult iswell-dened.Forinstance,thedeterminantof P = 0 B B @ 0100 1000 0010 0001 1 C C A couldbecomputedwithoneswap P 1 $ 2 )167(! 0 B B @ 1000 0100 0010 0001 1 C C A

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SectionI.Definition 313 orwiththree. P 3 $ 1 )167(! 0 B B @ 0010 1000 0100 0001 1 C C A 2 $ 3 )167(! 0 B B @ 0010 0100 1000 0001 1 C C A 1 $ 3 )167(! 0 B B @ 1000 0100 0010 0001 1 C C A Bothreductionshaveanoddnumberofswapssowegurethat j P j = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 buthowdoweknowthatthereisn'tsomewaytodoitwithanevennumberof swaps?Corollary4.6belowprovesthatthereisnopermutationmatrixthatcan berow-swappedtoanidentitymatrixintwoways,onewithanevennumberof swapsandtheotherwithanoddnumberofswaps. 4.1Denition Tworowsofapermutationmatrix 0 B B B B B B B @ k j 1 C C C C C C C A suchthat k>j areinan inversion oftheirnaturalorder. 4.2Example Thispermutationmatrix 0 @ 3 2 1 1 A = 0 @ 001 010 100 1 A hasthreeinversions: 3 precedes 1 3 precedes 2 ,and 2 precedes 1 4.3Lemma Arow-swapinapermutationmatrixchangesthenumberofinversionsfromeventoodd,orfromoddtoeven. Proof Consideraswapofrows j and k ,where k>j .Ifthetworowsare adjacent P = 0 B B B B @ j k 1 C C C C A k $ j )167(! 0 B B B B @ k j 1 C C C C A thentheswapchangesthetotalnumberofinversionsbyone|eitherremoving orproducingoneinversion,dependingonwhether j > k ornot,since inversionsinvolvingrowsnotinthispairarenotaected.Consequently,the totalnumberofinversionschangesfromoddtoevenorfromeventoodd.

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314 ChapterFour.Determinants Iftherowsarenotadjacentthentheycanbeswappedviaasequenceof adjacentswaps,rstbringingrow k up 0 B B B B B B B B B B B @ j j +1 j +2 k 1 C C C C C C C C C C C A k $ k )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 )167(! k )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 $ k )]TJ/F6 4.9813 Tf 5.396 0 Td [(2 )167(! ::: j +1 $ j )167(! 0 B B B B B B B B B B B @ k j j +1 k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 1 C C C C C C C C C C C A andthenbringingrow j down. j +1 $ j +2 )167(! j +2 $ j +3 )167(! ::: k )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 $ k )167(! 0 B B B B B B B B B B B @ k j +1 j +2 j 1 C C C C C C C C C C C A Eachoftheseadjacentswapschangesthenumberofinversionsfromoddtoeven orfromeventoodd.Thereareanoddnumber k )]TJ/F11 9.9626 Tf 10.144 0 Td [(j + k )]TJ/F11 9.9626 Tf 10.144 0 Td [(j )]TJ/F8 9.9626 Tf 10.144 0 Td [(1ofthem. Thetotalchangeinthenumberofinversionsisfromeventooddorfromodd toeven. QED 4.4Denition The signum ofapermutationsgn is+1ifthenumberof inversionsin P iseven,andis )]TJ/F8 9.9626 Tf 7.749 0 Td [(1ifthenumberofinversionsisodd. 4.5Example WiththesubscriptsfromExample3.8forthe3-permutations, sgn 1 =1whilesgn 2 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1. 4.6Corollary Ifapermutationmatrixhasanoddnumberofinversionsthen swappingittotheidentitytakesanoddnumberofswaps.Ifithasaneven numberofinversionsthenswappingtotheidentitytakesanevennumberof swaps. Proof Theidentitymatrixhaszeroinversions.Tochangeanoddnumberto zerorequiresanoddnumberofswaps,andtochangeanevennumbertozero requiresanevennumberofswaps. QED Westillhavenotshownthatthepermutationexpansioniswell-denedbecausewehavenotconsideredrowoperationsonpermutationmatricesotherthan

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SectionI.Definition 315 rowswaps.Wewillnessethisproblem:wewilldeneafunction d : M n n R byalteringthepermutationexpansionformula,replacing j P j withsgn d T = X permutations t 1 ; t 2 ; :::t n; n sgn thisgivesthesamevalueasthepermutationexpansionbecausethepriorresult showsthatdet P =sgn .Thisformula'sadvantageisthatthenumberof inversionsisclearlywell-dened|justcountthem.Therefore,wewillshow thatadeterminantfunctionexistsforallsizesbyshowingthat d isit,thatis, that d satisesthefourconditions. 4.7Lemma Thefunction d isadeterminant.Hencedeterminantsexistfor every n Proof We'llmustcheckthatithasthefourpropertiesfromthedenition. Propertyiseasy;in d I = X perms 1 ; 2 ; n; n sgn allofthesummandsarezeroexceptfortheproductdownthediagonal,which isone. Forpropertyconsider d ^ T where T k i )167(! ^ T X perms ^ t 1 ; ^ t i; i ^ t n; n sgn = X t 1 ; kt i; i t n; n sgn Factorthe k outofeachtermtogetthedesiredequality. = k X t 1 ; t i; i t n; n sgn = k d T For,let T i $ j )167(! ^ T d ^ T = X perms ^ t 1 ; ^ t i; i ^ t j; j ^ t n; n sgn Toconverttounhatted t 's,foreach considerthepermutation thatequals exceptthatthe i -thand j -thnumbersareinterchanged, i = j and j = i .Replacingthe in ^ t 1 ; ^ t i; i ^ t j; j ^ t n; n withthis gives t 1 ; t j; j t i; i t n; n .Nowsgn = )]TJ/F8 9.9626 Tf 9.409 0 Td [(sgn byLemma4.3 andsoweget = X t 1 ; t j; j t i; i t n; n )]TJ/F14 9.9626 Tf 4.566 -8.07 Td [()]TJ/F8 9.9626 Tf 9.409 0 Td [(sgn = )]TJ/F1 9.9626 Tf 9.41 9.465 Td [(X t 1 ; t j; j t i; i t n; n sgn

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316 ChapterFour.Determinants wherethesumisoverallpermutations derivedfromanotherpermutation byaswapofthe i -thand j -thnumbers.Butanypermutationcanbederived fromsomeotherpermutationbysuchaswap,inoneandonlyoneway,sothis summationisinfactasumoverallpermutations,takenonceandonlyonce. Thus d ^ T = )]TJ/F11 9.9626 Tf 7.749 0 Td [(d T Todopropertylet T k i + j )167(! ^ T andconsider d ^ T = X perms ^ t 1 ; ^ t i; i ^ t j; j ^ t n; n sgn = X t 1 ; t i; i kt i; j + t j; j t n; n sgn notice:that's kt i; j ,not kt j; j .Distribute,commute,andfactor. = X t 1 ; t i; i kt i; j t n; n sgn + t 1 ; t i; i t j; j t n; n sgn = X t 1 ; t i; i kt i; j t n; n sgn + X t 1 ; t i; i t j; j t n; n sgn = k X t 1 ; t i; i t i; j t n; n sgn + d T Wenishbyshowingthattheterms t 1 ; t i; i t i; j :::t n; n sgn addtozero.Thissumrepresents d S where S isamatrixequalto T except thatrow j of S isacopyofrow i of T becausethefactoris t i; j ,not t j; j Thus, S hastwoequalrows,rows i and j .Sincewehavealreadyshownthat d changessignonrowswaps,asinLemma2.3weconcludethat d S =0. QED Wehavenowshownthatdeterminantfunctionsexistforeachsize.We alreadyknowthatforeachsizethereisatmostonedeterminant.Therefore, thepermutationexpansioncomputestheoneandonlydeterminantvalueofa squarematrix. Weendthissubsectionbyprovingtheotherresultremainingfromtheprior subsection,thatthedeterminantofamatrixequalsthedeterminantofitstranspose. 4.8Example Writingoutthepermutationexpansionofthegeneral3 3matrix andofitstranspose,andcomparingcorrespondingterms abc def ghi = + cdh 001 100 010 +

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SectionI.Definition 317 termswiththesameletters adg beh cfi = + dhc 010 001 100 + showsthatthecorrespondingpermutationmatricesaretransposes.Thatis, thereisarelationshipbetweenthesecorrespondingpermutations.Exercise15 showsthattheyareinverses. 4.9Theorem Thedeterminantofamatrixequalsthedeterminantofits transpose. Proof Callthematrix T anddenotetheentriesof T trans with s 'ssothat t i;j = s j;i .Substitutiongivesthis j T j = X perms t 1 ; :::t n; n sgn = X s ; 1 :::s n ;n sgn andwecannishtheargumentbymanipulatingtheexpressionontheright toberecognizableasthedeterminantofthetranspose.Wehavewrittenall permutationexpansionsasinthemiddleexpressionabovewiththerowindices ascending.Torewritetheexpressionontherightinthisway,notethatbecause isapermutation,therowindicesinthetermontheright ,..., n are justthenumbers1,..., n ,rearranged.Wecanthuscommutetohavethese ascend,giving s 1 ; )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 s n; )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 n ifthecolumnindexis j andtherowindex is j then,wheretherowindexis i ,thecolumnindexis )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 i .Substituting ontherightgives = X )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 s 1 ; )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 s n; )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 n sgn )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Exercise14showsthatsgn )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 =sgn .Sinceeverypermutationisthe inverseofanother,asumoverall )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 isasumoverallpermutations = X perms s 1 ; 1 :::s n; n sgn = T trans asrequired. QED Exercises Thesesummarizethenotationusedinthisbookforthe 2 -and 3 -permutations. i 12 1 i 12 2 i 21 i 123 1 i 123 2 i 132 3 i 213 4 i 231 5 i 312 6 i 321

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318 ChapterFour.Determinants 4.10 Givethepermutationexpansionofageneral2 2matrixanditstranspose. X 4.11 Thisproblemappearsalsointhepriorsubsection. a Findtheinverseofeach2-permutation. b Findtheinverseofeach3-permutation. X 4.12a Findthesignumofeach2-permutation. b Findthesignumofeach3-permutation. 4.13 Whatisthesignumofthe n -permutation = h n;n )]TJ/F29 8.9664 Tf 8.74 0 Td [(1 ;:::; 2 ; 1 i ?[Strang80] 4.14 Provethese. a Everypermutationhasaninverse. b sgn )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 =sgn c Everypermutationistheinverseofanother. 4.15 Provethatthematrixofthepermutationinverseisthetransposeofthematrix ofthepermutation P )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 = P trans ,foranypermutation X 4.16 Showthatapermutationmatrixwith m inversionscanberowswappedto theidentityin m steps.ContrastthiswithCorollary4.6. X 4.17 Foranypermutation let g betheintegerdenedinthisway. g = Y i
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SectionII.GeometryofDeterminants 319 IIGeometryofDeterminants Thepriorsectiondevelopsthedeterminantalgebraically,byconsideringwhat formulassatisfycertainproperties.Thissectioncomplementsthatwithageometricapproach.Oneadvantageofthisapproachisthat,whilewehavesofar onlyconsideredwhetherornotadeterminantiszero,hereweshallgiveameaningtothevalueofthatdeterminant.Thepriorsectionhandlesdeterminants asfunctionsoftherows,butinthissectioncolumnsaremoreconvenient.The nalresultofthepriorsectionsaysthatwecanmaketheswitch. II.1DeterminantsasSizeFunctions Thisparallelogrampicture x 1 y 1 x 2 y 2 isfamiliarfromtheconstructionofthesumofthetwovectors.Onewayto computetheareathatitenclosesistodrawthisrectangleandsubtractthe areaofeachsubregion. y 1 y 2 x 2 x 1 A B C D E F areaofparallelogram =areaofrectangle )]TJ/F8 9.9626 Tf 9.963 0 Td [(areaof A )]TJ/F8 9.9626 Tf 9.963 0 Td [(areaof B )-222()]TJ/F8 9.9626 Tf 33.762 0 Td [(areaof F = x 1 + x 2 y 1 + y 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 2 y 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 1 y 1 = 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 2 y 2 = 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 2 y 2 = 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 1 y 1 = 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 2 y 1 = x 1 y 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 2 y 1 Thefactthattheareaequalsthevalueofthedeterminant x 1 x 2 y 1 y 2 = x 1 y 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 2 y 1 isnocoincidence.Thepropertiesinthedenitionofdeterminantsmakereasonablepostulatesforafunctionthatmeasuresthesizeoftheregionenclosed bythevectorsinthematrix. Forinstance,thisshowstheeectofmultiplyingoneofthebox-dening vectorsbyascalarthescalarusedis k =1 : 4. ~v ~w k~v ~w

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320 ChapterFour.Determinants Theregionformedby k~v and ~w isbigger,byafactorof k ,thantheshaded regionenclosedby ~v and ~w .Thatis,size k~v;~w = k size ~v;~w andingeneral weexpectofthesizemeasurethatsize :::;k~v;::: = k size :::;~v;::: .Of course,thispostulateisalreadyfamiliarasoneofthepropertiesinthedention ofdeterminants. Anotherpropertyofdeterminantsisthattheyareunaectedbypivoting. Herearebefore-pivotingandafter-pivotingboxesthescalarusedis k =0 : 35. ~v ~w ~v k~v + ~w Althoughtheregionontheright,theboxformedby v and k~v + ~w ,ismore slantedthantheshadedregion,thetwohavethesamebaseandthesameheight andhencethesamearea.Thisillustratesthatsize ~v;k~v + ~w =size ~v;~w Generalized,size :::;~v;:::;~w;::: =size :::;~v;:::;k~v + ~w;::: ,whichisa restatementofthedeterminantpostulate. Ofcourse,thispicture ~e 1 ~e 2 showsthatsize ~e 1 ;~e 2 =1,andwenaturallyextendthattoanynumberof dimensionssize ~e 1 ;:::;~e n =1,whichisarestatementofthepropertythatthe determinantoftheidentitymatrixisone. Withthat,becausepropertyofdeterminantsisredundantasremarked rightafterthedenition,wehavethatallofthepropertiesofdeterminantsare reasonabletoexpectofafunctionthatgivesthesizeofboxes.Wecannowcite theworkdoneinthepriorsectiontoshowthatthedeterminantexistsandis uniquetobeassuredthatthesepostulatesareconsistentandsucientwedo notneedanymorepostulates.Thatis,we'vegotanintuitivejusticationto interpretdet ~v 1 ;:::;~v n asthesizeoftheboxformedbythevectors. Comment. Anevenmorebasicapproach,whichalsoleadstothedenitionbelow,isin [Weston]. 1.1Example Thevolumeofthisparallelepiped,whichcanbefoundbythe usualformulafromhighschoolgeometry,is12. 2 0 2 0 3 1 )]TJ/F6 4.9813 Tf 5.397 0 Td [(1 0 1 20 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 030 211 =12

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SectionII.GeometryofDeterminants 321 1.2Remark Althoughpropertyofthedenitionofdeterminantsisredundant,itraisesanimportantpoint.Considerthesetwo. ~u ~v ~u ~v 41 23 =10 14 32 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 Theonlydierencebetweenthemisintheorderinwhichthevectorsaretaken. Ifwetake ~u rstandthengoto ~v ,followthecounterclockwisearcshown,then thesignispositive.Followingaclockwisearcgivesanegativesign.Thesign returnedbythesizefunctionreectsthe`orientation'or`sense'ofthebox.We seethesamethingifwepicturetheeectofscalarmultiplicationbyanegative scalar. Althoughitisbothinterestingandimportant,theideaoforientationturns outtobetricky.Itisnotneededforthedevelopmentbelow,andsowewillpass itby.SeeExercise27. 1.3Denition The box or parallelepiped formedby h ~v 1 ;:::;~v n i whereeach vectorisfrom R n includesalloftheset f t 1 ~v 1 + + t n ~v n t 1 ;:::;t n 2 [0 :: 1] g The volume ofaboxistheabsolutevalueofthedeterminantofthematrix withthosevectorsascolumns. 1.4Example Volume,becauseitisanabsolutevalue,doesnotdependon theorderinwhichthevectorsaregiven.Thevolumeoftheparallelepipedin Exercise1.1,canalsobecomputedastheabsolutevalueofthisdeterminant. 020 303 121 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 Thedenitionofvolumegivesageometricinterpretationtosomethingin thespace,boxesmadefromvectors.Thenextresultrelatesthegeometryto thefunctionsthatoperateonspaces. 1.5Theorem Atransformation t : R n R n changesthesizeofallboxesby thesamefactor,namelythesizeoftheimageofabox j t S j is j T j timesthe sizeofthebox j S j ,where T isthematrixrepresenting t withrespecttothe standardbasis.Thatis,forall n n matrices,thedeterminantofaproductis theproductofthedeterminants j TS j = j T jj S j Thetwosentencesstatethesameidea,rstinmaptermsandtheninmatrix terms.Althoughwetendtopreferamappointofview,thesecondsentence, thematrixversion,ismoreconvienentfortheproofandisalsothewaythat weshallusethisresultlater.AlternateproofsaregivenasExercise23and Exercise28.

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322 ChapterFour.Determinants Proof Thetwostatementsareequivalentbecause j t S j = j TS j ,asbothgive thesizeoftheboxthatistheimageoftheunitbox E n underthecomposition t s where s isthemaprepresentedby S withrespecttothestandardbasis. Firstconsiderthecasethat j T j =0.Amatrixhasazerodeterminantifand onlyifitisnotinvertible.Observethatif TS isinvertible,sothatthereisan M suchthat TS M = I ,thentheassociativepropertyofmatrixmultiplication T SM = I showsthat T isalsoinvertiblewithinverse SM .Therefore,if T isnotinvertiblethenneitheris TS |if j T j =0then j TS j =0,andtheresult holdsinthiscase. Nowconsiderthecasethat j T j6 =0,that T isnonsingular.Recallthatany nonsingularmatrixcanbefactoredintoaproductofelementarymatrices,so that TS = E 1 E 2 E r S .Intherestofthisargument,wewillverifythatif E isanelementarymatrixthen j ES j = j E jj S j .Theresultwillfollowbecause then j TS j = j E 1 E r S j = j E 1 jj E r jj S j = j E 1 E r jj S j = j T jj S j Iftheelementarymatrix E is M i k then M i k S equals S exceptthatrow i hasbeenmultipliedby k .Thethirdpropertyofdeterminantfunctionsthen givesthat j M i k S j = k j S j .But j M i k j = k ,againbythethirdproperty because M i k isderivedfromtheidentitybymultiplicationofrow i by k ,and so j ES j = j E jj S j holdsfor E = M i k .The E = P i;j = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1and E = C i;j k checksaresimilar. QED 1.6Example Applicationofthemap t representedwithrespecttothestandardbasesby 11 )]TJ/F8 9.9626 Tf 7.748 0 Td [(20 willdoublesizesofboxes,e.g.,fromthis ~w ~v 21 12 =3 tothis t ~w t ~v 33 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 =6 1.7Corollary Ifamatrixisinvertiblethenthedeterminantofitsinverseis theinverseofitsdeterminant j T )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 j =1 = j T j Proof 1= j I j = j TT )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 j = j T jj T )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 j QED Recallthatdeterminantsarenotadditivehomomorphisms,det A + B need notequaldet A +det B .Theabovetheoremsays,incontrast,thatdeterminantsaremultiplicativehomomorphisms:det AB doesequaldet A det B .

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SectionII.GeometryofDeterminants 323 Exercises 1.8 Findthevolumeoftheregionformed. a h 1 3 ; )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 4 i b h 2 1 0 ; 3 )]TJ/F29 8.9664 Tf 7.167 0 Td [(2 4 ; 8 )]TJ/F29 8.9664 Tf 7.167 0 Td [(3 8 i c h 0 B @ 1 2 0 1 1 C A ; 0 B @ 2 2 2 2 1 C A ; 0 B @ )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 3 0 5 1 C A ; 0 B @ 0 1 0 7 1 C A i X 1.9 Is 4 1 2 insideoftheboxformedbythesethree? 3 3 1 2 6 1 1 0 5 X 1.10 Findthevolumeofthisregion. X 1.11 Supposethat j A j =3.Bywhatfactordothesechangevolumes? a A b A 2 c A )]TJ/F31 5.9776 Tf 5.756 0 Td [(2 X 1.12 Bywhatfactordoeseachtransformationchangethesizeofboxes? a x y 7! 2 x 3 y b x y 7! 3 x )]TJ/F32 8.9664 Tf 9.215 0 Td [(y )]TJ/F29 8.9664 Tf 7.168 0 Td [(2 x + y c x y z 7! x )]TJ/F32 8.9664 Tf 9.216 0 Td [(y x + y + z y )]TJ/F29 8.9664 Tf 9.216 0 Td [(2 z 1.13 Whatistheareaoftheimageoftherectangle[2 :: 4] [2 :: 5]undertheaction ofthismatrix? 23 4 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 1.14 If t : R 3 R 3 changesvolumesbyafactorof7and s : R 3 R 3 changesvolumesbyafactorof3 = 2thenbywhatfactorwilltheircompositionchangesvolumes? 1.15 Inwhatwaydoesthedenitionofaboxdierfromthedentionofaspan? X 1.16 Whydoesn'tthispicturecontradictTheorem1.5? )]TJ/F31 5.9776 Tf 5.839 -5.44 Td [(21 01 )171(! areais2determinantis2areais5 X 1.17 Does j TS j = j ST j ? j T SP j = j TS P j ? 1.18a Supposethat j A j =3andthat j B j =2.Find j A 2 B trans B )]TJ/F31 5.9776 Tf 5.756 0 Td [(2 A trans j b Assumethat j A j =0.Provethat j 6 A 3 +5 A 2 +2 A j =0. X 1.19 Let T bethematrixrepresentingwithrespecttothestandardbasesthe mapthatrotatesplanevectorscounterclockwisethru radians.Bywhatfactor does T changesizes? X 1.20 Mustatransformation t : R 2 R 2 thatpreservesareasalsopreservelengths?

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324 ChapterFour.Determinants X 1.21 Whatisthevolumeofaparallelepipedin R 3 boundedbyalinearlydependent set? X 1.22 Findtheareaofthetrianglein R 3 withendpoints ; 2 ; 1, ; )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 ; 4,and ; 2 ; 2.Area,notvolume.Thetriangledenesaplane|whatistheareaofthe triangleinthatplane? X 1.23 AnalternateproofofTheorem1.5usesthedenitionofdeterminantfunctions. a Notethatthevectorsforming S makealinearlydependentsetifandonlyif j S j =0,andcheckthattheresultholdsinthiscase. b Forthe j S j6 =0case,toshowthat j TS j = j S j = j T j foralltransformations, considerthefunction d : M n n R givenby T 7!j TS j = j S j .Showthat d has therstpropertyofadeterminant. c Showthat d hastheremainingthreepropertiesofadeterminantfunction. d Concludethat j TS j = j T jj S j 1.24 Giveanon-identitymatrixwiththepropertythat A trans = A )]TJ/F31 5.9776 Tf 5.757 0 Td [(1 .Showthat if A trans = A )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 then j A j = 1.Doestheconversehold? 1.25 Thealgebraicpropertyofdeterminantsthatfactoringascalaroutofasingle rowwillmultiplythedeterminantbythatscalarshowsthatwhere H is3 3,the determinantof cH is c 3 timesthedeterminantof H .Explainthisgeometrically, thatis,usingTheorem1.5, X 1.26 Matrices H and G aresaidtobe similar ifthereisanonsingularmatrix P suchthat H = P )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 GP wewillstudythisrelationinChapterFive.Showthat similarmatriceshavethesamedeterminant. 1.27 Weusuallyrepresentvectorsin R 2 withrespecttothestandardbasisso vectorsintherstquadranthavebothcoordinatespositive. ~v Rep E 2 ~v = +3 +2 Movingcounterclockwisearoundtheorigin,wecyclethrufourregions: )196(! + + )195(! )]TJ/F7 6.9738 Tf 0.055 -7.971 Td [(+ )195(! )]TJ 0 -7.971 Td [()]TJ/F1 9.9626 Tf 6.226 14.147 Td [( )195(! + )]TJ/F1 9.9626 Tf 6.227 14.147 Td [( )195(! : Usingthisbasis B = h 0 1 ; )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 0 i ~ 2 ~ 1 givesthesamecounterclockwisecycle.Wesaythesetwobaseshavethesame orientation a Whydotheygivethesamecycle? b Whatothercongurationsofunitvectorsontheaxesgivethesamecycle? c Findthedeterminantsofthematricesformedfromthoseorderedbases. d Whatothercounterclockwisecyclesarepossible,andwhataretheassociated determinants? e Whathappensin R 1 ? f Whathappensin R 3 ? Afascinatinggeneral-audiencediscussionoforientationsisin[Gardner]. 1.28 ThisquestionusesmaterialfromtheoptionalDeterminantFunctionsExist subsection. ProveTheorem1.5byusingthepermutationexpansionformulafor thedeterminant.

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SectionII.GeometryofDeterminants 325 X 1.29a Showthatthisgivestheequationofalinein R 2 thru x 2 ;y 2 and x 3 ;y 3 xx 2 x 3 yy 2 y 3 111 =0 b [Petersen]Provethattheareaofatrianglewithvertices x 1 ;y 1 x 2 ;y 2 and x 3 ;y 3 is 1 2 x 1 x 2 x 3 y 1 y 2 y 3 111 : c [Math.Mag.,Jan.1973]Provethattheareaofatrianglewithverticesat x 1 ;y 1 x 2 ;y 2 ,and x 3 ;y 3 whosecoordinatesareintegershasanareaof N or N= 2forsomepositiveinteger N .

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326 ChapterFour.Determinants IIIOtherFormulas Thissectionisoptional.Latersectionsdonotdependonthismaterial. Determinantsareafountofinterestingandamusingformulas.Hereisone thatisoftenseenincalculusclassesandusedtocomputedeterminantsbyhand. III.1Laplace'sExpansion 1.1Example Inthispermutationexpansion t 1 ; 1 t 1 ; 2 t 1 ; 3 t 2 ; 1 t 2 ; 2 t 2 ; 3 t 3 ; 1 t 3 ; 2 t 3 ; 3 = t 1 ; 1 t 2 ; 2 t 3 ; 3 100 010 001 + t 1 ; 1 t 2 ; 3 t 3 ; 2 100 001 010 + t 1 ; 2 t 2 ; 1 t 3 ; 3 010 100 001 + t 1 ; 2 t 2 ; 3 t 3 ; 1 010 001 100 + t 1 ; 3 t 2 ; 1 t 3 ; 2 001 100 010 + t 1 ; 3 t 2 ; 2 t 3 ; 1 001 010 100 wecan,forinstance,factorouttheentriesfromtherstrow = t 1 ; 1 2 4 t 2 ; 2 t 3 ; 3 100 010 001 + t 2 ; 3 t 3 ; 2 100 001 010 3 5 + t 1 ; 2 2 4 t 2 ; 1 t 3 ; 3 010 100 001 + t 2 ; 3 t 3 ; 1 010 001 100 3 5 + t 1 ; 3 2 4 t 2 ; 1 t 3 ; 2 001 100 010 + t 2 ; 2 t 3 ; 1 001 010 100 3 5 andswaprowsinthepermutationmatricestogetthis. = t 1 ; 1 2 4 t 2 ; 2 t 3 ; 3 100 010 001 + t 2 ; 3 t 3 ; 2 100 001 010 3 5 )]TJ/F11 9.9626 Tf 9.962 0 Td [(t 1 ; 2 2 4 t 2 ; 1 t 3 ; 3 100 010 001 + t 2 ; 3 t 3 ; 1 100 001 010 3 5 + t 1 ; 3 2 4 t 2 ; 1 t 3 ; 2 100 010 001 + t 2 ; 2 t 3 ; 1 100 001 010 3 5

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SectionIII.OtherFormulas 327 Thepointoftheswappingoneswaptoeachofthepermutationmatriceson thesecondlineandtwoswapstoeachonthethirdlineisthatthethreelines simplifytothreeterms. = t 1 ; 1 t 2 ; 2 t 2 ; 3 t 3 ; 2 t 3 ; 3 )]TJ/F11 9.9626 Tf 9.963 0 Td [(t 1 ; 2 t 2 ; 1 t 2 ; 3 t 3 ; 1 t 3 ; 3 + t 1 ; 3 t 2 ; 1 t 2 ; 2 t 3 ; 1 t 3 ; 2 TheformulagiveninTheorem1.5,whichgeneralizesthisexample,isa recurrence |thedeterminantisexpressedasacombinationofdeterminants.This formulaisn'tcircularbecause,ashere,thedeterminantisexpressedintermsof determinantsofmatricesofsmallersize. 1.2Denition Forany n n matrix T ,the n )]TJ/F8 9.9626 Tf 10.106 0 Td [(1 n )]TJ/F8 9.9626 Tf 10.106 0 Td [(1matrixformed bydeletingrow i andcolumn j of T isthe i;j minor of T .The i;j cofactor T i;j of T is )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 i + j timesthedeterminantofthe i;j minorof T 1.3Example The1 ; 2cofactorofthematrixfromExample1.1isthenegative ofthesecond2 2determinant. T 1 ; 2 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 t 2 ; 1 t 2 ; 3 t 3 ; 1 t 3 ; 3 1.4Example Where T = 0 @ 123 456 789 1 A thesearethe1 ; 2and2 ; 2cofactors. T 1 ; 2 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1+2 46 79 =6 T 2 ; 2 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 2+2 13 79 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(12 1.5TheoremLaplaceExpansionofDeterminants Where T isan n n matrix,thedeterminantcanbefoundbyexpandingbycofactorsonrow i or column j j T j = t i; 1 T i; 1 + t i; 2 T i; 2 + + t i;n T i;n = t 1 ;j T 1 ;j + t 2 ;j T 2 ;j + + t n;j T n;j Proof Exercise27. QED 1.6Example Wecancomputethedeterminant j T j = 123 456 789 byexpandingalongtherstrow,asinExample1.1. j T j =1 +1 56 89 +2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 46 79 +3 +1 45 78 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3+12 )]TJ/F8 9.9626 Tf 9.963 0 Td [(9=0

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328 ChapterFour.Determinants Alternatively,wecanexpanddownthesecondcolumn. j T j =2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 46 79 +5 +1 13 79 +8 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 13 46 =12 )]TJ/F8 9.9626 Tf 9.963 0 Td [(60+48=0 1.7Example AroworcolumnwithmanyzeroessuggestsaLaplaceexpansion. 150 211 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 =0 +1 21 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 +1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 15 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 +0 +1 15 21 =16 Wenishbyapplyingthisresulttoderiveanewformulafortheinverse ofamatrix.WithTheorem1.5,thedeterminantofan n n matrix T can becalculatedbytakinglinearcombinationsofentriesfromarowandtheir associatedcofactors. t i; 1 T i; 1 + t i; 2 T i; 2 + + t i;n T i;n = j T j Recallthatamatrixwithtwoidenticalrowshasazerodeterminant.Thus,for anymatrix T ,weighingthecofactorsbyentriesfromthewrong"row|row k with k 6 = i |giveszero t i; 1 T k; 1 + t i; 2 T k; 2 + + t i;n T k;n =0 becauseitrepresentstheexpansionalongtherow k ofamatrixwithrow i equal torow k .Thisequationsummarizes and 0 B B B @ t 1 ; 1 t 1 ; 2 :::t 1 ;n t 2 ; 1 t 2 ; 2 :::t 2 ;n t n; 1 t n; 2 :::t n;n 1 C C C A 0 B B B @ T 1 ; 1 T 2 ; 1 :::T n; 1 T 1 ; 2 T 2 ; 2 :::T n; 2 T 1 ;n T 2 ;n :::T n;n 1 C C C A = 0 B B B @ j T j 0 ::: 0 0 j T j ::: 0 00 ::: j T j 1 C C C A Notethattheorderofthesubscriptsinthematrixofcofactorsisoppositeto theorderofsubscriptsintheothermatrix;e.g.,alongtherstrowofthematrix ofcofactorsthesubscriptsare1 ; 1then2 ; 1,etc. 1.8Denition Thematrix adjoint tothesquarematrix T is adj T = 0 B B B @ T 1 ; 1 T 2 ; 1 :::T n; 1 T 1 ; 2 T 2 ; 2 :::T n; 2 T 1 ;n T 2 ;n :::T n;n 1 C C C A where T j;i isthe j;i cofactor. 1.9Theorem Where T isasquarematrix, T adj T =adj T T = j T j I Proof Equations and QED

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SectionIII.OtherFormulas 329 1.10Example If T = 0 @ 104 21 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 101 1 A thentheadjointadj T is 0 @ T 1 ; 1 T 2 ; 1 T 3 ; 1 T 1 ; 2 T 2 ; 2 T 3 ; 2 T 1 ; 3 T 2 ; 3 T 3 ; 3 1 A = 0 B B B B B B B @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 01 )]TJ/F1 9.9626 Tf 9.409 14.446 Td [( 04 01 04 1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 )]TJ/F1 9.9626 Tf 9.409 14.446 Td [( 2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 11 14 11 )]TJ/F1 9.9626 Tf 9.409 14.446 Td [( 14 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 21 10 )]TJ/F1 9.9626 Tf 9.409 14.446 Td [( 10 10 10 21 1 C C C C C C C A = 0 @ 10 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 )]TJ/F8 9.9626 Tf 7.748 0 Td [(39 )]TJ/F8 9.9626 Tf 7.749 0 Td [(101 1 A andtakingtheproductwith T givesthediagonalmatrix j T j I 0 @ 104 21 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 101 1 A 0 @ 10 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(39 )]TJ/F8 9.9626 Tf 7.748 0 Td [(101 1 A = 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(300 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(30 00 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 1 A 1.11Corollary If j T j6 =0then T )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = = j T j adj T 1.12Example TheinverseofthematrixfromExample1.10is = )]TJ/F8 9.9626 Tf 8.426 0 Td [(3 adj T T )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = 0 @ 1 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(30 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 )]TJ/F8 9.9626 Tf 7.748 0 Td [(4 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(39 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(30 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(31 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 1 A = 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 304 = 3 11 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 1 = 30 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 = 3 1 A Theformulasfromthissectionareoftenusedforby-handcalculationand aresometimesusefulwithspecialtypesofmatrices.However,theyarenotthe bestchoiceforcomputationwitharbitrarymatricesbecausetheyrequiremore arithmeticthan,forinstance,theGauss-Jordanmethod. Exercises X 1.13 Findthecofactor. T = 102 )]TJ/F29 8.9664 Tf 7.167 0 Td [(113 02 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 a T 2 ; 3 b T 3 ; 2 c T 1 ; 3 X 1.14 Findthedeterminantbyexpanding 301 122 )]TJ/F29 8.9664 Tf 7.168 0 Td [(130 a ontherstrow b onthesecondrow c onthethirdcolumn. 1.15 FindtheadjointofthematrixinExample1.6. X 1.16 Findthematrixadjointtoeach.

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330 ChapterFour.Determinants a 214 )]TJ/F29 8.9664 Tf 7.167 0 Td [(102 101 b 3 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 24 c 11 50 d 143 )]TJ/F29 8.9664 Tf 7.168 0 Td [(103 189 X 1.17 FindtheinverseofeachmatrixinthepriorquestionwithTheorem1.9. 1.18 Findthematrixadjointtothisone. 0 B @ 2100 1210 0121 0012 1 C A X 1.19 Expandacrosstherstrowtoderivetheformulaforthedeterminantofa2 2 matrix. X 1.20 Expandacrosstherstrowtoderivetheformulaforthedeterminantofa3 3 matrix. X 1.21a Giveaformulafortheadjointofa2 2matrix. b Useittoderivetheformulafortheinverse. X 1.22 Canwecomputeadeterminantbyexpandingdownthediagonal? 1.23 Giveaformulafortheadjointofadiagonalmatrix. X 1.24 Provethatthetransposeoftheadjointistheadjointofthetranspose. 1.25 Proveordisprove:adjadj T = T 1.26 Asquarematrixis uppertriangular ifeach i;j entryiszerointhepartabove thediagonal,thatis,when i>j a Musttheadjointofanuppertriangularmatrixbeuppertriangular?Lower triangular? b Provethattheinverseofauppertriangularmatrixisuppertriangular,ifan inverseexists. 1.27 ThisquestionrequiresmaterialfromtheoptionalDeterminantsExistsubsection. ProveTheorem1.5byusingthepermutationexpansion. 1.28 Provethatthedeterminantofamatrixequalsthedeterminantofitstranspose usingLaplace'sexpansionandinductiononthesizeofthematrix. ? 1.29 Showthat F n = 1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(11 )]TJ/F29 8.9664 Tf 7.167 0 Td [(11 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 ::: 110101 ::: 011010 ::: 001101 ::: ::::::::: where F n isthe n -thtermof1 ; 1 ; 2 ; 3 ; 5 ;:::;x;y;x + y;::: ,theFibonaccisequence, andthedeterminantisoforder n )]TJ/F29 8.9664 Tf 9.215 0 Td [(1.[Am.Math.Mon.,Jun.1949]

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Topic:Cramer'sRule 331 Topic:Cramer'sRule Wehaveintroduceddeterminantfunctionsalgebraicallybylookingforaformula todecidewhetheramatrixisnonsingular.Afterthatintroductionwesawa geometricinterpretation,thatthedeterminantfunctiongivesthesizeofthebox withsidesformedbythecolumnsofthematrix.ThisTopicmakesaconnection betweenthetwoviews. First,alinearsystem x 1 +2 x 2 =6 3 x 1 + x 2 =8 isequivalenttoalinearrelationshipamongvectors. x 1 1 3 + x 2 2 1 = 6 8 Thepicturebelowshowsaparallelogramwithsidesformedfrom )]TJ/F7 6.9738 Tf 4.566 -3.649 Td [(1 3 and )]TJ/F7 6.9738 Tf 4.566 -3.649 Td [(2 1 nestedinsideaparallelogramwithsidesformedfrom x 1 )]TJ/F7 6.9738 Tf 4.566 -3.649 Td [(1 3 and x 2 )]TJ/F7 6.9738 Tf 4.566 -3.649 Td [(2 1 )]TJ/F6 4.9813 Tf 4.566 -3.786 Td [(2 1 x 2 )]TJ/F6 4.9813 Tf 4.566 -3.786 Td [(2 1 )]TJ/F6 4.9813 Tf 4.566 -3.786 Td [(1 3 x 1 )]TJ/F6 4.9813 Tf 4.567 -3.786 Td [(1 3 )]TJ/F6 4.9813 Tf 4.567 -3.786 Td [(6 8 Soevenwithoutdeterminantswecanstatethealgebraicissuethatopenedthis book,ndingthesolutionofalinearsystem,ingeometricterms:bywhatfactors x 1 and x 2 mustwedilatethevectorstoexpandthesmallparallegramtollthe largerone? However,byemployingthegeometricsignicanceofdeterminantswecan getsomethingthatisnotjustarestatement,butalsogivesusanewinsightand sometimesallowsustocomputeanswersquickly.Comparethesizesofthese shadedboxes. )]TJ/F6 4.9813 Tf 4.566 -3.786 Td [(2 1 )]TJ/F6 4.9813 Tf 4.566 -3.786 Td [(1 3 )]TJ/F6 4.9813 Tf 4.566 -3.786 Td [(2 1 x 1 )]TJ/F6 4.9813 Tf 4.567 -3.786 Td [(1 3 )]TJ/F6 4.9813 Tf 4.566 -3.786 Td [(2 1 )]TJ/F6 4.9813 Tf 4.566 -3.786 Td [(6 8 Thesecondisformedfrom x 1 )]TJ/F7 6.9738 Tf 4.566 -3.649 Td [(1 3 and )]TJ/F7 6.9738 Tf 4.567 -3.649 Td [(2 1 ,andoneofthepropertiesofthesize function|thedeterminant|isthatitssizeistherefore x 1 timesthesizeofthe

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332 ChapterFour.Determinants rstbox.Sincethethirdboxisformedfrom x 1 )]TJ/F7 6.9738 Tf 4.566 -3.649 Td [(1 3 + x 2 )]TJ/F7 6.9738 Tf 4.567 -3.649 Td [(2 1 = )]TJ/F7 6.9738 Tf 4.566 -3.649 Td [(6 8 and )]TJ/F7 6.9738 Tf 4.567 -3.649 Td [(2 1 ,and thedeterminantisunchangedbyadding x 2 timesthesecondcolumntotherst column,thesizeofthethirdboxequalsthatofthesecond.Wehavethis. 62 81 = x 1 12 x 1 31 = x 1 12 31 Solvinggivesthevalueofoneofthevariables. x 1 = 62 81 12 31 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(10 )]TJ/F8 9.9626 Tf 7.748 0 Td [(5 =2 Thetheoremthatgeneralizesthisexample, Cramer'sRule ,is:if j A j6 =0 thenthesystem A~x = ~ b hastheuniquesolution x i = j B i j = j A j wherethematrix B i isformedfrom A byreplacingcolumn i withthevector ~ b .Exercise3asks foraproof. Forinstance,tosolvethissystemfor x 2 0 @ 104 21 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 101 1 A 0 @ x 1 x 2 x 3 1 A = 0 @ 2 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 A wedothiscomputation. x 2 = 124 21 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 104 21 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 101 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(18 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 Cramer'sRuleallowsustosolvemanytwoequations/twounknownssystems byeye.Itisalsosometimesusedforthreeequations/threeunknownssystems. Butcomputinglargedeterminantstakesalongtime,sosolvinglargesystems byCramer'sRuleisnotpractical. Exercises 1 UseCramer'sRuletosolveeachforeachofthevariables. a x )]TJ/F32 8.9664 Tf 13.823 0 Td [(y =4 )]TJ/F32 8.9664 Tf 7.168 0 Td [(x +2 y = )]TJ/F29 8.9664 Tf 7.167 0 Td [(7 b )]TJ/F29 8.9664 Tf 7.168 0 Td [(2 x + y = )]TJ/F29 8.9664 Tf 7.168 0 Td [(2 x )]TJ/F29 8.9664 Tf 9.216 0 Td [(2 y = )]TJ/F29 8.9664 Tf 7.168 0 Td [(2 2 UseCramer'sRuletosolvethissystemfor z 2 x + y + z =1 3 x + z =4 x )]TJ/F32 8.9664 Tf 9.216 0 Td [(y )]TJ/F32 8.9664 Tf 9.216 0 Td [(z =2 3 ProveCramer'sRule.

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Topic:Cramer'sRule 333 4 Supposethatalinearsystemhasasmanyequationsasunknowns,thatallof itscoecientsandconstantsareintegers,andthatitsmatrixofcoecientshas determinant1.Provethattheentriesinthesolutionareallintegers. Remark. Thisisoftenusedtoinventlinearsystemsforexercises.Ifaninstructormakes thelinearsystemwiththispropertythenthesolutionisnotsomedisagreeable fraction. 5 UseCramer'sRuletogiveaformulaforthesolutionofatwoequations/two unknownslinearsystem. 6 CanCramer'sRuletellthedierencebetweenasystemwithnosolutionsand onewithinnitelymany? 7 TherstpictureinthisTopictheonethatdoesn'tusedeterminantsshows auniquesolutioncase.Produceasimilarpictureforthecaseofinntelymany solutions,andthecaseofnosolutions.

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334 ChapterFour.Determinants Topic:SpeedofCalculatingDeterminants Thepermutationexpansionformulaforcomputingdeterminantsisusefulfor provingtheorems,butthemethodofusingrowoperationsisamuchbetterfor ndingthedeterminantsofalargematrix.Wecanmakethisstatementprecise byconsidering,ascomputeralgorithmdesignersdo,thenumberofarithmetic operationsthateachmethoduses. Thespeedofanalgorithmismeasuredbyndinghowthetimetakenby thecomputergrowsasthesizeofitsinputdatasetgrows.Forinstance,how muchlongerwillthealgorithmtakeifweincreasethesizeoftheinputdata byafactoroften,froma1000rowmatrixtoa10 ; 000rowmatrixorfrom 10 ; 000to100 ; 000?Doesthetimetakengrowbyafactoroften,orbyafactor ofahundred,orbyafactorofathousand?Thatis,isthetimetakenbythe algorithmproportionaltothesizeofthedataset,ortothesquareofthatsize, ortothecubeofthatsize,etc.? Recallthepermutationexpansionformulafordeterminants. t 1 ; 1 t 1 ; 2 :::t 1 ;n t 2 ; 1 t 2 ; 2 :::t 2 ;n t n; 1 t n; 2 :::t n;n = X permutations t 1 ; t 2 ; t n; n j P j = t 1 ; 1 t 2 ; 1 t n; 1 n j P 1 j + t 1 ; 2 t 2 ; 2 t n; 2 n j P 2 j + t 1 ; k t 2 ; k t n; k n j P k j Thereare n != n n )]TJ/F8 9.9626 Tf 9.208 0 Td [(1 n )]TJ/F8 9.9626 Tf 9.208 0 Td [(2 2 1dierent n -permutations.Fornumbers n ofanysizeatall,thisisalargevalue;forinstance,evenif n isonly10 thentheexpansionhas10!=3 ; 628 ; 800terms,allofwhichareobtainedby multiplying n entriestogether.Thisisaverylargenumberofmultiplications forinstance,[Knuth]suggests10!stepsasaroughboundaryforthelimit ofpracticalcalculation.Thefactorialfunctiongrowsfasterthanthesquare function.Itgrowsfasterthanthecubefunction,thefourthpowerfunction, oranypolynomialfunction.Onewaytoseethatthefactorialfunctiongrows fasterthanthesquareistonotethatmultiplyingthersttwofactorsin n gives n n )]TJ/F8 9.9626 Tf 10.324 0 Td [(1,whichforlarge n isapproximately n 2 ,andthenmultiplying inmorefactorswillmakeitevenlarger.Thesameargumentworksforthe cubefunction,etc.Soacomputerthatisprogrammedtousethepermutation expansionformula,andthustoperformanumberofoperationsthatisgreater thanorequaltothefactorialofthenumberofrows,wouldtakeverylongtimes asitsinputdatasetgrows. Incontrast,thetimetakenbytherowreductionmethoddoesnotgrowso fast.Thisfragmentofrow-reductioncodeisinthecomputerlanguageFORTRAN.Thematrixisstoredinthe N N array A .Foreach ROW between1 and N partsoftheprogramnotshownherehavealreadyfoundthepivotentry

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Topic:SpeedofCalculatingDeterminants 335 A ROW;COL .Nowtheprogramdoesarowpivot. )]TJ/F46 9.9626 Tf 7.749 0 Td [(PIVINV ROW + i Thiscodefragmentisforillustrationonlyandisincomplete.Still,analysisof anishedversionthatincludesallofthetestsandsubcasesismessierbutgives essentiallythesameconclusion. PIVINV=1.0/AROW,COL DO10I=ROW+1,N DO20J=I,N AI,J=AI,J-PIVINV*AROW,J 20CONTINUE 10CONTINUE Theoutermostloopnotshownrunsthrough N )]TJ/F8 9.9626 Tf 10.351 0 Td [(1rows.Foreachrow, thenested I and J loopsshownperformarithmeticontheentriesin A thatare belowandtotherightofthepivotentry.Assumethatthepivotisfoundin theexpectedplace,thatis,that COL = ROW .Thenthereare N )]TJ/F46 9.9626 Tf 10.172 0 Td [(ROW 2 entriesbelowandtotherightofthepivot.Onaverage, ROW willbe N= 2. Thus,weestimatethatthearithmeticwillbeperformedabout N= 2 2 times, thatis,willruninatimeproportionaltothesquareofthenumberofequations. Takingintoaccounttheouterloopthatisnotshown,wegettheestimatethat therunningtimeofthealgorithmisproportionaltothecubeofthenumberof equations. Findingthefastestalgorithmtocomputethedeterminantisatopicofcurrentresearch.Algorithmsareknownthatrunintimebetweenthesecondand thirdpower. Speedestimateslikethesehelpustounderstandhowquicklyorslowlyan algorithmwillrun.Algorithmsthatrunintimeproportionaltothesizeof thedatasetarefast,algorithmsthatrunintimeproportionaltothesquareof thesizeofthedatasetarelessfast,buttypicallyquiteusable,andalgorithms thatrunintimeproportionaltothecubeofthesizeofthedatasetarestill reasonableinspeedfornot-too-biginputdata.However,algorithmsthatrunin timegreaterthanorequaltothefactorialofthesizeofthedatasetarenot practicalforinputofanyappreciablesize. Thereareothermethodsbesidesthetwodiscussedherethatarealsoused forcomputationofdeterminants.Thoselieoutsideofourscope.Nonetheless, thiscontrastofthetwomethodsforcomputingdeterminantsmakesthepoint thatalthoughinprincipletheygivethesameanswer,inpracticetheideaisto selecttheonethatisfast. Exercises Mostoftheseproblemspresumeaccesstoacomputer. 1 Computersystemsgeneraterandomnumbersofcourse,theseareonlypseudorandom,inthattheyaregeneratedbyanalgorithm,buttheypassanumberof reasonablestatisticaltestsforrandomness. a Filla5 5arraywithrandomnumberssay,intherange[0 :: 1.Seeifitis singular.Repeatthatexperimentafewtimes.Aresingularmatricesfrequent orrareinthissense?

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336 ChapterFour.Determinants b Timeyourcomputeralgebrasystematndingthedeterminantoften5 5 arraysofrandomnumbers.Findtheaveragetimeperarray.Repeattheprior itemfor15 15arrays,25 25arrays,and35 35arrays.Noticethat,whenan arrayissingular,itcansometimesbefoundtobesoquitequickly,forinstance iftherstrowequalsthesecond.Inthelightofyouranswertotherstpart, doyouexpectthatsingularsystemsplayalargeroleinyouraverage? c Graphtheinputsizeversustheaveragetime. 2 Computethedeterminantofeachofthesebyhandusingthetwomethodsdiscussedabove. a 21 5 )]TJ/F29 8.9664 Tf 7.167 0 Td [(3 b 311 )]TJ/F29 8.9664 Tf 7.167 0 Td [(105 )]TJ/F29 8.9664 Tf 7.167 0 Td [(12 )]TJ/F29 8.9664 Tf 7.168 0 Td [(2 c 2100 1320 0 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(21 00 )]TJ/F29 8.9664 Tf 7.168 0 Td [(21 Countthenumberofmultiplicationsanddivisionsusedineachcase,foreachof themethods.Onacomputer,multiplicationsanddivisionstakemuchlongerthan additionsandsubtractions,soalgorithmdesignersworryaboutthemmore. 3 What10 10arraycanyouinventthattakesyourcomputersystemthelongest toreduce?Theshortest? 4 WritetherestoftheFORTRANprogramtodoastraightforwardimplementation ofcalculatingdeterminantsviaGauss'method.Don'ttestforazeropivot. Comparethespeedofyourcodetothatusedinyourcomputeralgebrasystem. 5 TheFORTRANlanguagespecicationrequiresthatarraysbestoredbycolumn",thatis,theentirerstcolumnisstoredcontiguously,thenthesecondcolumn,etc.Doesthecodefragmentgiventakeadvantageofthis,orcanitbe rewrittentomakeitfaster,bytakingadvantageofthefactthatcomputerfetches arefasterfromcontiguouslocations?

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Topic:ProjectiveGeometry 337 Topic:ProjectiveGeometry TherearegeometriesotherthanthefamiliarEuclideanone.Onesuchgeometry aroseinart,whereitwasobservedthatwhataviewerseesisnotnecessarily whatisthere.ThisisLeonardodaVinci's TheLastSupper Whatisthereintheroom,forinstancewheretheceilingmeetstheleftand rightwalls,arelinesthatareparallel.However,whataviewerseesislines that,ifextended,wouldintersect.Theintersectionpointiscalledthe vanishing point .Thisaspectofperspectiveisalsofamiliarastheimageofalongstretch ofrailroadtracksthatappeartoconvergeatthehorizon. Todepicttheroom,daVincihasadoptedamodelofhowwesee,ofhow weprojectthethreedimensionalscenetoatwodimensionalimage.Thismodel isonlyarstapproximation|itdoesnottakeintoaccountthatourretinais curvedandourlensbendsthelight,thatwehavebinocularvision,orthatour brain'sprocessinggreatlyaectswhatwesee|butnonethelessitisinteresting, bothartisticallyandmathematically. Theprojectionisnotorthogonal,itisa centralprojection fromasingle point,totheplaneofthecanvas. A B C Itisnotanorthogonalprojectionsincethelinefromtheviewerto C isnot orthogonaltotheimageplane.Asthepicturesuggests,theoperationofcentral projectionpreservessomegeometricproperties|linesprojecttolines.However,itfailstopreservesomeothers|equallengthsegmentscanprojectto segmentsofunequallength;thelengthof AB isgreaterthanthelengthof

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338 ChapterFour.Determinants BC becausethesegmentprojectedto AB isclosertotheviewerandcloser thingslookbigger.Thestudyoftheeectsofcentralprojectionsisprojective geometry.Wewillseehowlinearalgebracanbeusedinthisstudy. Therearethreecasesofcentralprojection.Therstistheprojectiondone byamovieprojector. projector P source S image I Wecanthinkthateachsourcepointispushed"fromthedomainplaneoutwardtotheimagepointinthecodomainplane.Thiscaseofprojectionhasa somewhatdierentcharacterthanthesecondcase,thatoftheartistpulling" thesourcebacktothecanvas. painter P image I source S Intherstcase S isinthemiddlewhileinthesecondcase I isinthemiddle. Onemorecongurationispossible,with P inthemiddle.Anexampleofthis iswhenweuseapinholetoshinetheimageofasolareclipseontoapieceof paper. source S pinhole P image I Weshalltakeeachofthethreetobeacentralprojectionby P of S to I .

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Topic:ProjectiveGeometry 339 Consideragaintheeectofrailroadtracksthatappeartoconvergetoa point.Wemodelthiswithparallellinesinadomainplane S andaprojection viaa P toacodomainplane I .Thegraylinesareparallelto S and I S I P Allthreeprojectioncasesappearhere.Therstpicturebelowshows P acting likeamovieprojectorbypushingpointsfrompartof S outtoimagepointson thelowerhalfof I .Themiddlepictureshows P actingliketheartistbypulling pointsfromanotherpartof S backtoimagepointsinthemiddleof I .Inthe thirdpicture, P actslikethepinhole,projectingpointsfrom S totheupper partof I .Thispictureisthetrickiest|thepointsthatareprojectednearto thevanishingpointaretheonesthatarefaroutonthebottomleftof S .Points in S thatareneartotheverticalgraylinearesenthighupon I S I P S I P S I P Therearetwoawkwardthingsaboutthissituation.Therstisthatneither ofthetwopointsinthedomainnearesttotheverticalgraylineseebelow hasanimagebecauseaprojectionfromthosetwoisalongthegraylinethatis paralleltothecodomainplanewesometimessaythatthesetwoareprojected toinnity".Thesecondawkwardthingisthatthevanishingpointin I isn't theimageofanypointfrom S becauseaprojectiontothispointwouldbealong thegraylinethatisparalleltothedomainplanewesometimessaythatthe vanishingpointistheimageofaprojectionfrominnity". S I P

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340 ChapterFour.Determinants Forabettermodel,puttheprojector P attheorigin.Imaginethat P is coveredbyaglasshemisphericdome.As P looksoutward,anythingintheline ofvisionisprojectedtothesamespotonthedome.Thisincludesthingson thelinebetween P andthedome,asinthecaseofprojectionbythemovie projector.Itincludesthingsonthelinefurtherfrom P thanthedome,asin thecaseofprojectionbythepainter.Italsoincludesthingsonthelinethatlie behind P ,asinthecaseofprojectionbyapinhole. ` = f k 1 2 3 k 2 R g Fromthisperspective P ,allofthespotsonthelineareseenasthesamepoint. Accordingly,foranynonzerovector ~v 2 R 3 ,wedenetheassociated point v intheprojectiveplane tobetheset f k~v k 2 R and k 6 =0 g ofnonzerovectors lyingonthesamelinethroughtheoriginas ~v .Todescribeaprojectivepoint wecangiveanyrepresentativememberoftheline,sothattheprojectivepoint shownabovecanberepresentedinanyofthesethreeways. 0 @ 1 2 3 1 A 0 @ 1 = 3 2 = 3 1 1 A 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6 1 A Eachoftheseisa homogeneouscoordinatevector for v Thispicture,andtheabovedenitionthatarisesfromit,clariesthedescriptionofcentralprojectionbutthereissomethingawkwardaboutthedome model:whatiftheviewerlooksdown?Ifwedraw P 'slineofsightsothat thepartcomingtowardus,outofthepage,goesdownbelowthedomethen wecantracethelineofsightbackward,uppast P andtowardthepartofthe hemispherethatisbehindthepage.Sointhedomemodel,lookingdowngives aprojectivepointthatisbehindtheviewer.Therefore,iftheviewerinthe pictureabovedropsthelineofsighttowardthebottomofthedomethenthe projectivepointdropsalsoandasthelineofsightcontinuesdownpastthe equator,theprojectivepointsuddenlyshiftsfromthefrontofthedometothe backofthedome.Thisdiscontinuityinthedrawingmeansthatweoftenhave totreatequatorialpointsasaseparatecase.Thatis,whiletherailroadtrack discussionofcentralprojectionhasthreecases,thedomemodelhastwo. Wecandobetterthanthis.Consideraspherecenteredattheorigin.Any linethroughtheoriginintersectsthesphereintwospots,whicharesaidtobe antipodal .Becauseweassociateeachlinethroughtheoriginwithapointinthe projectiveplane,wecandrawsuchapointasapairofantipodalspotsonthe sphere.Below,thetwoantipodalspotsareshownconnectedbyadashedline

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Topic:ProjectiveGeometry 341 toemphasizethattheyarenottwodierentpoints,thepairofspotstogether makeoneprojectivepoint. Whiledrawingapointasapairofantipodalspotsisnotasnaturalastheonespot-per-pointdomemode,ontheotherhandtheawkwardnessofthedome modelisgone,inthatifasalineofviewslidesfromnorthtosouth,nosudden changeshappenonthepicture.Thismodelofcentralprojectionisuniform| thethreecasesarereducedtoone. Sofarwehavedescribedpointsinprojectivegeometry.Whataboutlines? Whataviewerattheoriginseesasalineisshownbelowasagreatcircle,the intersectionofthemodelspherewithaplanethroughtheorigin. Oneoftheprojectivepointsonthislineisshowntobringoutasubtlety. Becausetwoantipodalspotstogethermakeupasingleprojectivepoint,the greatcircle'sbehind-the-paperpartisthesamesetofprojectivepointsasits in-front-of-the-paperpart.Justaswedidwitheachprojectivepoint,wewill alsodescribeaprojectivelinewithatripleofreals.Forinstance,themembers ofthisplanethroughtheoriginin R 3 f 0 @ x y z 1 A x + y )]TJ/F11 9.9626 Tf 9.962 0 Td [(z =0 g projecttoalinethatwecandescribedwiththetriple )]TJ/F8 9.9626 Tf 4.566 -7.97 Td [(11 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 weuserow vectorstotypographicallydistinguishlinesfrompoints.Ingeneral,forany nonzerothree-widerowvector ~ L wedenetheassociated lineintheprojective plane ,tobetheset L = f k ~ L k 2 R and k 6 =0 g ofnonzeromultiplesof ~ L Thereasonthatthisdescriptionofalineasatripleisconvienentisthat intheprojectiveplane,apoint v andaline L are incident |thepointlies ontheline,thelinepassesthroughtthepoint|ifandonlyifadotproduct oftheirrepresentatives v 1 L 1 + v 2 L 2 + v 3 L 3 iszeroExercise4showsthatthis isindependentofthechoiceofrepresentatives ~v and ~ L .Forinstance,the projectivepointdescribedabovebythecolumnvectorwithcomponents1,2, and3liesintheprojectivelinedescribedby )]TJ/F8 9.9626 Tf 4.566 -7.97 Td [(11 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ,simplybecauseany

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342 ChapterFour.Determinants vectorin R 3 whosecomponentsareinratio1:2:3liesintheplanethroughthe originwhoseequationisoftheform1 k x +1 k y )]TJ/F8 9.9626 Tf 9.245 0 Td [(1 k z =0foranynonzero k Thatis,theincidenceformulaisinheritedfromthethree-spacelinesandplanes ofwhich v and L areprojections. Thus,wecandoanalyticprojectivegeometry.Forinstance,theprojective line L = )]TJ/F8 9.9626 Tf 4.566 -7.97 Td [(11 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 hastheequation1 v 1 +1 v 2 )]TJ/F8 9.9626 Tf 10.731 0 Td [(1 v 3 =0,becausepoints incidentonthelinearecharacterizedbyhavingthepropertythattheirrepresentativessatisfythisequation.OnedierencefromfamiliarEuclideananlaytic geometryisthatinprojectivegeometrywetalkabouttheequationofapoint. Foraxedpointlike v = 0 @ 1 2 3 1 A thepropertythatcharacterizeslinesthroughthispointthatis,linesincident onthispointisthatthecomponentsofanyrepresentativessatisfy1 L 1 +2 L 2 + 3 L 3 =0andsothisistheequationof v Thissymmetryofthestatementsaboutlinesandpointsbringsupthe Duality Principle ofprojectivegeometry:inanytruestatement,interchanging`point' with`line'resultsinanothertruestatement.Forexample,justastwodistinct pointsdetermineoneandonlyoneline,intheprojectiveplane,twodistinct linesdetermineoneandonlyonepoint.Hereisapictureshowingtwolinesthat crossinantipodalspotsandthuscrossatoneprojectivepoint. ContrastthiswithEuclideangeometry,wheretwodistinctlinesmayhavea uniqueintersectionormaybeparallel.Inthisway,projectivegeometryis simpler,moreuniform,thanEuclideangeometry. Thatsimplicityisrelevantbecausethereisarelationshipbetweenthetwo spaces:theprojectiveplanecanbeviewedasanextensionoftheEuclidean plane.Takethespheremodeloftheprojectiveplanetobetheunitspherein R 3 andtakeEuclideanspacetobetheplane z =1.Thisgivesusawayof viewingsomepointsinprojectivespaceascorrespondingtopointsinEuclidean space,becauseallofthepointsontheplaneareprojectionsofantipodalspots fromthesphere.

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Topic:ProjectiveGeometry 343 Notethoughthatprojectivepointsontheequatordon'tprojectuptotheplane. Instead,theseproject`outtoinnity'.Wecanthusthinkofprojectivespace asconsistingoftheEuclideanplanewithsomeextrapointsadjoined|theEuclideanplaneisembeddedintheprojectiveplane.Theseextrapoints,the equatorialpoints,arethe idealpoints or pointsatinnity andtheequatoris the idealline or lineatinnity notethatitisnotaEuclideanline,itisa projectiveline. Theadvantageoftheextensiontotheprojectiveplaneisthatsomeofthe awkwardnessofEuclideangeometrydisappears.Forinstance,theprojective linesshownabovein crossatantipodalspots,asingleprojectivepoint,on thesphere'sequator.Ifweputthoselinesinto thentheycorrespondto Euclideanlinesthatareparallel.Thatis,inmovingfromtheEuclideanplaneto theprojectiveplane,wemovefromhavingtwocases,thatlineseitherintersect orareparallel,tohavingonlyonecase,thatlinesintersectpossiblyatapoint atinnity. TheprojectivecaseisnicerinmanywaysthantheEuclideancasebuthas theproblemthatwedon'thavethesameexperienceorintuitionswithit.That's oneadvantageofdoinganalyticgeometry,wheretheequationscanleadusto therightconclusions.Analyticprojectivegeometryuseslinearalgebra.For instance,forthreepointsoftheprojectiveplane t u ,and v ,settingupthe equationsforthosepointsbyxingvectorsrepresentingeach,showsthatthe threearecollinear|incidentinasingleline|ifandonlyiftheresultingthreeequationsystemhasinnitelymanyrowvectorsolutionsrepresentingthatline. That,inturn,holdsifandonlyifthisdeterminantiszero. t 1 u 1 v 1 t 2 u 2 v 2 t 3 u 3 v 3 Thus,threepointsintheprojectiveplanearecollinearifandonlyifanythree representativecolumnvectorsarelinearlydependent.Similarlyandillustrating theDualityPrinciple,threelinesintheprojectiveplaneareincidentona singlepointifandonlyifanythreerowvectorsrepresentingthemarelinearly dependent. Thefollowingresultismoreevidenceofthe`niceness'ofthegeometryofthe projectiveplane,comparedtotheEuclideancase.Thesetwotrianglesaresaid tobe inperspective from P becausetheircorrespondingverticesarecollinear. O T 1 U 1 V 1 T 2 U 2 V 2 Considerthepairsofcorrespondingsides:thesides T 1 U 1 and T 2 U 2 ,thesides T 1 V 1 and T 2 V 2 ,andthesides U 1 V 1 and U 2 V 2 .Desargue'sTheoremisthat

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344 ChapterFour.Determinants whenthethreepairsofcorrespondingsidesareextendedtolines,theyintersect shownhereasthepoint TU ,thepoint TV ,andthepoint UV ,andfurther, thosethreeintersectionpointsarecollinear. TU TV UV Wewillprovethistheorem,usingprojectivegeometry.Thesearedrawnas Euclideanguresbecauseitisthemorefamiliarimage.Toconsiderthemas projectivegures,wecanimaginethat,althoughthelinesegmentsshownare partsofgreatcirclesandsoarecurved,themodelhassuchalargeradius comparedtothesizeoftheguresthatthesidesappearinthissketchtobe straight. Forthisproof,weneedapreliminarylemma[Coxeter]:if W X Y Z are fourpointsintheprojectiveplanenothreeofwhicharecollinearthenthere arehomogeneouscoordinatevectors ~w ~x ~y ,and ~z fortheprojectivepoints, andabasis B for R 3 ,satisfyingthis. Rep B ~w = 0 @ 1 0 0 1 A Rep B ~x = 0 @ 0 1 0 1 A Rep B ~y = 0 @ 0 0 1 1 A Rep B ~z = 0 @ 1 1 1 1 A Theproofisstraightforward.Because W;X;Y arenotonthesameprojective line,anyhomogeneouscoordinatevectors ~w 0 ;~x 0 ;~y 0 donotlineonthesame planethroughtheoriginin R 3 andsoformaspanningsetfor R 3 .Thusany homogeneouscoordinatevectorfor Z canbewrittenasacombination ~z 0 = a ~w 0 + b ~x 0 + c ~y 0 .Then,wecantake ~w = a ~w 0 ~x = b ~x 0 ~y = c ~y 0 ,and ~z = ~z 0 ,wherethebasisis B = h ~w;~x;~y i Now,toproveofDesargue'sTheorem,usethelemmatoxhomogeneous coordinatevectorsandabasis. Rep B ~ t 1 = 0 @ 1 0 0 1 A Rep B ~u 1 = 0 @ 0 1 0 1 A Rep B ~v 1 = 0 @ 0 0 1 1 A Rep B ~o = 0 @ 1 1 1 1 A Becausetheprojectivepoint T 2 isincidentontheprojectiveline OT 1 ,any homogeneouscoordinatevectorfor T 2 liesintheplanethroughtheoriginin R 3 thatisspannedbyhomogeneouscoordinatevectorsof O and T 1 : Rep B ~ t 2 = a 0 @ 1 1 1 1 A + b 0 @ 1 0 0 1 A

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Topic:ProjectiveGeometry 345 forsomescalars a and b .Thatis,thehomogenouscoordinatevectorsofmembers T 2 oftheline OT 1 areoftheformontheleftbelow,andtheformsfor U 2 and V 2 aresimilar. Rep B ~ t 2 = 0 @ t 2 1 1 1 A Rep B ~u 2 = 0 @ 1 u 2 1 1 A Rep B ~v 2 = 0 @ 1 1 v 2 1 A Theprojectiveline T 1 U 1 istheimageofaplanethroughtheoriginin R 3 .A quickwaytogetitsequationistonotethatanyvectorinitislinearlydependent onthevectorsfor T 1 and U 1 andsothisdeterminantiszero. 10 x 01 y 00 z =0= z =0 Theequationoftheplanein R 3 whoseimageistheprojectiveline T 2 U 2 isthis. t 2 1 x 1 u 2 y 11 z =0= )]TJ/F11 9.9626 Tf 9.963 0 Td [(u 2 x + )]TJ/F11 9.9626 Tf 9.963 0 Td [(t 2 y + t 2 u 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 z =0 Findingtheintersectionofthetwoisroutine. T 1 U 1 T 2 U 2 = 0 @ t 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(u 2 0 1 A Thisis,ofcourse,thehomogeneouscoordinatevectorofaprojectivepoint. Theothertwointersectionsaresimilar. T 1 V 1 T 2 V 2 = 0 @ 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(t 2 0 v 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 1 A U 1 V 1 U 2 V 2 = 0 @ 0 u 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 1 )]TJ/F11 9.9626 Tf 9.962 0 Td [(v 2 1 A Theproofisnishedbynotingthattheseprojectivepointsareononeprojective linebecausethesumofthethreehomogeneouscoordinatevectorsiszero. EveryprojectivetheoremhasatranslationtoaEuclideanversion,although theEuclideanresultisoftenmessiertostateandprove.Desargue'stheorem illustratesthis.InthetranslationtoEuclideanspace,thecasewhere O lieson theideallinemustbetreatedseparatelyforthenthelines T 1 T 2 U 1 U 2 ,and V 1 V 2 areparallel. TheparentheticalremarkfollowingthestatementofDesargue'sTheorem suggeststhinkingoftheEuclideanpicturesasguresfromprojectivegeometry foramodelofverylargeradius.Thatis,justasasmallareaoftheearthappears attopeoplelivingthere,theprojectiveplaneisalso`locallyEuclidean'. AlthoughitslocalpropertiesarethefamiliarEuclideanones,thereisaglobal propertyoftheprojectiveplanethatisquitedierent.Thepicturebelowshows

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346 ChapterFour.Determinants aprojectivepoint.Atthatpointisdrawnan xy -axis.Thereissomething interestingaboutthewaythisaxisappearsattheantipodalendsofthesphere. Inthenorthernhemisphere,wheretheaxisaredrawninblack,arighthandput downwithngersonthe x -axiswillhavethethumbpointalongthe y -axis.But theantipodalaxishasjusttheopposite:arighthandplacedwithitsngerson the x -axiswillhavethethumbpointinthewrongway,instead,itisalefthand thatworks.Briey,theprojectiveplaneisnotorientable:inthisgeometry,left andrighthandednessarenotxedpropertiesofgures. Thesequenceofpicturesbelowdramatizesthisnon-orientability.Theysketch atriparoundthisspaceinthedirectionofthe y partofthe xy -axis.Warning: thetripshownisnothalfwayaround,itisafullcircuit.True,ifwemadethis intoamoviethenwecouldwatchthenorthernhemispherespotsinthedrawing abovegraduallyrotateabouthalfwayaroundthespheretothelastpicture below.Andwecouldwatchthesouthernhemispherespotsinthepictureabove slidethroughthesouthpoleandupthroughtheequatortothelastpicture. But:thespotsateitherendofthedashedlinearethesameprojectivepoint. Wedon'tneedtocontinueonmuchfurther;weareprettymuchbacktothe projectivepointwherewestartedbythelastpicture. = = Attheendofthecircuit,the x partofthe xy -axessticksoutintheother direction.Thus,intheprojectiveplanewecannotdescribeagureasright-or left-handedanotherwaytomakethispointisthatwecannotdescribeaspiral asclockwiseorcounterclockwise. Thisexhibitionoftheexistenceofanon-orientablespaceraisesthequestion ofwhetherouruniverseisorientable:isispossibleforanastronauttoleave right-handedandreturnleft-handed?Anexcellentnontechnicalreferenceis [Gardner].Anclassicsciencectionstoryaboutorientationreversalis[Clarke]. Soprojectivegeometryismathematicallyinteresting,inadditiontothenaturalwayinwhichitarisesinart.Itismorethanjustatechnicaldeviceto shortensomeproofs.Foranoverview,see[Courant&Robbins].Theapproach we'vetakenhere,theanalyticapproach,leadstoquicktheoremsand|most importantlyforus|illustratesthepoweroflinearalgebrasee[Hanes],[Ryan], and[Eggar].Butanotherapproach,thesyntheticapproachofderivingthe

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Topic:ProjectiveGeometry 347 resultsfromanaxiomsystem,isbothextraordinarilybeautifulandisalsothe historicalrouteofdevelopment.Twonesourcesforthisapproachare[Coxeter] or[Seidenberg].Aninterestingandeasyapplicationis[Davies] Exercises 1 Whatistheequationofthispoint? 1 0 0 2a Findthelineincidentonthesepointsintheprojectiveplane. 1 2 3 ; 4 5 6 b Findthepointincidentonbothoftheseprojectivelines. )]TJ/F29 8.9664 Tf 4.566 -7.771 Td [(123 ; )]TJ/F29 8.9664 Tf 4.566 -7.771 Td [(456 3 Findtheformulaforthelineincidentontwoprojectivepoints.Findtheformula forthepointincidentontwoprojectivelines. 4 Provethatthedenitionofincidenceisindependentofthechoiceoftherepresentativesof p and L .Thatis,if p 1 p 2 p 3 ,and q 1 q 2 q 3 aretwotriplesof homogeneouscoordinatesfor p ,and L 1 L 2 L 3 ,and M 1 M 2 M 3 aretwotriples ofhomogeneouscoordinatesfor L ,provethat p 1 L 1 + p 2 L 2 + p 3 L 3 =0ifandonly if q 1 M 1 + q 2 M 2 + q 3 M 3 =0. 5 Giveadrawingtoshowthatcentralprojectiondoesnotpreservecircles,thata circlemayprojecttoanellipse.Cananon-circularellipseprojecttoacircle? 6 Givetheformulaforthecorrespondencebetweenthenon-equatorialpartofthe antipodalmodaloftheprojectiveplane,andtheplane z =1. 7 Pappus'sTheoremAssumethat T 0 U 0 ,and V 0 arecollinearandthat T 1 U 1 and V 1 arecollinear.Considerthesethreepoints:itheintersection V 2 ofthelines T 0 U 1 and T 1 U 0 ,iitheintersection U 2 ofthelines T 0 V 1 and T 1 V 0 ,andiiithe intersection T 2 of U 0 V 1 and U 1 V 0 a DrawaEuclideanpicture. b ApplythelemmausedinDesargue'sTheoremtogetsimplehomogeneous coordinatevectorsforthe T 'sand V 0 c Findtheresultinghomogeneouscoordinatevectorsfor U 'sthesemusteach involveaparameteras,e.g., U 0 couldbeanywhereonthe T 0 V 0 line. d Findtheresultinghomogeneouscoordinatevectorsfor V 1 Hint: itinvolves twoparameters. e Findtheresultinghomogeneouscoordinatevectorsfor V 2 .Italsoinvolves twoparameters. f Showthattheproductofthethreeparametersis1. g Verifythat V 2 isonthe T 2 U 2 line..

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ChapterFive Similarity Whilestudyingmatrixequivalence,wehaveshownthatforanyhomomorphism therearebases B and D suchthattherepresentationmatrixhasablockpartialidentityform. Rep B;D h = Identity Zero Zero Zero Thisrepresentationdescribesthemapassending c 1 ~ 1 + + c n ~ n to c 1 ~ 1 + + c k ~ k + ~ 0+ + ~ 0,where n isthedimensionofthedomainand k isthe dimensionoftherange.So,underthisrepresentationtheactionofthemapis easytounderstandbecausemostofthematrixentriesarezero. Thischapterconsidersthespecialcasewherethedomainandthecodomain areequal,thatis,wherethehomomorphismisatransformation.Inthiscase wenaturallyasktondasinglebasis B sothatRep B;B t isassimpleas possiblewewilltake`simple'tomeanthatithasmanyzeroes.Amatrix havingtheaboveblockpartial-identityformisnotalwayspossiblehere.Butwe willdevelopaformthatcomesclose,arepresentationthatisnearlydiagonal. IComplexVectorSpaces Thischapterrequiresthatwefactorpolynomials.Ofcourse,manypolynomials donotfactorovertherealnumbers;forinstance, x 2 +1doesnotfactorinto theproductoftwolinearpolynomialswithrealcoecients.Forthatreason,we shallfromnowontakeourscalarsfromthecomplexnumbers. Thatis,weareshiftingfromstudyingvectorspacesovertherealnumbers tovectorspacesoverthecomplexnumbers|inthischaptervectorandmatrix entriesarecomplex. Anyrealnumberisacomplexnumberandaglancethroughthischapter showsthatmostoftheexamplesuseonlyrealnumbers.Nonetheless,thecritical theoremsrequirethatthescalarsbecomplexnumbers,sotherstsectionbelow isaquickreviewofcomplexnumbers. 349

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350 ChapterFive.Similarity Inthisbookwearemovingtothemoregeneralcontextoftakingscalarsto becomplexonlyforthepragmaticreasonthatwemustdosoinordertodevelop therepresentation.Wewillnotgointousingothersetsofscalarsinmoredetail becauseitcoulddistractfromourgoal.However,theideaoftakingscalars fromastructureotherthantherealnumbersisaninterestingone.Delightful presentationstakingthisapproacharein[Halmos]and[Homan&Kunze]. I.1FactoringandComplexNumbers;AReview Thissubsectionisareviewonlyandwetakethemainresultsasknown.For proofs,see[Birkho&MacLane]or[Ebbinghaus]. Justasintegershaveadivisionoperation|e.g.,`4goes5timesinto21with remainder1'|sodopolynomials. 1.1TheoremDivisionTheoremforPolynomials Let c x beapolynomial.If m x isanon-zeropolynomialthenthereare quotient and remainder polynomials q x and r x suchthat c x = m x q x + r x wherethedegreeof r x isstrictlylessthanthedegreeof m x Inthisbookconstantpolynomials,includingthezeropolynomial,aresaidto havedegree0.Thisisnotthestandarddenition,butitisconvienenthere. Thepointoftheintegerdivisionstatement`4goes5timesinto21with remainder1'isthattheremainderislessthan4|while4goes5times,itdoes notgo6times.Inthesameway,thepointofthepolynomialdivisionstatement isitsnalclause. 1.2Example If c x =2 x 3 )]TJ/F8 9.9626 Tf 9.743 0 Td [(3 x 2 +4 x and m x = x 2 +1then q x =2 x )]TJ/F8 9.9626 Tf 9.743 0 Td [(3 and r x =2 x +3.Notethat r x hasalowerdegreethan m x 1.3Corollary Theremainderwhen c x isdividedby x )]TJ/F11 9.9626 Tf 10.268 0 Td [( istheconstant polynomial r x = c Proof Theremaindermustbeaconstantpolynomialbecauseitisofdegreeless thanthedivisor x )]TJ/F11 9.9626 Tf 9.517 0 Td [( ,Todeterminetheconstant,take m x fromthetheorem tobe x )]TJ/F11 9.9626 Tf 9.963 0 Td [( andsubstitute for x toget c = )]TJ/F11 9.9626 Tf 9.962 0 Td [( q + r x QED Ifadivisor m x goesintoadividend c x evenly,meaningthat r x isthe zeropolynomial,then m x isa factor of c x .Any root ofthefactorany 2 R suchthat m =0isarootof c x since c = m q =0.The priorcorollaryimmediatelyyieldsthefollowingconverse. 1.4Corollary If isarootofthepolynomial c x then x )]TJ/F11 9.9626 Tf 10.396 0 Td [( divides c x evenly,thatis, x )]TJ/F11 9.9626 Tf 9.962 0 Td [( isafactorof c x .

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SectionI.ComplexVectorSpaces 351 Findingtherootsandfactorsofahigh-degreepolynomialcanbehard.But forsecond-degreepolynomialswehavethequadraticformula:therootsof ax 2 + bx + c are 1 = )]TJ/F11 9.9626 Tf 7.749 0 Td [(b + p b 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 ac 2 a 2 = )]TJ/F11 9.9626 Tf 7.749 0 Td [(b )]TJ 9.962 8.303 Td [(p b 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 ac 2 a ifthediscriminant b 2 )]TJ/F8 9.9626 Tf 9.059 0 Td [(4 ac isnegativethenthepolynomialhasnorealnumber roots.Apolynomialthatcannotbefactoredintotwolower-degreepolynomials withrealnumbercoecientsis irreducibleoverthereals 1.5Theorem Anyconstantorlinearpolynomialisirreducibleoverthereals. Aquadraticpolynomialisirreducibleovertherealsifandonlyifitsdiscriminantisnegative.Nocubicorhigher-degreepolynomialisirreducibleoverthe reals. 1.6Corollary Anypolynomialwithrealcoecientscanbefactoredintolinear andirreduciblequadraticpolynomials.Thatfactorizationisunique;anytwo factorizationshavethesamepowersofthesamefactors. Notetheanalogywiththeprimefactorizationofintegers.Inbothcases,the uniquenessclauseisveryuseful. 1.7Example Becauseofuniquenessweknow,withoutmultiplyingthemout, that x +3 2 x 2 +1 3 doesnotequal x +3 4 x 2 + x +1 2 1.8Example Byuniqueness,if c x = m x q x thenwhere c x = x )]TJ/F8 9.9626 Tf -335.962 -11.955 Td [(3 2 x +2 3 and m x = x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x +2 2 ,weknowthat q x = x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x +2. While x 2 +1hasnorealrootsandsodoesn'tfactorovertherealnumbers, ifweimaginearoot|traditionallydenoted i sothat i 2 +1=0|then x 2 +1 factorsintoaproductoflinears x )]TJ/F11 9.9626 Tf 9.962 0 Td [(i x + i Soweadjointhisroot i totherealsandclosethenewsystemwithrespect toaddition,multiplication,etc.i.e.,wealsoadd3+ i ,and2 i ,and3+2 i ,etc., puttinginalllinearcombinationsof1and i .Wethengetanewstructure,the complexnumbers ,denoted C In C wecanfactorobviously,atleastsomequadraticsthatwouldbeirreducibleifweweretosticktotherealnumbers.Surprisingly,in C wecannot onlyfactor x 2 +1anditscloserelatives,wecanfactoranyquadratic. ax 2 + bx + c = a )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(x )]TJ 11.158 6.74 Td [()]TJ/F11 9.9626 Tf 7.749 0 Td [(b + p b 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 ac 2 a )]TJ/F11 9.9626 Tf 4.567 -8.069 Td [(x )]TJ 11.158 6.739 Td [()]TJ/F11 9.9626 Tf 7.749 0 Td [(b )]TJ 9.962 8.303 Td [(p b 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 ac 2 a 1.9Example Theseconddegreepolynomial x 2 + x +1factorsoverthecomplex numbersintotheproductoftworstdegreepolynomials. )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(x )]TJ 11.158 6.739 Td [()]TJ/F8 9.9626 Tf 7.749 0 Td [(1+ p )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 2 )]TJ/F11 9.9626 Tf 9.132 -8.069 Td [(x )]TJ 11.158 6.739 Td [()]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ 9.963 7.827 Td [(p )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 2 = )]TJ/F11 9.9626 Tf 4.566 -8.069 Td [(x )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 8.944 6.739 Td [(1 2 + p 3 2 i )]TJ/F11 9.9626 Tf 9.132 -8.069 Td [(x )]TJ/F8 9.9626 Tf 9.963 0 Td [( )]TJ/F8 9.9626 Tf 8.944 6.739 Td [(1 2 )]TJ 11.158 14.981 Td [(p 3 2 i 1.10CorollaryFundamentalTheoremofAlgebra Polynomialswith complexcoecientsfactorintolinearpolynomialswithcomplexcoecients. Thefactorizationisunique.

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352 ChapterFive.Similarity I.2ComplexRepresentations Recallthedenitionsofthecomplexnumberaddition a + bi + c + di = a + c + b + d i andmultiplication. a + bi c + di = ac + adi + bci + bd )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = ac )]TJ/F11 9.9626 Tf 9.963 0 Td [(bd + ad + bc i 2.1Example Forinstance, )]TJ/F8 9.9626 Tf 9.072 0 Td [(2 i ++4 i =6+2 i and )]TJ/F8 9.9626 Tf 9.072 0 Td [(3 i )]TJ/F8 9.9626 Tf 9.072 0 Td [(0 : 5 i = 6 : 5 )]TJ/F8 9.9626 Tf 9.962 0 Td [(13 i Handlingscalaroperationswiththoserules,alloftheoperationsthatwe've coveredforrealvectorspacescarryoverunchanged. 2.2Example Matrixmultiplicationisthesame,althoughthescalararithmetic involvesmorebookkeeping. 1+1 i 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(0 i i )]TJ/F8 9.9626 Tf 7.749 0 Td [(2+3 i 1+0 i 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(0 i 3 i )]TJ/F11 9.9626 Tf 7.749 0 Td [(i = +1 i +0 i + )]TJ/F8 9.9626 Tf 9.962 0 Td [(0 i i +1 i )]TJ/F8 9.9626 Tf 9.963 0 Td [(0 i + )]TJ/F8 9.9626 Tf 9.963 0 Td [(0 i )]TJ/F11 9.9626 Tf 7.749 0 Td [(i i +0 i + )]TJ/F8 9.9626 Tf 7.748 0 Td [(2+3 i i i )]TJ/F8 9.9626 Tf 9.963 0 Td [(0 i + )]TJ/F8 9.9626 Tf 7.749 0 Td [(2+3 i )]TJ/F11 9.9626 Tf 7.749 0 Td [(i = 1+7 i 1 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 i )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 )]TJ/F8 9.9626 Tf 9.963 0 Td [(5 i 3+3 i Everythingelsefrompriorchaptersthatwecan,weshallalsocarryover unchanged.Forinstance,weshallcallthis h 0 B B B @ 1+0 i 0+0 i 0+0 i 1 C C C A ;:::; 0 B B B @ 0+0 i 0+0 i 1+0 i 1 C C C A i the standardbasis for C n asavectorspaceover C andagaindenoteit E n .

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SectionII.Similarity 353 IISimilarity II.1DenitionandExamples We'vedened H and ^ H tobematrix-equivalentiftherearenonsingularmatrices P and Q suchthat ^ H = PHQ .Thatdenitionismotivatedbythis diagram V w.r.t. B h )333()223()222()333(! H W w.r.t. D id ? ? y id ? ? y V w.r.t. ^ B h )333()223()222()333(! ^ H W w.r.t. ^ D showingthat H and ^ H bothrepresent h butwithrespecttodierentpairsof bases.Wenowspecializethatsetuptothecasewherethecodomainequalsthe domain,andwherethecodomain'sbasisequalsthedomain'sbasis. V w.r.t. B t )333()223()222()333(! V w.r.t. B id ? ? y id ? ? y V w.r.t. D t )333()223()222()333(! V w.r.t. D Tomovefromthelowerlefttothelowerrightwecaneithergostraightover,or up,over,andthendown.Inmatrixterms, Rep D;D t =Rep B;D idRep B;B t )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(Rep B;D id )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 recallthatarepresentationofcompositionlikethisonereadsrighttoleft. 1.1Denition Thematrices T and S are similar ifthereisanonsingular P suchthat T = PSP )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Sincenonsingularmatricesaresquare,thesimilarmatrices T and S mustbe squareandofthesamesize. 1.2Example Withthesetwo, P = 21 11 S = 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 calculationgivesthat S issimilartothismatrix. T = 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 11

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354 ChapterFive.Similarity 1.3Example Theonlymatrixsimilartothezeromatrixisitself: PZP )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = PZ = Z .Theonlymatrixsimilartotheidentitymatrixisitself: PIP )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = PP )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = I Sincematrixsimilarityisaspecialcaseofmatrixequivalence,iftwomatricesaresimilarthentheyareequivalent.Whatabouttheconverse:must matrixequivalentsquarematricesbesimilar?Theanswerisno.Theprior exampleshowsthatthesimilarityclassesaredierentfromthematrixequivalenceclasses,becausethematrixequivalenceclassoftheidentityconsistsof allnonsingularmatricesofthatsize.Thus,forinstance,thesetwoarematrix equivalentbutnotsimilar. T = 10 01 S = 12 03 Sosomematrixequivalenceclassessplitintotwoormoresimilarityclasses| similaritygivesanerpartitionthandoesequivalence.Thispictureshowssome matrixequivalenceclassessubdividedintosimilarityclasses. ... A B Tounderstandthesimilarityrelationweshallstudythesimilarityclasses. Weapproachthisquestioninthesamewaythatwe'vestudiedboththerow equivalenceandmatrixequivalencerelations,byndingacanonicalformfor representatives ofthesimilarityclasses,calledJordanform.Withthiscanonicalform,wecandecideiftwomatricesaresimilarbycheckingwhetherthey reducetothesamerepresentative.We'vealsoseenwithbothrowequivalence andmatrixequivalencethatacanonicalformgivesusinsightintothewaysin whichmembersofthesameclassarealikee.g.,twoidentically-sizedmatrices arematrixequivalentifandonlyiftheyhavethesamerank. Exercises 1.4 For S = 13 )]TJ/F29 8.9664 Tf 7.168 0 Td [(2 )]TJ/F29 8.9664 Tf 7.168 0 Td [(6 T = 00 )]TJ/F29 8.9664 Tf 7.168 0 Td [(11 = 2 )]TJ/F29 8.9664 Tf 7.168 0 Td [(5 P = 42 )]TJ/F29 8.9664 Tf 7.167 0 Td [(32 checkthat T = PSP )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 X 1.5 Example1.3showsthattheonlymatrixsimilartoazeromatrixisitselfand thattheonlymatrixsimilartotheidentityisitself. a Showthatthe1 1matrix,also,issimilaronlytoitself. b Isamatrixoftheform cI forsomescalar c similaronlytoitself? c Isadiagonalmatrixsimilaronlytoitself? 1.6 Showthatthesematricesarenotsimilar. 104 113 217 101 011 312 Moreinformationonrepresentativesisintheappendix.

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SectionII.Similarity 355 1.7 Considerthetransformation t : P 2 !P 2 describedby x 2 7! x +1, x 7! x 2 )]TJ/F29 8.9664 Tf 8.963 0 Td [(1, and1 7! 3. a Find T =Rep B;B t where B = h x 2 ;x; 1 i b Find S =Rep D;D t where D = h 1 ; 1+ x; 1+ x + x 2 i c Findthematrix P suchthat T = PSP )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 X 1.8 Exhibitannontrivialsimilarityrelationshipinthisway:let t : C 2 C 2 actby 1 2 7! 3 0 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 1 7! )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 2 andpicktwobases,andrepresent t withrespecttothen T =Rep B;B t and S =Rep D;D t .Thencomputethe P and P )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 tochangebasesfrom B to D and backagain. 1.9 ExplainExample1.3intermsofmaps. X 1.10 Aretheretwomatrices A and B thataresimilarwhile A 2 and B 2 arenot similar?[Halmos] X 1.11 Provethatiftwomatricesaresimilarandoneisinvertiblethensoistheother. X 1.12 Showthatsimilarityisanequivalencerelation. 1.13 Consideramatrixrepresenting,withrespecttosome B;B ,reectionacross the x -axisin R 2 .Consideralsoamatrixrepresenting,withrespecttosome D;D reectionacrossthe y -axis.Musttheybesimilar? 1.14 Provethatsimilaritypreservesdeterminantsandrank.Doestheconverse hold? 1.15 Isthereamatrixequivalenceclasswithonlyonematrixsimilarityclassinside? Onewithinnitelymanysimilarityclasses? 1.16 Cantwodierentdiagonalmatricesbeinthesamesimilarityclass? X 1.17 Provethatiftwomatricesaresimilarthentheir k -thpowersaresimilarwhen k> 0.Whatif k 0? X 1.18 Let p x bethepolynomial c n x n + + c 1 x + c 0 .Showthatif T issimilarto S then p T = c n T n + + c 1 T + c 0 I issimilarto p S = c n S n + + c 1 S + c 0 I 1.19 Listallofthematrixequivalenceclassesof1 1matrices.Alsolistthesimilarityclasses,anddescribewhichsimilarityclassesarecontainedinsideofeach matrixequivalenceclass. 1.20 Doessimilaritypreservesums? 1.21 Showthatif T )]TJ/F32 8.9664 Tf 9.293 0 Td [(I and N aresimilarmatricesthen T and N + I arealso similar. II.2Diagonalizability Thepriorsubsectiondenestherelationofsimilarityandshowsthat,although similarmatricesarenecessarilymatrixequivalent,theconversedoesnothold. Somematrix-equivalenceclassesbreakintotwoormoresimilarityclassesthe nonsingular n n matrices,forinstance.Thismeansthatthecanonicalform formatrixequivalence,ablockpartial-identity,cannotbeusedasacanonical formformatrixsimilaritybecausethepartial-identitiescannotbeinmorethan

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356 ChapterFive.Similarity onesimilarityclass,sotherearesimilarityclasseswithoutone.Thispicture illustrates.Asearlierinthisbook,classrepresentativesareshownwithstars. ... ? ? ? ? ? ? ? ? ? Wearedevelopingacanonicalformforrepresentativesofthesimilarityclasses. Wenaturallytrytobuildonourpreviouswork,meaningrstthatthepartial identitymatricesshouldrepresentthesimilarityclassesintowhichtheyfall, andbeyondthat,thattherepresentativesshouldbeassimpleaspossible.The simplestextensionofthepartial-identityformisadiagonalform. 2.1Denition Atransformationis diagonalizable ifithasadiagonalrepresentationwithrespecttothesamebasisforthecodomainasforthedomain. A diagonalizablematrix isonethatissimilartoadiagonalmatrix: T isdiagonalizableifthereisanonsingular P suchthat PTP )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 isdiagonal. 2.2Example Thematrix 4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 11 isdiagonalizable. 20 03 = )]TJ/F8 9.9626 Tf 7.748 0 Td [(12 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 11 )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 2.3Example Noteverymatrixisdiagonalizable.Thesquareof N = 00 10 isthezeromatrix.Thus,foranymap n that N representswithrespecttothe samebasisforthedomainasforthecodomain,thecomposition n n isthe zeromap.Thisimpliesthatnosuchmap n canbediagonallyrepresentedwith respecttoany B;B becausenopowerofanonzerodiagonalmatrixiszero. Thatis,thereisnodiagonalmatrixin N 'ssimilarityclass. Thatexampleshowsthatadiagonalformwillnotdoforacanonicalform| wecannotndadiagonalmatrixineachmatrixsimilarityclass.However,the canonicalformthatwearedevelopinghasthepropertythatifamatrixcan bediagonalizedthenthediagonalmatrixisthecanonicalrepresentativeofthe similarityclass.Thenextresultcharacterizeswhichmapscanbediagonalized. 2.4Corollary Atransformation t isdiagonalizableifandonlyifthereisa basis B = h ~ 1 ;:::; ~ n i andscalars 1 ;:::; n suchthat t ~ i = i ~ i foreach i .

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SectionII.Similarity 357 Proof Thisfollowsfromthedenitionbyconsideringadiagonalrepresentation matrix. Rep B;B t = 0 B B @ Rep B t ~ 1 Rep B t ~ n 1 C C A = 0 B @ 1 0 . 0 n 1 C A Thisrepresentationisequivalenttotheexistenceofabasissatisfyingthestated conditionssimplybythedenitionofmatrixrepresentation. QED 2.5Example Todiagonalize T = 32 01 wetakeitastherepresentationofatransformationwithrespecttothestandard basis T =Rep E 2 ; E 2 t andwelookforabasis B = h ~ 1 ; ~ 2 i suchthat Rep B;B t = 1 0 0 2 thatis,suchthat t ~ 1 = 1 ~ 1 and t ~ 2 = 2 ~ 2 32 01 ~ 1 = 1 ~ 1 32 01 ~ 2 = 2 ~ 2 Wearelookingforscalars x suchthatthisequation 32 01 b 1 b 2 = x b 1 b 2 hassolutions b 1 and b 2 ,whicharenotbothzero.Rewritethatasalinearsystem. )]TJ/F11 9.9626 Tf 9.963 0 Td [(x b 1 +2 b 2 =0 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x b 2 =0 Inthebottomequationthetwonumbersmultiplytogivezeroonlyifatleast oneofthemiszerosotherearetwopossibilities, b 2 =0and x =1.Inthe b 2 =0 possibility,therstequationgivesthateither b 1 =0or x =3.Sincethecase ofboth b 1 =0and b 2 =0isdisallowed,weareleftlookingatthepossibilityof x =3.Withit,therstequationin is0 b 1 +2 b 2 =0andsoassociated with3arevectorswithasecondcomponentofzeroandarstcomponentthat isfree. 32 01 b 1 0 =3 b 1 0 Thatis,onesolutionto is 1 =3,andwehavearstbasisvector. ~ 1 = 1 0

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358 ChapterFive.Similarity Inthe x =1possibility,therstequationin is2 b 1 +2 b 2 =0,andso associatedwith1arevectorswhosesecondcomponentisthenegativeoftheir rstcomponent. 32 01 b 1 )]TJ/F11 9.9626 Tf 7.749 0 Td [(b 1 =1 b 1 )]TJ/F11 9.9626 Tf 7.749 0 Td [(b 1 Thus,anothersolutionis 2 =1andasecondbasisvectoristhis. ~ 2 = 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Tonish,drawingthesimilaritydiagram R 2 w.r.t. E 2 t )333()223()222()333(! T R 2 w.r.t. E 2 id ? ? y id ? ? y R 2 w.r.t. B t )333()223()222()333(! D R 2 w.r.t. B andnotingthatthematrixRep B; E 2 idiseasyleadstothisdiagonalization. 30 01 = 11 0 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 32 01 11 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 Inthenextsubsection,wewillexpandonthatexamplebyconsideringmore closelythepropertyofCorollary2.4.Thisincludesseeinganotherway,theway thatwewillroutinelyuse,tondthe 's. Exercises X 2.6 RepeatExample2.5forthematrixfromExample2.2. 2.7 Diagonalizetheseuppertriangularmatrices. a )]TJ/F29 8.9664 Tf 7.168 0 Td [(21 02 b 54 01 X 2.8 Whatformdothepowersofadiagonalmatrixhave? 2.9 Givetwosame-sizeddiagonalmatricesthatarenotsimilar.Mustanytwo dierentdiagonalmatricescomefromdierentsimilarityclasses? 2.10 Giveanonsingulardiagonalmatrix.Canadiagonalmatrixeverbesingular? X 2.11 Showthattheinverseofadiagonalmatrixisthediagonalofthetheinverses, ifnoelementonthatdiagonaliszero.Whathappenswhenadiagonalentryis zero? 2.12 TheequationendingExample2.5 11 0 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 32 01 11 0 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 = 30 01 isabitjarringbecausefor P wemusttaketherstmatrix,whichisshownasan inverse,andfor P )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 wetaketheinverseoftherstmatrix,sothatthetwo )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 powerscancelandthismatrixisshownwithoutasuperscript )]TJ/F29 8.9664 Tf 7.167 0 Td [(1. a Checkthatthisnicer-appearingequationholds. 30 01 = 11 0 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 32 01 11 0 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 )]TJ/F31 5.9776 Tf 5.756 0 Td [(1

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SectionII.Similarity 359 b Isthepreviousitemacoincidence?Orcanwealwaysswitchthe P andthe P )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 ? 2.13 Showthatthe P usedtodiagonalizeinExample2.5isnotunique. 2.14 Findaformulaforthepowersofthismatrix Hint :seeExercise8. )]TJ/F29 8.9664 Tf 7.167 0 Td [(31 )]TJ/F29 8.9664 Tf 7.167 0 Td [(42 X 2.15 Diagonalizethese. a 11 00 b 01 10 2.16 Wecanaskhowdiagonalizationinteractswiththematrixoperations.Assume that t;s : V V areeachdiagonalizable.Is ct diagonalizableforallscalars c ? Whatabout t + s ? t s ? X 2.17 Showthatmatricesofthisformarenotdiagonalizable. 1 c 01 c 6 =0 2.18 Showthateachoftheseisdiagonalizable. a 12 21 b xy yz x;y;z scalars II.3EigenvaluesandEigenvectors InthissubsectionwewillfocusonthepropertyofCorollary2.4. 3.1Denition Atransformation t : V V hasascalar eigenvalue ifthere isanonzero eigenvector ~ 2 V suchthat t ~ = ~ Eigen"isGermanforcharacteristicof"orpeculiarto";someauthorscall these characteristic valuesandvectors.Noauthorscallthempeculiar". 3.2Example Theprojectionmap 0 @ x y z 1 A 7)167(! 0 @ x y 0 1 A x;y;z 2 C hasaneigenvalueof1associatedwithanyeigenvectoroftheform 0 @ x y 0 1 A where x and y arenon-0scalars.Ontheotherhand,2isnotaneigenvalueof sincenonon~ 0vectorisdoubled. Thatexampleshowswhythe`non~ 0'appearsinthedenition.Disallowing ~ 0asaneigenvectoreliminatestrivialeigenvalues.

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360 ChapterFive.Similarity 3.3Example Theonlytransformationonthetrivialspace f ~ 0 g is ~ 0 7! ~ 0.This maphasnoeigenvaluesbecausetherearenonon~ 0vectors ~v mappedtoascalar multiple ~v ofthemselves. 3.4Example Considerthehomomorphism t : P 1 !P 1 givenby c 0 + c 1 x 7! c 0 + c 1 + c 0 + c 1 x .Therangeof t isone-dimensional.Thusanapplicationof t toavectorintherangewillsimplyrescalethatvector: c + cx 7! c + c x Thatis, t hasaneigenvalueof2associatedwitheigenvectorsoftheform c + cx where c 6 =0. Thismapalsohasaneigenvalueof0associatedwitheigenvectorsoftheform c )]TJ/F11 9.9626 Tf 9.962 0 Td [(cx where c 6 =0. 3.5Denition Asquarematrix T hasascalar eigenvalue associatedwith thenon~ 0 eigenvector ~ if T ~ = ~ 3.6Remark Althoughthisextensionfrommapstomatricesisobvious,there isapointthatmustbemade.Eigenvaluesofamaparealsotheeigenvaluesof matricesrepresentingthatmap,andsosimilarmatriceshavethesameeigenvalues.Buttheeigenvectorsaredierent|similarmatricesneednothavethe sameeigenvectors. Forinstance,consideragainthetransformation t : P 1 !P 1 givenby c 0 + c 1 x 7! c 0 + c 1 + c 0 + c 1 x .Ithasaneigenvalueof2associatedwitheigenvectors oftheform c + cx where c 6 =0.Ifwerepresent t withrespectto B = h 1+ 1 x; 1 )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 x i T =Rep B;B t = 20 00 then2isaneigenvalueof T ,associatedwiththeseeigenvectors. f c 0 c 1 20 00 c 0 c 1 = 2 c 0 2 c 1 g = f c 0 0 c 0 2 C ;c 0 6 =0 g Ontheotherhand,representing t withrespectto D = h 2+1 x; 1+0 x i gives S =Rep D;D t = 31 )]TJ/F8 9.9626 Tf 7.748 0 Td [(3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 andtheeigenvectorsof S associatedwiththeeigenvalue2arethese. f c 0 c 1 31 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 c 0 c 1 = 2 c 0 2 c 1 g = f 0 c 1 c 1 2 C ;c 1 6 =0 g Thussimilarmatricescanhavedierenteigenvectors. Hereisaninformaldescriptionofwhat'shappening.Theunderlyingtransformationdoublestheeigenvectors ~v 7! 2 ~v .Butwhenthematrixrepresenting thetransformationis T =Rep B;B t thenitassumes"thatcolumnvectorsare representationswithrespectto B .Incontrast, S =Rep D;D t assumes"that columnvectorsarerepresentationswithrespectto D .Sothevectorsthatget doubledbyeachmatrixlookdierent.

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SectionII.Similarity 361 Thenextexampleillustratesthebasictoolforndingeigenvectorsandeigenvalues. 3.7Example Whataretheeigenvaluesandeigenvectorsofthismatrix? T = 0 @ 121 20 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(123 1 A Tondthescalars x suchthat T ~ = x ~ fornon~ 0eigenvectors ~ ,bringeverythingtotheleft-handside 0 @ 121 20 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(123 1 A 0 @ z 1 z 2 z 3 1 A )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 0 @ z 1 z 2 z 3 1 A = ~ 0 andfactor T )]TJ/F11 9.9626 Tf 8.636 0 Td [(xI ~ = ~ 0.Notethatitsays T )]TJ/F11 9.9626 Tf 8.636 0 Td [(xI ;theexpression T )]TJ/F11 9.9626 Tf 8.636 0 Td [(x doesn't makesensebecause T isamatrixwhile x isascalar.Thishomogeneouslinear system 0 @ 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 21 20 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(123 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 1 A 0 @ z 1 z 2 z 3 1 A = 0 @ 0 0 0 1 A hasanon~ 0solutionifandonlyifthematrixissingular.Wecandetermine whenthathappens. 0= j T )]TJ/F11 9.9626 Tf 9.962 0 Td [(xI j = 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 21 20 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(123 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x = x 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [(4 x 2 +4 x = x x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 2 Theeigenvaluesare 1 =0and 2 =2.Tondtheassociatedeigenvectors, plugineacheigenvalue.Pluggingin 1 =0gives 0 @ 1 )]TJ/F8 9.9626 Tf 9.962 0 Td [(021 20 )]TJ/F8 9.9626 Tf 9.962 0 Td [(0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(123 )]TJ/F8 9.9626 Tf 9.963 0 Td [(0 1 A 0 @ z 1 z 2 z 3 1 A = 0 @ 0 0 0 1 A = 0 @ z 1 z 2 z 3 1 A = 0 @ a )]TJ/F11 9.9626 Tf 7.749 0 Td [(a a 1 A forascalarparameter a 6 =0 a isnon-0becauseeigenvectorsmustbenon~ 0. Inthesameway,pluggingin 2 =2gives 0 @ 1 )]TJ/F8 9.9626 Tf 9.962 0 Td [(221 20 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(123 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 1 A 0 @ z 1 z 2 z 3 1 A = 0 @ 0 0 0 1 A = 0 @ z 1 z 2 z 3 1 A = 0 @ b 0 b 1 A with b 6 =0.

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362 ChapterFive.Similarity 3.8Example If S = 1 03 here isnotaprojectionmap,itisthenumber3 : 14 ::: then )]TJ/F11 9.9626 Tf 9.963 0 Td [(x 1 03 )]TJ/F11 9.9626 Tf 9.963 0 Td [(x = x )]TJ/F11 9.9626 Tf 9.963 0 Td [( x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 so S haseigenvaluesof 1 = and 2 =3.Tondassociatedeigenvectors,rst plugin 1 for x : )]TJ/F11 9.9626 Tf 9.962 0 Td [( 1 03 )]TJ/F11 9.9626 Tf 9.962 0 Td [( z 1 z 2 = 0 0 = z 1 z 2 = a 0 forascalar a 6 =0,andthenplugin 2 : )]TJ/F8 9.9626 Tf 9.962 0 Td [(31 03 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 z 1 z 2 = 0 0 = z 1 z 2 = )]TJ/F11 9.9626 Tf 7.749 0 Td [(b= )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 b where b 6 =0. 3.9Denition The characteristicpolynomial ofasquarematrix T isthe determinantofthematrix T )]TJ/F11 9.9626 Tf 10.388 0 Td [(xI ,where x isavariable.The characteristic equation is j T )]TJ/F11 9.9626 Tf 10.093 0 Td [(xI j =0.Thecharacteristicpolynomialofatransformation t isthepolynomialofanyRep B;B t Exercise30checksthatthecharacteristicpolynomialofatransformationis well-dened,thatis,anychoiceofbasisyieldsthesamepolynomial. 3.10Lemma Alineartransformationonanontrivialvectorspacehasatleast oneeigenvalue. Proof Anyrootofthecharacteristicpolynomialisaneigenvalue.Overthe complexnumbers,anypolynomialofdegreeoneorgreaterhasaroot.Thisis thereasonthatinthischapterwe'vegonetoscalarsthatarecomplex. QED Noticethefamiliarformofthesetsofeigenvectorsintheaboveexamples. 3.11Denition The eigenspace ofatransformation t associatedwiththe eigenvalue is V = f ~ t ~ = ~ g[f ~ 0 g .Theeigenspaceofamatrixis denedanalogously. 3.12Lemma Aneigenspaceisasubspace. Proof Aneigenspacemustbenonempty|foronethingitcontainsthezero vector|andsoweneedonlycheckclosure.Takevectors ~ 1 ;:::; ~ n from V ,to showthatanylinearcombinationisin V t c 1 ~ 1 + c 2 ~ 2 + + c n ~ n = c 1 t ~ 1 + + c n t ~ n = c 1 ~ 1 + + c n ~ n = c 1 ~ 1 + + c n ~ n thesecondequalityholdsevenifany ~ i is ~ 0since t ~ 0= ~ 0= ~ 0. QED

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SectionII.Similarity 363 3.13Example InExample3.8theeigenspaceassociatedwiththeeigenvalue andtheeigenspaceassociatedwiththeeigenvalue3arethese. V = f a 0 a 2 R g V 3 = f )]TJ/F11 9.9626 Tf 7.749 0 Td [(b= )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 b b 2 R g 3.14Example InExample3.7,thesearetheeigenspacesassociatedwiththe eigenvalues0and2. V 0 = f 0 @ a )]TJ/F11 9.9626 Tf 7.748 0 Td [(a a 1 A a 2 R g ;V 2 = f 0 @ b 0 b 1 A b 2 R g : 3.15Remark Thecharacteristicequationis0= x x )]TJ/F8 9.9626 Tf 8.548 0 Td [(2 2 soinsomesense2is aneigenvaluetwice".Howevertherearenottwice"asmanyeigenvectors,in thatthedimensionoftheeigenspaceisone,nottwo.Thenextexampleshows acasewhereanumber,1,isadoublerootofthecharacteristicequationand thedimensionoftheassociatedeigenspaceistwo. 3.16Example Withrespecttothestandardbases,thismatrix 0 @ 100 010 000 1 A representsprojection. 0 @ x y z 1 A 7)167(! 0 @ x y 0 1 A x;y;z 2 C Itseigenspaceassociatedwiththeeigenvalue0anditseigenspaceassociated withtheeigenvalue1areeasytond. V 0 = f 0 @ 0 0 c 3 1 A c 3 2 C g V 1 = f 0 @ c 1 c 2 0 1 A c 1 ;c 2 2 C g Bythelemma,iftwoeigenvectors ~v 1 and ~v 2 areassociatedwiththesame eigenvaluethenanylinearcombinationofthosetwoisalsoaneigenvectorassociatedwiththatsameeigenvalue.But,iftwoeigenvectors ~v 1 and ~v 2 are associatedwithdierenteigenvaluesthenthesum ~v 1 + ~v 2 neednotberelated totheeigenvalueofeitherone.Infact,justtheopposite.Iftheeigenvaluesare dierentthentheeigenvectorsarenotlinearlyrelated. 3.17Theorem Foranysetofdistincteigenvaluesofamapormatrix,aset ofassociatedeigenvectors,onepereigenvalue,islinearlyindependent.

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364 ChapterFive.Similarity Proof Wewilluseinductiononthenumberofeigenvalues.Ifthereisnoeigenvalueoronlyoneeigenvaluethenthesetofassociatedeigenvectorsisemptyoris asingletonsetwithanon~ 0member,andineithercaseislinearlyindependent. Forinduction,assumethatthetheoremistrueforanysetof k distincteigenvalues,supposethat 1 ;:::; k +1 aredistincteigenvalues,andlet ~v 1 ;:::;~v k +1 beassociatedeigenvectors.If c 1 ~v 1 + + c k ~v k + c k +1 ~v k +1 = ~ 0thenaftermultiplyingbothsidesofthedisplayedequationby k +1 ,applyingthemapormatrix tobothsidesofthedisplayedequation,andsubtractingtherstresultfromthe second,wehavethis. c 1 k +1 )]TJ/F11 9.9626 Tf 9.963 0 Td [( 1 ~v 1 + + c k k +1 )]TJ/F11 9.9626 Tf 9.963 0 Td [( k ~v k + c k +1 k +1 )]TJ/F11 9.9626 Tf 9.963 0 Td [( k +1 ~v k +1 = ~ 0 Theinductionhypothesisnowapplies: c 1 k +1 )]TJ/F11 9.9626 Tf 8.377 0 Td [( 1 =0 ;:::;c k k +1 )]TJ/F11 9.9626 Tf 8.377 0 Td [( k =0. Thus,asalltheeigenvaluesaredistinct, c 1 ;:::;c k areall0.Finally,now c k +1 mustbe0becauseweareleftwiththeequation ~v k +1 6 = ~ 0. QED 3.18Example Theeigenvaluesof 0 @ 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(22 011 )]TJ/F8 9.9626 Tf 7.748 0 Td [(483 1 A aredistinct: 1 =1, 2 =2,and 3 =3.Asetofassociatedeigenvectorslike f 0 @ 2 1 0 1 A ; 0 @ 9 4 4 1 A ; 0 @ 2 1 2 1 A g islinearlyindependent. 3.19Corollary An n n matrixwith n distincteigenvaluesisdiagonalizable. Proof Formabasisofeigenvectors.ApplyCorollary2.4. QED Exercises 3.20 Foreach,ndthecharacteristicpolynomialandtheeigenvalues. a 10 )]TJ/F29 8.9664 Tf 7.168 0 Td [(9 4 )]TJ/F29 8.9664 Tf 7.168 0 Td [(2 b 12 43 c 03 70 d 00 00 e 10 01 X 3.21 Foreachmatrix,ndthecharacteristicequation,andtheeigenvaluesand associatedeigenvectors. a 30 8 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 b 32 )]TJ/F29 8.9664 Tf 7.168 0 Td [(10 3.22 Findthecharacteristicequation,andtheeigenvaluesandassociatedeigenvectorsforthismatrix. Hint. Theeigenvaluesarecomplex. )]TJ/F29 8.9664 Tf 7.167 0 Td [(2 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 52

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SectionII.Similarity 365 3.23 Findthecharacteristicpolynomial,theeigenvalues,andtheassociatedeigenvectorsofthismatrix. 111 001 001 X 3.24 Foreachmatrix,ndthecharacteristicequation,andtheeigenvaluesand associatedeigenvectors. a 3 )]TJ/F29 8.9664 Tf 7.168 0 Td [(20 )]TJ/F29 8.9664 Tf 7.168 0 Td [(230 005 b 010 001 4 )]TJ/F29 8.9664 Tf 7.168 0 Td [(178 X 3.25 Let t : P 2 !P 2 be a 0 + a 1 x + a 2 x 2 7! a 0 +6 a 1 +2 a 2 )]TJ/F29 8.9664 Tf 9.216 0 Td [( a 1 +8 a 2 x + a 0 )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 a 2 x 2 : Finditseigenvaluesandtheassociatedeigenvectors. 3.26 Findtheeigenvaluesandeigenvectorsofthismap t : M 2 !M 2 ab cd 7! 2 ca + c b )]TJ/F29 8.9664 Tf 9.216 0 Td [(2 cd X 3.27 Findtheeigenvaluesandassociatedeigenvectorsofthedierentiationoperator d=dx : P 3 !P 3 3.28 Provethattheeigenvaluesofatriangularmatrixupperorlowertriangular aretheentriesonthediagonal. X 3.29 Findtheformulaforthecharacteristicpolynomialofa2 2matrix. 3.30 Provethatthecharacteristicpolynomialofatransformationiswell-dened. X 3.31a Cananynon~ 0vectorinanynontrivialvectorspacebeaeigenvector? Thatis,givena ~v 6 = ~ 0fromanontrivial V ,isthereatransformation t : V V andascalar 2 R suchthat t ~v = ~v ? b Givenascalar ,cananynon~ 0vectorinanynontrivialvectorspacebean eigenvectorassociatedwiththeeigenvalue ? X 3.32 Supposethat t : V V and T =Rep B;B t .Provethattheeigenvectorsof T associatedwith arethenon~ 0vectorsinthekernelofthemaprepresentedwith respecttothesamebasesby T )]TJ/F32 8.9664 Tf 9.215 0 Td [(I 3.33 Provethatif a;:::;d areallintegersand a + b = c + d then ab cd hasintegraleigenvalues,namely a + b and a )]TJ/F32 8.9664 Tf 9.215 0 Td [(c X 3.34 Provethatif T isnonsingularandhaseigenvalues 1 ;:::; n then T )]TJ/F31 5.9776 Tf 5.757 0 Td [(1 has eigenvalues1 = 1 ;:::; 1 = n .Istheconversetrue? X 3.35 Supposethat T is n n and c;d arescalars. a Provethatif T hastheeigenvalue withanassociatedeigenvector ~v then ~v isaneigenvectorof cT + dI associatedwitheigenvalue c + d b Provethatif T isdiagonalizablethensois cT + dI X 3.36 Showthat isaneigenvalueof T ifandonlyifthemaprepresentedby T )]TJ/F32 8.9664 Tf 8.179 0 Td [(I isnotanisomorphism. 3.37 [Strang80] a Showthatif isaneigenvalueof A then k isaneigenvalueof A k b Whatiswrongwiththisproofgeneralizingthat?If isaneigenvalueof A and isaneigenvaluefor B ,then isaneigenvaluefor AB ,for,if A~x = ~x and B~x = ~x then AB~x = A~x = A~x~x "?

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366 ChapterFive.Similarity 3.38 Domatrix-equivalentmatriceshavethesameeigenvalues? 3.39 Showthatasquarematrixwithrealentriesandanoddnumberofrowshas atleastonerealeigenvalue. 3.40 Diagonalize. )]TJ/F29 8.9664 Tf 7.167 0 Td [(122 222 )]TJ/F29 8.9664 Tf 7.167 0 Td [(3 )]TJ/F29 8.9664 Tf 7.168 0 Td [(6 )]TJ/F29 8.9664 Tf 7.168 0 Td [(6 3.41 Supposethat P isanonsingular n n matrix.Showthatthe similaritytransformation map t P : M n n !M n n sending T 7! PTP )]TJ/F31 5.9776 Tf 5.757 0 Td [(1 isanisomorphism. ? 3.42 Showthatif A isan n squarematrixandeachrowcolumnsumsto c then c isacharacteristicrootof A .[Math.Mag.,Nov.1967]

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SectionIII.Nilpotence 367 IIINilpotence Thegoalofthischapteristoshowthateverysquarematrixissimilartoone thatisasumoftwokindsofsimplematrices.Thepriorsectionfocusedonthe rstkind,diagonalmatrices.Wenowconsidertheotherkind. III.1Self-Composition Thissubsectionisoptional,althoughitisnecessaryforlatermaterialinthis sectionandinthenextone. Alineartransformations t : V V ,becauseithasthesamedomainand codomain,canbeiterated. Thatis,compositionsof t withitselfsuchas t 2 = t t and t 3 = t t t aredened. ~v t ~v t 2 ~v Notethatthispowernotationforthelineartransformationfunctionsdovetails withthenotationthatwe'veusedearlierfortheirsquarematrixrepresentations becauseifRep B;B t = T thenRep B;B t j = T j 1.1Example Forthederivativemap d=dx : P 3 !P 3 givenby a + bx + cx 2 + dx 3 d=dx 7)167(! b +2 cx +3 dx 2 thesecondpoweristhesecondderivative a + bx + cx 2 + dx 3 d 2 =dx 2 7)167(! 2 c +6 dx thethirdpoweristhethirdderivative a + bx + cx 2 + dx 3 d 3 =dx 3 7)167(! 6 d andanyhigherpoweristhezeromap. 1.2Example Thistransformationofthespaceof2 2matrices ab cd t 7)167(! ba d 0 Moreinformationonfunctioninterationisintheappendix.

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368 ChapterFive.Similarity hasthissecondpower ab cd t 2 7)167(! ab 00 andthisthirdpower. ab cd t 3 7)167(! ba 00 Afterthat, t 4 = t 2 and t 5 = t 3 ,etc. Theseexamplessuggestthatoniterationmoreandmorezerosappearuntil thereisasettlingdown.Thenextresultmakesthisprecise. 1.3Lemma Foranytransformation t : V V ,therangespacesofthepowers formadescendingchain V R t R t 2 andthenullspacesformanascendingchain. f ~ 0 g N t N t 2 Further,thereisa k suchthatforpowerslessthan k thesubsetsareproperif j
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SectionIII.Nilpotence 369 andthischainofnullspaces. f ~ 0 gP 0 P 1 P 2 P 3 = P 3 = 1.5Example Thetransformation : C 3 C 3 projectingontothersttwo coordinates 0 @ c 1 c 2 c 3 1 A 7)167(! 0 @ c 1 c 2 0 1 A has C 3 R = R 2 = and f ~ 0 g N = N 2 = 1.6Example Let t : P 2 !P 2 bethemap c 0 + c 1 x + c 2 x 2 7! 2 c 0 + c 2 x: Asthe lemmadescribes,oniterationtherangespaceshrinks R t 0 = P 2 R t = f a + bx a;b 2 C g R t 2 = f a a 2 C g andthenstabilizes R t 2 = R t 3 = ,whilethenullspacegrows N t 0 = f 0 g N t = f cx c 2 C g N t 2 = f cx + d c;d 2 C g andthenstabilizes N t 2 = N t 3 = ThisgraphillustratesLemma1.3.Thehorizontalaxisgivesthepower j ofatransformation.Theverticalaxisgivesthedimensionoftherangespace of t j asthedistanceabovezero|andthusalsoshowsthedimensionofthe nullspaceasthedistancebelowthegrayhorizontalline,becausethetwoaddto thedimension n ofthedomain. 0 1 2 j n n rank t j Power j ofthetransformation Assketched,oniterationtherankfallsandwithitthenullitygrowsuntilthe tworeachasteadystate.Thisstatemustbereachedbythe n -thiterate.The steadystate'sdistanceabovezeroisthedimensionofthegeneralizedrangespace anditsdistancebelow n isthedimensionofthegeneralizednullspace. 1.7Denition Let t beatransformationonan n -dimensionalspace.The generalizedrangespace orthe closureoftherangespace is R 1 t = R t n The generalizednullspace orthe closureofthenullspace is N 1 t = N t n

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370 ChapterFive.Similarity Exercises 1.8 Givethechainsofrangespacesandnullspacesforthezeroandidentitytransformations. 1.9 Foreachmap,givethechainofrangespacesandthechainofnullspaces,and thegeneralizedrangespaceandthegeneralizednullspace. a t 0 : P 2 !P 2 a + bx + cx 2 7! b + cx 2 b t 1 : R 2 R 2 a b 7! 0 a c t 2 : P 2 !P 2 a + bx + cx 2 7! b + cx + ax 2 d t 3 : R 3 R 3 a b c 7! a a b 1.10 Provethatfunctioncompositionisassociative t t t = t t t andsowe canwrite t 3 withoutspecifyingagrouping. 1.11 Checkthatasubspacemustbeofdimensionlessthanorequaltothedimensionofitssuperspace.Checkthatifthesubspaceisproperthesubspacedoesnot equalthesuperspacethenthedimensionisstrictlyless. Thisisusedintheproof ofLemma1.3. 1.12 Provethatthegeneralizedrangespace R 1 t istheentirespace,andthe generalizednullspace N 1 t istrivial,ifthetransformation t isnonsingular.Is this`onlyif'also? 1.13 VerifythenullspacehalfofLemma1.3. 1.14 Giveanexampleofatransformationonathreedimensionalspacewhose rangehasdimensiontwo.Whatisitsnullspace?Iterateyourexampleuntilthe rangespaceandnullspacestabilize. 1.15 Showthattherangespaceandnullspaceofalineartransformationneednot bedisjoint.Aretheyeverdisjoint? III.2Strings Thissubsectionisoptional,andrequiresmaterialfromtheoptionalDirectSum subsection. Thepriorsubsectionshowsthatas j increases,thedimensionsofthe R t j 's fallwhilethedimensionsofthe N t j 'srise,insuchawaythatthisrankand nullitysplitthedimensionof V .Canwesaymore;dothetwosplitabasis|is V = R t j N t j ? Theanswerisyesforthesmallestpower j =0since V = R t 0 N t 0 = V f ~ 0 g .Theanswerisalsoyesattheotherextreme. 2.1Lemma Where t : V V isalineartransformation,thespaceisthedirect sum V = R 1 t N 1 t .Thatis,bothdim V =dim R 1 t +dim N 1 t and R 1 t N 1 t = f ~ 0 g .

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SectionIII.Nilpotence 371 Proof Wewillverifythesecondsentence,whichisequivalenttotherst.The rstclause,thatthedimension n ofthedomainof t n equalstherankof t n plus thenullityof t n ,holdsforanytransformationandsoweneedonlyverifythe secondclause. Assumethat ~v 2 R 1 t N 1 t = R t n N t n ,toprovethat ~v is ~ 0. Because ~v isinthenullspace, t n ~v = ~ 0.Ontheotherhand,because R t n = R t n +1 ,themap t : R 1 t R 1 t isadimension-preservinghomomorphism andthereforeisone-to-one.Acompositionofone-to-onemapsisone-to-one, andso t n : R 1 t R 1 t isone-to-one.Butnow|becauseonly ~ 0issentby aone-to-onelinearmapto ~ 0|thefactthat t n ~v = ~ 0impliesthat ~v = ~ 0. QED 2.2Note Technicallyweshoulddistinguishthemap t : V V fromthemap t : R 1 t R 1 t becausethedomainsorcodomainsmightdier.Thesecond oneissaidtobethe restriction of t to R t k .Weshalluselaterapointfrom thatproofabouttherestrictionmap,namelythatitisnonsingular. Incontrasttothe j =0and j = n cases,forintermediatepowersthespace V mightnotbethedirectsumof R t j and N t j .Thenextexampleshows thatthetwocanhaveanontrivialintersection. 2.3Example Considerthetransformationof C 2 denedbythisactiononthe elementsofthestandardbasis. 1 0 n 7)167(! 0 1 0 1 n 7)167(! 0 0 N =Rep E 2 ; E 2 n = 00 10 Thevector ~e 2 = 0 1 isinboththerangespaceandnullspace.Anotherwaytodepictthismap's actioniswitha string ~e 1 7! ~e 2 7! ~ 0 2.4Example Amap^ n : C 4 C 4 whoseactionon E 4 isgivenbythestring ~e 1 7! ~e 2 7! ~e 3 7! ~e 4 7! ~ 0 has R ^ n N ^ n equaltothespan[ f ~e 4 g ],has R ^ n 2 N ^ n 2 =[ f ~e 3 ;~e 4 g ],and has R ^ n 3 N ^ n 3 =[ f ~e 4 g ].Thematrixrepresentationisallzerosexceptfor somesubdiagonalones. ^ N =Rep E 4 ; E 4 ^ n = 0 B B @ 0000 1000 0100 0010 1 C C A Moreinformationonmaprestrictionsisintheappendix.

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372 ChapterFive.Similarity 2.5Example Transformationscanactviamorethanonestring.Atransformation t actingonabasis B = h ~ 1 ;:::; ~ 5 i by ~ 1 7! ~ 2 7! ~ 3 7! ~ 0 ~ 4 7! ~ 5 7! ~ 0 isrepresentedbyamatrixthatisallzerosexceptforblocksofsubdiagonalones Rep B;B t = 0 B B B B @ 000 00 100 00 010 00 000 00 000 10 1 C C C C A thelinesjustvisuallyorganizetheblocks. Inthosethreeexamplesallvectorsareeventuallytransformedtozero. 2.6Denition A nilpotent transformationisonewithapowerthatisthe zeromap.A nilpotentmatrix isonewithapowerthatisthezeromatrix.In eithercase,theleastsuchpoweristhe indexofnilpotency 2.7Example InExample2.3theindexofnilpotencyistwo.InExample2.4 itisfour.InExample2.5itisthree. 2.8Example Thedierentiationmap d=dx : P 2 !P 2 isnilpotentofindex threesincethethirdderivativeofanyquadraticpolynomialiszero.Thismap's actionisdescribedbythestring x 2 7! 2 x 7! 2 7! 0andtakingthebasis B = h x 2 ; 2 x; 2 i givesthisrepresentation. Rep B;B d=dx = 0 @ 000 100 010 1 A Notallnilpotentmatricesareallzerosexceptforblocksofsubdiagonalones. 2.9Example Withthematrix ^ N fromExample2.4,andthisfour-vectorbasis D = h 0 B B @ 1 0 1 0 1 C C A ; 0 B B @ 0 2 1 0 1 C C A ; 0 B B @ 1 1 1 0 1 C C A ; 0 B B @ 0 0 0 1 1 C C A i achangeofbasisoperationproducesthisrepresentationwithrespectto D;D 0 B B @ 1010 0210 1110 0001 1 C C A 0 B B @ 0000 1000 0100 0010 1 C C A 0 B B @ 1010 0210 1110 0001 1 C C A )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 0 B B @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(1010 )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 )]TJ/F8 9.9626 Tf 7.748 0 Td [(250 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(130 21 )]TJ/F8 9.9626 Tf 7.749 0 Td [(20 1 C C A

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SectionIII.Nilpotence 373 Thenewmatrixisnilpotent;it'sfourthpoweristhezeromatrixsince P ^ NP )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 4 = P ^ NP )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 P ^ NP )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 P ^ NP )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 P ^ NP )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = P ^ N 4 P )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 and ^ N 4 isthezeromatrix. ThegoalofthissubsectionisTheorem2.13,whichshowsthattheprior exampleisprototypicalinthateverynilpotentmatrixissimilartoonethatis allzerosexceptforblocksofsubdiagonalones. 2.10Denition Let t beanilpotenttransformationon V .A t -stringgeneratedby ~v 2 V isasequence h ~v;t ~v ;:::;t k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ~v i .Thissequencehas length k A t -stringbasis isabasisthatisaconcatenationof t -strings. 2.11Example InExample2.5,the t -strings h ~ 1 ; ~ 2 ; ~ 3 i and h ~ 4 ; ~ 5 i ,oflength threeandtwo,canbeconcatenatedtomakeabasisforthedomainof t 2.12Lemma Ifaspacehasa t -stringbasisthenthelongeststringinithas lengthequaltotheindexofnilpotencyof t Proof Supposenot.Thosestringscannotbelonger;iftheindexis k then t k sendsanyvector|includingthosestartingthestring|to ~ 0.Sosuppose insteadthatthereisatransformation t ofindex k onsomespace,suchthat thespacehasa t -stringbasiswhereallofthestringsareshorterthanlength k .Because t hasindex k ,thereisavector ~v suchthat t k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ~v 6 = ~ 0.Represent ~v asalinearcombinationofbasiselementsandapply t k )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 .Wearesupposing that t k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 sendseachbasiselementto ~ 0butthatitdoesnotsend ~v to ~ 0.That isimpossible. QED Weshallshowthateverynilpotentmaphasanassociatedstringbasis.Then ourgoaltheorem,thateverynilpotentmatrixissimilartoonethatisallzeros exceptforblocksofsubdiagonalones,isimmediate,asinExample2.5. Lookingforacounterexample,anilpotentmapwithoutanassociatedstring basisthatisdisjoint,willsuggesttheideafortheproof.Considerthemap t : C 5 C 5 withthisaction. ~e 1 ~e 2 7! 7! ~e 3 7! ~ 0 ~e 4 7! ~e 5 7! ~ 0 Rep E 5 ; E 5 t = 0 B B B B @ 00000 00000 11000 00000 00010 1 C C C C A Evenafterommittingthezerovector,thesethreestringsaren'tdisjoint,but thatdoesn'tendhopeofndinga t -stringbasis.Itonlymeansthat E 5 willnot doforthestringbasis. Tondabasisthatwilldo,werstndthenumberandlengthsofits strings.Since t 'sindexofnilpotencyistwo,Lemma2.12saysthatatleastone

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374 ChapterFive.Similarity stringinthebasishaslengthtwo.Thusthemapmustactonastringbasisin oneofthesetwoways. ~ 1 7! ~ 2 7! ~ 0 ~ 3 7! ~ 4 7! ~ 0 ~ 5 7! ~ 0 ~ 1 7! ~ 2 7! ~ 0 ~ 3 7! ~ 0 ~ 4 7! ~ 0 ~ 5 7! ~ 0 Now,thekeypoint.Atransformationwiththeleft-handactionhasanullspace ofdimensionthreesincethat'showmanybasisvectorsaresenttozero.A transformationwiththeright-handactionhasanullspaceofdimensionfour. Usingthematrixrepresentationabove,calculationof t 'snullspace N t = f 0 B B B B @ x )]TJ/F11 9.9626 Tf 7.748 0 Td [(x z 0 r 1 C C C C A x;z;r 2 C g showsthatitisthree-dimensional,meaningthatwewanttheleft-handaction. Toproduceastringbasis,rstpick ~ 2 and ~ 4 from R t N t ~ 2 = 0 B B B B @ 0 0 1 0 0 1 C C C C A ~ 4 = 0 B B B B @ 0 0 0 0 1 1 C C C C A otherchoicesarepossible,justbesurethat f ~ 2 ; ~ 4 g islinearlyindependent. For ~ 5 pickavectorfrom N t thatisnotinthespanof f ~ 2 ; ~ 4 g ~ 5 = 0 B B B B @ 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 0 0 0 1 C C C C A Finally,take ~ 1 and ~ 3 suchthat t ~ 1 = ~ 2 and t ~ 3 = ~ 4 ~ 1 = 0 B B B B @ 0 1 0 0 0 1 C C C C A ~ 3 = 0 B B B B @ 0 0 0 1 0 1 C C C C A

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SectionIII.Nilpotence 375 Now,withrespectto B = h ~ 1 ;:::; ~ 5 i ,thematrixof t isasdesired. Rep B;B t = 0 B B B B @ 00 00 0 10 00 0 00 00 0 00 10 0 00 00 0 1 C C C C A 2.13Theorem Anynilpotenttransformation t isassociatedwitha t -string basis.Whilethebasisisnotunique,thenumberandthelengthofthestrings isdeterminedby t Thisillustratestheproof.Basisvectorsarecategorizedintokind1,kind2,and kind3.Theyarealsoshownassquaresorcircles,accordingtowhetherthey areinthenullspaceornot. k 3 7! k 1 7!7! k 1 7! 1 7! ~ 0 k 3 7! k 1 7!7! k 1 7! 1 7! ~ 0 k 3 7! k 1 7!7! k 1 7! 1 7! ~ 0 2 7! ~ 0 2 7! ~ 0 Proof Fixavectorspace V ;wewillarguebyinductionontheindexofnilpotencyof t : V V .Ifthatindexis1then t isthezeromapandanybasisis astringbasis ~ 1 7! ~ 0,..., ~ n 7! ~ 0.Fortheinductivestep,assumethatthe theoremholdsforanytransformationwithanindexofnilpotencybetween1 and k )]TJ/F8 9.9626 Tf 9.963 0 Td [(1andconsidertheindex k case. Firstobservethattherestrictiontotherangespace t : R t R t isalso nilpotent,ofindex k )]TJ/F8 9.9626 Tf 10.153 0 Td [(1.Applytheinductivehypothesistogetastringbasis for R t ,wherethenumberandlengthofthestringsisdeterminedby t B = h ~ 1 ;t ~ 1 ;:::;t h 1 ~ 1 i h ~ 2 ;:::;t h 2 ~ 2 i h ~ i ;:::;t h i ~ i i Intheillustrationthesearethebasisvectorsofkind1,sothereare i strings shownwiththiskindofbasisvector. Second,notethattakingthenalnonzerovectorineachstringgivesabasis C = h t h 1 ~ 1 ;:::;t h i ~ i i for R t N t .Theseareillustratedwith1'sin squares.For,amemberof R t ismappedtozeroifandonlyifitisalinear combinationofthosebasisvectorsthataremappedtozero.Extend C toa basisforallof N t ^ C = C h ~ 1 ;:::; ~ p i The ~ 'sarethevectorsofkind2sothat ^ C isthesetofsquares.Whilemany choicesarepossibleforthe ~ 's,theirnumber p isdeterminedbythemap t asit isthedimensionof N t minusthedimensionof R t N t .

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376 ChapterFive.Similarity Finally, B ^ C isabasisfor R t + N t becauseanysumofsomethinginthe rangespacewithsomethinginthenullspacecanberepresentedusingelements of B fortherangespacepartandelementsof ^ C forthepartfromthenullspace. Notethat dim )]TJ/F15 9.9626 Tf 4.566 -8.07 Td [(R t + N t =dim R t +dim N t )]TJ/F8 9.9626 Tf 9.963 0 Td [(dim R t N t =rank t +nullity t )]TJ/F11 9.9626 Tf 9.963 0 Td [(i =dim V )]TJ/F11 9.9626 Tf 9.963 0 Td [(i andso B ^ C canbeextendedtoabasisforallof V bytheadditionof i more vectors.Specically,rememberthateachof ~ 1 ;:::; ~ i isin R t ,andextend B ^ C withvectors ~v 1 ;:::;~v i suchthat t ~v 1 = ~ 1 ;:::;t ~v i = ~ i .Inthe illustration,thesearethe3's.Thecheckthatlinearindependenceispreserved bythisextensionisExercise29. QED 2.14Corollary Everynilpotentmatrixissimilartoamatrixthatisallzeros exceptforblocksofsubdiagonalones.Thatis,everynilpotentmapisrepresentedwithrespecttosomebasisbysuchamatrix. Thisformisuniqueinthesensethatifanilpotentmatrixissimilartotwo suchmatricesthenthosetwosimplyhavetheirblocksordereddierently.Thus thisisacanonicalformforthesimilarityclassesofnilpotentmatricesprovided thatweordertheblocks,say,fromlongesttoshortest. 2.15Example Thematrix M = 1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 hasanindexofnilpotencyoftwo,asthiscalculationshows. p M p N M p 1 M = 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 f x x x 2 C g 2 M 2 = 00 00 C 2 Thecalculationalsodescribeshowamap m representedby M mustactonany stringbasis.Withonemapapplicationthenullspacehasdimensiononeandso onevectorofthebasisissenttozero.Onasecondapplication,thenullspace hasdimensiontwoandsotheotherbasisvectorissenttozero.Thus,theaction ofthemapis ~ 1 7! ~ 2 7! ~ 0andthecanonicalformofthematrixisthis. 00 10 Wecanexhibitsucha m -stringbasisandthechangeofbasismatriceswitnessingthematrixsimilarity.Forthebasis,take M torepresent m withrespect

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SectionIII.Nilpotence 377 tothestandardbases,picka ~ 2 2 N m andalsopicka ~ 1 sothat m ~ 1 = ~ 2 ~ 2 = 1 1 ~ 1 = 1 0 Ifwetake M tobearepresentativewithrespecttosomenonstandardbases thenthispickingstepisjustmoremessy.Recallthesimilaritydiagram. C 2 w.r.t. E 2 m )333()223()222()333(! M C 2 w.r.t. E 2 id ? ? y P id ? ? y P C 2 w.r.t. B m )333()223()222()333(! C 2 w.r.t. B ThecanonicalformequalsRep B;B m = PMP )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ,where P )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 =Rep B; E 2 id= 11 01 P = P )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 1 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 01 andthevericationofthematrixcalculationisroutine. 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 01 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 11 01 = 00 10 2.16Example Thematrix 0 B B B B @ 00000 10000 )]TJ/F8 9.9626 Tf 7.748 0 Td [(111 )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 01000 10 )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 1 C C C C A isnilpotent.Thesecalculationsshowthenullspacesgrowing. p N p N N p 1 0 B B B B @ 00000 10000 )]TJ/F8 9.9626 Tf 7.749 0 Td [(111 )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 01000 10 )]TJ/F8 9.9626 Tf 7.749 0 Td [(11 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 C C C C A f 0 B B B B @ 0 0 u )]TJ/F11 9.9626 Tf 9.963 0 Td [(v u v 1 C C C C A u;v 2 C g 2 0 B B B B @ 00000 00000 10000 10000 00000 1 C C C C A f 0 B B B B @ 0 y z u v 1 C C C C A y;z;u;v 2 C g 3 {zeromatrix{ C 5 Thattableshowsthatanystringbasismustsatisfy:thenullspaceafteronemap applicationhasdimensiontwosotwobasisvectorsaresentdirectlytozero,

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378 ChapterFive.Similarity thenullspaceafterthesecondapplicationhasdimensionfoursotwoadditional basisvectorsaresenttozerobytheseconditeration,andthenullspaceafter threeapplicationsisofdimensionvesothenalbasisvectorissenttozeroin threehops. ~ 1 7! ~ 2 7! ~ 3 7! ~ 0 ~ 4 7! ~ 5 7! ~ 0 Toproducesuchabasis,rstpicktwoindependentvectorsfrom N n ~ 3 = 0 B B B B @ 0 0 1 1 0 1 C C C C A ~ 5 = 0 B B B B @ 0 0 0 1 1 1 C C C C A thenadd ~ 2 ; ~ 4 2 N n 2 suchthat n ~ 2 = ~ 3 and n ~ 4 = ~ 5 ~ 2 = 0 B B B B @ 0 1 0 0 0 1 C C C C A ~ 4 = 0 B B B B @ 0 1 0 1 0 1 C C C C A andnishbyadding ~ 1 2 N n 3 = C 5 suchthat n ~ 1 = ~ 2 ~ 1 = 0 B B B B @ 1 0 1 0 0 1 C C C C A Exercises X 2.17 Whatistheindexofnilpotencyofthe left-shift operator,hereactingonthe spaceoftriplesofreals? x;y;z 7! ;x;y X 2.18 Foreachstringbasisstatetheindexofnilpotencyandgivethedimensionof therangespaceandnullspaceofeachiterationofthenilpotentmap. a ~ 1 7! ~ 2 7! ~ 0 ~ 3 7! ~ 4 7! ~ 0 b ~ 1 7! ~ 2 7! ~ 3 7! ~ 0 ~ 4 7! ~ 0 ~ 5 7! ~ 0 ~ 6 7! ~ 0 c ~ 1 7! ~ 2 7! ~ 3 7! ~ 0 Alsogivethecanonicalformofthematrix. 2.19 Decidewhichofthesematricesarenilpotent.

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SectionIII.Nilpotence 379 a )]TJ/F29 8.9664 Tf 7.167 0 Td [(24 )]TJ/F29 8.9664 Tf 7.167 0 Td [(12 b 31 13 c )]TJ/F29 8.9664 Tf 7.167 0 Td [(321 )]TJ/F29 8.9664 Tf 7.167 0 Td [(321 )]TJ/F29 8.9664 Tf 7.167 0 Td [(321 d 114 30 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 527 e 45 )]TJ/F29 8.9664 Tf 7.168 0 Td [(22 )]TJ/F29 8.9664 Tf 7.167 0 Td [(19 33 )]TJ/F29 8.9664 Tf 7.168 0 Td [(16 )]TJ/F29 8.9664 Tf 7.167 0 Td [(14 69 )]TJ/F29 8.9664 Tf 7.168 0 Td [(34 )]TJ/F29 8.9664 Tf 7.167 0 Td [(29 X 2.20 Findthecanonicalformofthismatrix. 0 B B B @ 01101 00111 00000 00000 00000 1 C C C A X 2.21 ConsiderthematrixfromExample2.16. a Usetheactionofthemaponthestringbasistogivethecanonicalform. b Findthechangeofbasismatricesthatbringthematrixtocanonicalform. c Usetheanswerintheprioritemtochecktheanswerintherstitem. X 2.22 Eachofthesematricesisnilpotent. a 1 = 2 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 = 2 1 = 2 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 = 2 b 000 0 )]TJ/F29 8.9664 Tf 7.168 0 Td [(11 0 )]TJ/F29 8.9664 Tf 7.168 0 Td [(11 c )]TJ/F29 8.9664 Tf 7.168 0 Td [(11 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 101 1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(11 Puteachincanonicalform. 2.23 Describetheeectofleftorrightmultiplicationbyamatrixthatisinthe canonicalformfornilpotentmatrices. 2.24 Isnilpotenceinvariantundersimilarity?Thatis,mustamatrixsimilartoa nilpotentmatrixalsobenilpotent?Ifso,withthesameindex? X 2.25 Showthattheonlyeigenvalueofanilpotentmatrixiszero. 2.26 Isthereanilpotenttransformationofindexthreeonatwo-dimensionalspace? 2.27 IntheproofofTheorem2.13,whyisn'ttheproof'sbasecasethattheindex ofnilpotencyiszero? X 2.28 Let t : V V bealineartransformationandsuppose ~v 2 V issuchthat t k ~v = ~ 0but t k )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 ~v 6 = ~ 0.Considerthe t -string h ~v;t ~v ;:::;t k )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 ~v i a Provethat t isatransformationonthespanofthesetofvectorsinthestring, thatis,provethat t restrictedtothespanhasarangethatisasubsetofthe span.Wesaythatthespanisa t -invariant subspace. b Provethattherestrictionisnilpotent. c Provethatthe t -stringislinearlyindependentandsoisabasisforitsspan. d Representtherestrictionmapwithrespecttothe t -stringbasis. 2.29 FinishtheproofofTheorem2.13. 2.30 Showthattheterms`nilpotenttransformation'and`nilpotentmatrix',as giveninDenition2.6,twitheachother:amapisnilpotentifandonlyifitis representedbyanilpotentmatrix.Isitthatatransformationisnilpotentifan onlyifthereisabasissuchthatthemap'srepresentationwithrespecttothat basisisanilpotentmatrix,orthatanyrepresentationisanilpotentmatrix? 2.31 Let T benilpotentofindexfour.Howbigcantherangespaceof T 3 be? 2.32 Recallthatsimilarmatriceshavethesameeigenvalues.Showthattheconverse doesnothold. 2.33 Proveanilpotentmatrixissimilartoonethatisallzerosexceptforblocksof super-diagonalones.

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380 ChapterFive.Similarity X 2.34 Provethatifatransformationhasthesamerangespaceasnullspace.thenthe dimensionofitsdomainiseven. 2.35 Provethatiftwonilpotentmatricescommutethentheirproductandsumare alsonilpotent. 2.36 Considerthetransformationof M n n givenby t S T = ST )]TJ/F32 8.9664 Tf 9.29 0 Td [(TS where S is an n n matrix.Provethatif S isnilpotentthensois t S 2.37 Showthatif N isnilpotentthen I )]TJ/F32 8.9664 Tf 9.215 0 Td [(N isinvertible.Isthat`onlyif'also?

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SectionIV.JordanForm 381 IVJordanForm Thissectionusesmaterialfromthreeoptionalsubsections:DirectSum,DeterminantsExist,andOtherFormulasfortheDeterminant. Thechapteronlinearmapsshowsthatevery h : V W canberepresented byapartial-identitymatrixwithrespecttosomebases B V and D W Thischapterrevisitsthisissueinthespecialcasethatthemapisalinear transformation t : V V .Ofcourse,thegeneralresultstillappliesbutwith thecodomainanddomainequalwenaturallyaskabouthavingthetwobases alsobeequal.Thatis,wewantacanonicalformtorepresenttransformations asRep B;B t Afterabriefreviewsection,webeganbynotingthatablockpartialidentity formmatrixisnotalwaysobtainableinthis B;B case.Wethereforeconsidered thenaturalgeneralization,diagonalmatrices,andshowedthatifitseigenvalues aredistinctthenamapormatrixcanbediagonalized.Butwealsogavean exampleofamatrixthatcannotbediagonalizedandinthesectionpriortothis onewedevelopedthatexample.Weshowedthatalinearmapisnilpotent| ifwetakehigherandhigherpowersofthemapormatrixthenweeventually getthezeromapormatrix|ifandonlyifthereisabasisonwhichitactsvia disjointstrings.Thatledtoacanonicalformfornilpotentmatrices. Now,thissectionconcludesthechapter.Wewillshowthatthetwocases we'vestudiedareexhaustiveinthatforanylineartransformationthereisa basissuchthatthematrixrepresentationRep B;B t isthesumofadiagonal matrixandanilpotentmatrixinitscanonicalform. IV.1PolynomialsofMapsandMatrices Recallthatthesetofsquarematricesisavectorspaceunderentry-by-entry additionandscalarmultiplicationandthatthisspace M n n hasdimension n 2 Thus,forany n n matrix T the n 2 +1-memberset f I;T;T 2 ;:::;T n 2 g islinearly dependentandsotherearescalars c 0 ;:::;c n 2 suchthat c n 2 T n 2 + + c 1 T + c 0 I isthezeromatrix. 1.1Remark Thisobservationissmallbutimportant.Itsaysthatevery transformationexhibitsageneralizednilpotency:thepowersofasquarematrix cannotclimbforeverwithoutarepeat". 1.2Example Rotationofplanevectors = 6radianscounterclockwiseisrepresentedwithrespecttothestandardbasisby T = p 3 = 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 1 = 2 p 3 = 2 andverifyingthat0 T 4 +0 T 3 +1 T 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 T )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 I equalsthezeromatrixiseasy.

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382 ChapterFive.Similarity 1.3Denition Foranypolynomial f x = c n x n + + c 1 x + c 0 ,where t isa lineartransformationthen f t isthetransformation c n t n + + c 1 t + c 0 id onthesamespaceandwhere T isasquarematrixthen f T isthematrix c n T n + + c 1 T + c 0 I 1.4Remark If,forinstance, f x = x )]TJ/F8 9.9626 Tf 10.486 0 Td [(3,thenmostauthorswriteinthe identitymatrix: f T = T )]TJ/F8 9.9626 Tf 10.235 0 Td [(3 I .Butmostauthorsdon'twriteintheidentity map: f t = t )]TJ/F8 9.9626 Tf 9.963 0 Td [(3.Inthisbookweshallalsoobservethisconvention. Ofcourse,if T =Rep B;B t then f T =Rep B;B f t ,whichfollowsfrom therelationships T j =Rep B;B t j ,and cT =Rep B;B ct ,and T 1 + T 2 = Rep B;B t 1 + t 2 AsExample1.2shows,theremaybepolynomialsofdegreesmallerthan n 2 thatzerothemapormatrix. 1.5Denition The minimalpolynomial m x ofatransformation t ora squarematrix T isthepolynomialofleastdegreeandwithleadingcoecient 1suchthat m t isthezeromapor m T isthezeromatrix. Aminimalpolynomialalwaysexistsbytheobservationopeningthissubsection.Aminimalpolynomialisuniquebythe`withleadingcoecient1'clause. Thisisbecauseiftherearetwopolynomials m x and^ m x thatarebothofthe minimaldegreetomakethemapormatrixzeroandthusareofequaldegree, andbothhaveleading1's,thentheirdierence m x )]TJ/F8 9.9626 Tf 11.896 0 Td [(^ m x hasasmallerdegreethaneitherandstillsendsthemapormatrixtozero.Thus m x )]TJ/F8 9.9626 Tf 11.02 0 Td [(^ m x is thezeropolynomialandthetwoareequal.Theleadingcoecientrequirement alsopreventsaminimalpolynomialfrombeingthezeropolynomial. 1.6Example Wecanseethat m x = x 2 )]TJ/F8 9.9626 Tf 10.051 0 Td [(2 x )]TJ/F8 9.9626 Tf 10.051 0 Td [(1isminimalforthematrix ofExample1.2bycomputingthepowersof T uptothepower n 2 =4. T 2 = 1 = 2 )]TJ 7.748 8.241 Td [(p 3 = 2 p 3 = 21 = 2 T 3 = 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 10 T 4 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 )]TJ 7.749 8.241 Td [(p 3 = 2 p 3 = 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 2 Next,put c 4 T 4 + c 3 T 3 + c 2 T 2 + c 1 T + c 0 I equaltothezeromatrix )]TJ/F8 9.9626 Tf 7.748 0 Td [( = 2 c 4 + = 2 c 2 + p 3 = 2 c 1 + c 0 =0 )]TJ/F8 9.9626 Tf 7.749 0 Td [( p 3 = 2 c 4 )]TJ/F11 9.9626 Tf 9.963 0 Td [(c 3 )]TJ/F8 9.9626 Tf 9.963 0 Td [( p 3 = 2 c 2 )]TJ/F8 9.9626 Tf 18.265 0 Td [( = 2 c 1 =0 p 3 = 2 c 4 + c 3 + p 3 = 2 c 2 + = 2 c 1 =0 )]TJ/F8 9.9626 Tf 7.748 0 Td [( = 2 c 4 + = 2 c 2 + p 3 = 2 c 1 + c 0 =0 anduseGauss'method. c 4 )]TJ/F11 9.9626 Tf 23.246 0 Td [(c 2 )]TJ 9.963 8.242 Td [(p 3 c 1 )]TJ/F8 9.9626 Tf 18.264 0 Td [(2 c 0 =0 c 3 + p 3 c 2 +2 c 1 + p 3 c 0 =0 Setting c 4 c 3 ,and c 2 tozeroforces c 1 and c 0 toalsocomeoutaszero.Toget aleadingone,themostwecandoistoset c 4 and c 3 tozero.Thustheminimal polynomialisquadratic.

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SectionIV.JordanForm 383 Usingthemethodofthatexampletondtheminimalpolynomialofa3 3 matrixwouldmeandoingGaussianreductiononasystemwithnineequations intenunknowns.Weshalldevelopanalternative.Tobegin,notethatwecan breakapolynomialofamaporamatrixintoitscomponents. 1.7Lemma Supposethatthepolynomial f x = c n x n + + c 1 x + c 0 factors as k x )]TJ/F11 9.9626 Tf 10.212 0 Td [( 1 q 1 x )]TJ/F11 9.9626 Tf 10.212 0 Td [( ` q ` .If t isalineartransformationthenthesetwoare equalmaps. c n t n + + c 1 t + c 0 = k t )]TJ/F11 9.9626 Tf 9.963 0 Td [( 1 q 1 t )]TJ/F11 9.9626 Tf 9.962 0 Td [( ` q ` Consequently,if T isasquarematrixthen f T and k T )]TJ/F11 9.9626 Tf 8.51 0 Td [( 1 I q 1 T )]TJ/F11 9.9626 Tf 8.51 0 Td [( ` I q ` areequalmatrices. Proof Thisargumentisbyinductiononthedegreeofthepolynomial.The caseswherethepolynomialisofdegree0and1areclear.Thefullinduction argumentisExercise1.7butthedegreetwocasegivesitssense. Aquadraticpolynomialfactorsintotwolinearterms f x = k x )]TJ/F11 9.9626 Tf 9.316 0 Td [( 1 x )]TJ/F11 9.9626 Tf -335.963 -11.955 Td [( 2 = k x 2 + 1 + 2 x + 1 2 theroots 1 and 2 mightbeequal.Wecan checkthatsubstituting t for x inthefactoredandunfactoredversionsgivesthe samemap. )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(k t )]TJ/F11 9.9626 Tf 9.963 0 Td [( 1 t )]TJ/F11 9.9626 Tf 9.963 0 Td [( 2 ~v = )]TJ/F11 9.9626 Tf 4.566 -8.07 Td [(k t )]TJ/F11 9.9626 Tf 9.963 0 Td [( 1 t ~v )]TJ/F11 9.9626 Tf 9.963 0 Td [( 2 ~v = k )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(t t ~v )]TJ/F11 9.9626 Tf 9.962 0 Td [(t 2 ~v )]TJ/F11 9.9626 Tf 9.963 0 Td [( 1 t ~v )]TJ/F11 9.9626 Tf 9.963 0 Td [( 1 2 ~v = k )]TJ/F11 9.9626 Tf 4.567 -8.07 Td [(t t ~v )]TJ/F8 9.9626 Tf 9.963 0 Td [( 1 + 2 t ~v + 1 2 ~v = k t 2 )]TJ/F8 9.9626 Tf 9.962 0 Td [( 1 + 2 t + 1 2 ~v Thethirdequalityholdsbecausethescalar 2 comesoutofthesecondterm,as t islinear. QED Inparticular,ifaminimialpolynomial m x foratransformation t factors as m x = x )]TJ/F11 9.9626 Tf 10.382 0 Td [( 1 q 1 x )]TJ/F11 9.9626 Tf 10.383 0 Td [( ` q ` then m t = t )]TJ/F11 9.9626 Tf 10.382 0 Td [( 1 q 1 t )]TJ/F11 9.9626 Tf 10.382 0 Td [( ` q ` is thezeromap.Since m t sendseveryvectortozero,atleastoneofthemaps t )]TJ/F11 9.9626 Tf 9.825 0 Td [( i sendssomenonzerovectorstozero.So,too,inthematrixcase|if m is minimalfor T then m T = T )]TJ/F11 9.9626 Tf 9.097 0 Td [( 1 I q 1 T )]TJ/F11 9.9626 Tf 9.097 0 Td [( ` I q ` isthezeromatrixandat leastoneofthematrices T )]TJ/F11 9.9626 Tf 8.739 0 Td [( i I sendssomenonzerovectorstozero.Rewording bothcases:atleastsomeofthe i areeigenvalues.SeeExercise29. Recallhowwehaveearlierfoundeigenvalues.Wehavelookedfor suchthat T~v = ~v byconsideringtheequation ~ 0= T~v )]TJ/F11 9.9626 Tf 7.959 0 Td [(x~v = T )]TJ/F11 9.9626 Tf 7.959 0 Td [(xI ~v andcomputingthe determinantofthematrix T )]TJ/F11 9.9626 Tf 10.006 0 Td [(xI .Thatdeterminantisapolynomialin x ,the characteristicpolynomial,whoserootsaretheeigenvalues.Themajorresult ofthissubsection,thenextresult,isthatthereisaconnectionbetweenthis characteristicpolynomialandtheminimalpolynomial.Thisresultsexpands onthepriorparagraph'sinsightthatsomerootsoftheminimalpolynomial areeigenvaluesbyassertingthateveryrootoftheminimalpolynomialisan eigenvalueandfurtherthateveryeigenvalueisarootoftheminimalpolynomial thisisbecauseitsays`1 q i 'andnotjust`0 q i '.

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384 ChapterFive.Similarity 1.8TheoremCayley-Hamilton Ifthecharacteristicpolynomialofa transformationorsquarematrixfactorsinto k x )]TJ/F11 9.9626 Tf 9.963 0 Td [( 1 p 1 x )]TJ/F11 9.9626 Tf 9.962 0 Td [( 2 p 2 x )]TJ/F11 9.9626 Tf 9.962 0 Td [( ` p ` thenitsminimalpolynomialfactorsinto x )]TJ/F11 9.9626 Tf 9.962 0 Td [( 1 q 1 x )]TJ/F11 9.9626 Tf 9.962 0 Td [( 2 q 2 x )]TJ/F11 9.9626 Tf 9.963 0 Td [( ` q ` where1 q i p i foreach i between1and ` Theprooftakesupthenextthreelemmas.Althoughtheyarestatedonlyin matrixterms,theyapplyequallywelltomaps.Wegivethematrixversiononly becauseitisconvenientfortherstproof. Therstresultisthekey|someauthorscallittheCayley-HamiltonTheoremandcallTheorem1.8aboveacorollary.Fortheproof,observethatamatrix ofpolynomialscanbethoughtofasapolynomialwithmatrixcoecients. 2 x 2 +3 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 x 2 +2 3 x 2 +4 x +14 x 2 + x +1 = 21 34 x 2 + 30 41 x + )]TJ/F8 9.9626 Tf 7.749 0 Td [(12 11 1.9Lemma If T isasquarematrixwithcharacteristicpolynomial c x then c T isthezeromatrix. Proof Let C be T )]TJ/F11 9.9626 Tf 10.449 0 Td [(xI ,thematrixwhosedeterminantisthecharacteristic polynomial c x = c n x n + + c 1 x + c 0 C = 0 B B B @ t 1 ; 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(xt 1 ; 2 ::: t 2 ; 1 t 2 ; 2 )]TJ/F11 9.9626 Tf 9.962 0 Td [(x t n;n )]TJ/F11 9.9626 Tf 9.962 0 Td [(x 1 C C C A Recallthattheproductoftheadjointofamatrixwiththematrixitselfisthe determinantofthatmatrixtimestheidentity. c x I =adj C C =adj C T )]TJ/F11 9.9626 Tf 9.962 0 Td [(xI =adj C T )]TJ/F8 9.9626 Tf 9.962 0 Td [(adj C x Theentriesofadj C arepolynomials,eachofdegreeatmost n )]TJ/F8 9.9626 Tf 10.266 0 Td [(1sincethe minorsofamatrixdroparowandcolumn.Rewriteit,assuggestedabove,as adj C = C n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 + + C 1 x + C 0 whereeach C i isamatrixofscalars.The leftandrightendsofequation abovegivethis. c n Ix n + c n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 Ix n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 + + c 1 Ix + c 0 I = C n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 T x n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 + + C 1 T x + C 0 T )]TJ/F11 9.9626 Tf 9.963 0 Td [(C n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 x n )]TJ/F11 9.9626 Tf 9.962 0 Td [(C n )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 x n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 )-222()]TJ/F11 9.9626 Tf 33.762 0 Td [(C 0 x

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SectionIV.JordanForm 385 Equatethecoecientsof x n ,thecoecientsof x n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ,etc. c n I = )]TJ/F11 9.9626 Tf 7.749 0 Td [(C n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 c n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 I = )]TJ/F11 9.9626 Tf 7.749 0 Td [(C n )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 + C n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 T c 1 I = )]TJ/F11 9.9626 Tf 7.749 0 Td [(C 0 + C 1 T c 0 I = C 0 T Multiplyfromtherightbothsidesoftherstequationby T n ,bothsides ofthesecondequationby T n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ,etc.Add.Theresultontheleftis c n T n + c n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 T n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 + + c 0 I ,andtheresultontherightisthezeromatrix. QED Wesometimesrefertothatlemmabysayingthatamatrixormap satises itscharacteristicpolynomial. 1.10Lemma Where f x isapolynomial,if f T isthezeromatrixthen f x isdivisiblebytheminimalpolynomialof T .Thatis,anypolynomialsatised by T isdivisableby T 'sminimalpolynomial. Proof Let m x beminimalfor T .TheDivisionTheoremforPolynomials gives f x = q x m x + r x wherethedegreeof r isstrictlylessthanthe degreeof m .Plugging T inshowsthat r T isthezeromatrix,because T satisesboth f and m .Thatcontradictstheminimalityof m unless r isthe zeropolynomial. QED Combiningthepriortwolemmasgivesthattheminimalpolynomialdivides thecharacteristicpolynomial.Thus,anyrootoftheminimalpolynomialis alsoarootofthecharacteristicpolynomial.Thatis,sofarwehavethatif m x = x )]TJ/F11 9.9626 Tf 9.79 0 Td [( 1 q 1 ::: x )]TJ/F11 9.9626 Tf 9.79 0 Td [( i q i then c x musthastheform x )]TJ/F11 9.9626 Tf 9.79 0 Td [( 1 p 1 ::: x )]TJ/F11 9.9626 Tf -335.962 -11.955 Td [( i p i x )]TJ/F11 9.9626 Tf 10.012 0 Td [( i +1 p i +1 ::: x )]TJ/F11 9.9626 Tf 10.012 0 Td [( ` p ` whereeach q j islessthanorequalto p j .The proofoftheCayley-HamiltonTheoremisnishedbyshowingthatinfactthe characteristicpolynomialhasnoextraroots i +1 ,etc. 1.11Lemma Eachlinearfactorofthecharacteristicpolynomialofasquare matrixisalsoalinearfactoroftheminimalpolynomial. Proof Let T beasquarematrixwithminimalpolynomial m x andassume that x )]TJ/F11 9.9626 Tf 9.431 0 Td [( isafactorofthecharacteristicpolynomialof T ,thatis,assumethat isaneigenvalueof T .Wemustshowthat x )]TJ/F11 9.9626 Tf 9.637 0 Td [( isafactorof m ,thatis,that m =0. Ingeneral,where isassociatedwiththeeigenvector ~v ,foranypolynomialfunction f x ,applicationofthematrix f T to ~v equalstheresultof multiplying ~v bythescalar f .Forinstance,if T haseigenvalue associatedwiththeeigenvector ~v and f x = x 2 +2 x +3then T 2 +2 T +3 ~v = T 2 ~v +2 T ~v +3 ~v = 2 ~v +2 ~v +3 ~v = 2 +2 +3 ~v .Now,as m T is thezeromatrix, ~ 0= m T ~v = m ~v andtherefore m =0. QED

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386 ChapterFive.Similarity 1.12Example WecanusetheCayley-HamiltonTheoremtohelpndthe minimalpolynomialofthismatrix. T = 0 B B @ 2001 1202 002 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 0001 1 C C A First,itscharacteristicpolynomial c x = x )]TJ/F8 9.9626 Tf 9.21 0 Td [(1 x )]TJ/F8 9.9626 Tf 9.209 0 Td [(2 3 canbefoundwiththe usualdeterminant.Now,theCayley-HamiltonTheoremsaysthat T 'sminimal polynomialiseither x )]TJ/F8 9.9626 Tf 9.411 0 Td [(1 x )]TJ/F8 9.9626 Tf 9.41 0 Td [(2or x )]TJ/F8 9.9626 Tf 9.411 0 Td [(1 x )]TJ/F8 9.9626 Tf 9.41 0 Td [(2 2 or x )]TJ/F8 9.9626 Tf 9.411 0 Td [(1 x )]TJ/F8 9.9626 Tf 9.41 0 Td [(2 3 .Wecan decideamongthechoicesjustbycomputing: T )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 I T )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 I = 0 B B @ 1001 1102 001 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 0000 1 C C A 0 B B @ 0001 1002 000 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 000 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 1 C C A = 0 B B @ 0000 1001 0000 0000 1 C C A and T )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 I T )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 I 2 = 0 B B @ 0000 1001 0000 0000 1 C C A 0 B B @ 0001 1002 000 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 000 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 1 C C A = 0 B B @ 0000 0000 0000 0000 1 C C A andso m x = x )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 2 Exercises X 1.13 Whatarethepossibleminimalpolynomialsifamatrixhasthegivencharacteristicpolynomial? a 8 x )]TJ/F29 8.9664 Tf 9.439 0 Td [(3 4 b = 3 x +1 3 x )]TJ/F29 8.9664 Tf 9.44 0 Td [(4 c )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 x )]TJ/F29 8.9664 Tf 9.44 0 Td [(2 2 x )]TJ/F29 8.9664 Tf 9.44 0 Td [(5 2 d 5 x +3 2 x )]TJ/F29 8.9664 Tf 9.215 0 Td [(1 x )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 2 Whatisthedegreeofeachpossibility? X 1.14 Findtheminimalpolynomialofeachmatrix. a 300 130 004 b 300 130 003 c 300 130 013 d 201 062 002 e 221 062 002 f 0 B B B @ )]TJ/F29 8.9664 Tf 7.167 0 Td [(14000 03000 0 )]TJ/F29 8.9664 Tf 7.167 0 Td [(4 )]TJ/F29 8.9664 Tf 7.167 0 Td [(100 3 )]TJ/F29 8.9664 Tf 7.167 0 Td [(9 )]TJ/F29 8.9664 Tf 7.167 0 Td [(42 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 15414 1 C C C A 1.15 Findtheminimalpolynomialofthismatrix. 010 001 100 X 1.16 Whatistheminimalpolynomialofthedierentiationoperator d=dx on P n ?

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SectionIV.JordanForm 387 X 1.17 Findtheminimalpolynomialofmatricesofthisform 0 B B B B B B @ 00 ::: 0 1 00 01 0 00 ::: 1 1 C C C C C C A wherethescalar isxedi.e.,isnotavariable. 1.18 Whatistheminimalpolynomialofthetransformationof P n thatsends p x to p x +1? 1.19 Whatistheminimalpolynomialofthemap : C 3 C 3 projectingontothe rsttwocoordinates? 1.20 Finda3 3matrixwhoseminimalpolynomialis x 2 1.21 WhatiswrongwiththisclaimedproofofLemma1.9:if c x = j T )]TJ/F32 8.9664 Tf 8.28 0 Td [(xI j then c T = j T )]TJ/F32 8.9664 Tf 9.215 0 Td [(TI j =0"?[Cullen] 1.22 VerifyLemma1.9for2 2matricesbydirectcalculation. X 1.23 Provethattheminimalpolynomialofan n n matrixhasdegreeatmost n not n 2 asmightbeguessedfromthissubsection'sopening.Verifythatthis maximum, n ,canhappen. X 1.24 Theonlyeigenvalueofanilpotentmapiszero.Showthattheconversestatementholds. 1.25 Whatistheminimalpolynomialofazeromapormatrix?Ofanidentitymap ormatrix? X 1.26 InterprettheminimalpolynomialofExample1.2geometrically. 1.27 Whatistheminimalpolynomialofadiagonalmatrix? X 1.28 A projection isanytransformation t suchthat t 2 = t .Forinstance,the transformationoftheplane R 2 projectingeachvectorontoitsrstcoordinatewill, ifdonetwice,resultinthesamevalueasifitisdonejustonce.Whatisthe minimalpolynomialofaprojection? 1.29 Thersttwoitemsofthisquestionarereview. a Provethatthecompositionofone-to-onemapsisone-to-one. b Provethatifalinearmapisnotone-to-onethenatleastonenonzerovector fromthedomainissenttothezerovectorinthecodomain. c Verifythestatement,excerptedhere,thatpreceedsTheorem1.8. ...ifaminimialpolynomial m x foratransformation t factorsas m x = x )]TJ/F32 8.9664 Tf 8.782 0 Td [( 1 q 1 x )]TJ/F32 8.9664 Tf 8.782 0 Td [( ` q ` then m t = t )]TJ/F32 8.9664 Tf 8.782 0 Td [( 1 q 1 t )]TJ/F32 8.9664 Tf 8.782 0 Td [( ` q ` isthezeromap.Since m t sendseveryvectortozero,atleastone ofthemaps t )]TJ/F32 8.9664 Tf 9.349 0 Td [( i sendssomenonzerovectorstozero....Rewording ...:atleastsomeofthe i areeigenvalues. 1.30 Trueorfalse:foratransformationonan n dimensionalspace,iftheminimal polynomialhasdegree n thenthemapisdiagonalizable. 1.31 Let f x beapolynomial.Provethatif A and B aresimilarmatricesthen f A issimilarto f B a Nowshowthatsimilarmatriceshavethesamecharacteristicpolynomial. b Showthatsimilarmatriceshavethesameminimalpolynomial.

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388 ChapterFive.Similarity c Decideifthesearesimilar. 13 23 4 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 11 1.32a Showthatamatrixisinvertibleifandonlyiftheconstantterminits minimalpolynomialisnot0. b Showthatifasquarematrix T isnotinvertiblethenthereisanonzero matrix S suchthat ST and TS bothequalthezeromatrix. X 1.33a FinishtheproofofLemma1.7. b Giveanexampletoshowthattheresultdoesnotholdif t isnotlinear. 1.34 Anytransformationorsquarematrixhasaminimalpolynomial.Doesthe conversehold? IV.2JordanCanonicalForm Thissubsectionmovesfromthecanonicalformfornilpotentmatricestothe oneforallmatrices. Wehaveshownthatifamapisnilpotentthenallofitseigenvaluesarezero. Wecannowprovetheconverse. 2.1Lemma Alineartransformationwhoseonlyeigenvalueiszeroisnilpotent. Proof Ifatransformation t onan n -dimensionalspacehasonlythesingle eigenvalueofzerothenitscharacteristicpolynomialis x n .TheCayley-Hamilton Theoremsaysthatamapsatisesitscharacteristicpolynimialso t n isthezero map.Thus t isnilpotent. QED Wehaveacanonicalformfornilpotentmatrices,thatis,foreachmatrix whosesingleeigenvalueiszero:eachsuchmatrixissimilartoonethatisall zeroesexceptforblocksofsubdiagonalones.Tomakethisrepresentation uniquewecanxsomearrangementoftheblocks,say,fromlongesttoshortest. Wenextextendthistoallsingle-eigenvaluematrices. Observethatif t 'sonlyeigenvalueis then t )]TJ/F11 9.9626 Tf 10.836 0 Td [( 'sonlyeigenvalueis0 because t ~v = ~v ifandonlyif t )]TJ/F11 9.9626 Tf 10.101 0 Td [( ~v =0 ~v .Thenaturalwaytoextend theresultsfornilpotentmatricesistorepresent t )]TJ/F11 9.9626 Tf 9.85 0 Td [( inthecanonicalform N andtrytousethattogetasimplerepresentation T for t .Thenextresultsays thatthistryworks. 2.2Lemma Ifthematrices T )]TJ/F11 9.9626 Tf 9.779 0 Td [(I and N aresimilarthen T and N + I are alsosimilar,viathesamechangeofbasismatrices. Proof With N = P T )]TJ/F11 9.9626 Tf 11.327 0 Td [(I P )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = PTP )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F11 9.9626 Tf 11.327 0 Td [(P I P )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 wehave N = PTP )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F11 9.9626 Tf 10.761 0 Td [(PP )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 I sincethediagonalmatrix I commuteswithanything, andso N = PTP )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(I .Therefore N + I = PTP )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 ,asrequired. QED

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SectionIV.JordanForm 389 2.3Example Thecharacteristicpolynomialof T = 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 14 is x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 2 andso T hasonlythesingleeigenvalue3.Thusfor T )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 I = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 11 theonlyeigenvalueis0,and T )]TJ/F8 9.9626 Tf 10.322 0 Td [(3 I isnilpotent.Thenullspacesareroutine tond;toeasethiscomputationwetake T torepresentthetransformation t : C 2 C 2 withrespecttothestandardbasisweshallmaintainthisconvention fortherestofthechapter. N t )]TJ/F8 9.9626 Tf 9.963 0 Td [(3= f )]TJ/F11 9.9626 Tf 7.749 0 Td [(y y y 2 C g N t )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 2 = C 2 Thedimensionsofthesenullspacesshowthattheactionofanassociatedmap t )]TJ/F8 9.9626 Tf 9.968 0 Td [(3onastringbasisis ~ 1 7! ~ 2 7! ~ 0.Thus,thecanonicalformfor t )]TJ/F8 9.9626 Tf 9.968 0 Td [(3with onechoiceforastringbasisis Rep B;B t )]TJ/F8 9.9626 Tf 9.962 0 Td [(3= N = 00 10 B = h 1 1 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(2 2 i andbyLemma2.2, T issimilartothismatrix. Rep t B;B = N +3 I = 30 13 Wecanproducethesimilaritycomputation.RecallfromtheNilpotence sectionhowtondthechangeofbasismatrices P and P )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 toexpress N as P T )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 I P )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 .Thesimilaritydiagram C 2 w.r.t. E 2 t )]TJ/F7 6.9738 Tf 6.226 0 Td [(3 )333()223()222()333(! T )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 I C 2 w.r.t. E 2 id ? ? y P id ? ? y P C 2 w.r.t. B t )]TJ/F7 6.9738 Tf 6.227 0 Td [(3 )333()223()222()333(! N C 2 w.r.t. B describesthattomovefromthelowerlefttotheupperleftwemultiplyby P )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = )]TJ/F8 9.9626 Tf 4.566 -8.07 Td [(Rep E 2 ;B id )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 =Rep B; E 2 id= 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 12 andtomovefromtheupperrighttothelowerrightwemultiplybythismatrix. P = 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 12 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 = 1 = 21 = 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = 41 = 4

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390 ChapterFive.Similarity Sothesimilarityisexpressedby 30 13 = 1 = 21 = 2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 = 41 = 4 2 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 14 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 12 whichiseasilychecked. 2.4Example Thismatrixhascharacteristicpolynomial x )]TJ/F8 9.9626 Tf 9.962 0 Td [(4 4 T = 0 B B @ 410 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 0301 0040 1005 1 C C A andsohasthesingleeigenvalue4.Thenullitiesof t )]TJ/F8 9.9626 Tf 9.998 0 Td [(4are:thenullspaceof t )]TJ/F8 9.9626 Tf 9.553 0 Td [(4hasdimensiontwo,thenullspaceof t )]TJ/F8 9.9626 Tf 9.553 0 Td [(4 2 hasdimensionthree,andthe nullspaceof t )]TJ/F8 9.9626 Tf 9.608 0 Td [(4 3 hasdimensionfour.Thus, t )]TJ/F8 9.9626 Tf 9.608 0 Td [(4hastheactiononastring basisof ~ 1 7! ~ 2 7! ~ 3 7! ~ 0and ~ 4 7! ~ 0.Thisgivesthecanonicalform N for t )]TJ/F8 9.9626 Tf 9.963 0 Td [(4,whichinturngivestheformfor t N +4 I = 0 B B @ 4000 1400 0140 0004 1 C C A Anarraythatisallzeroes,exceptforsomenumber downthediagonal andblocksofsubdiagonalones,isa Jordanblock .WehaveshownthatJordan blockmatricesarecanonicalrepresentativesofthesimilarityclassesofsingleeigenvaluematrices. 2.5Example The3 3matriceswhoseonlyeigenvalueis1 = 2separateinto threesimilarityclasses.Thethreeclasseshavethesecanonicalrepresentatives. 0 @ 1 = 200 01 = 20 001 = 2 1 A 0 @ 1 = 200 11 = 20 001 = 2 1 A 0 @ 1 = 200 11 = 20 011 = 2 1 A Inparticular,thismatrix 0 @ 1 = 200 01 = 20 011 = 2 1 A belongstothesimilarityclassrepresentedbythemiddleone,becausewehave adoptedtheconventionoforderingtheblocksofsubdiagonalonesfromthe longestblocktotheshortest. Wewillnownishtheprogramofthischapterbyextendingthisworkto covermapsandmatriceswithmultipleeigenvalues.Thebestpossibilityfor generalmapsandmatriceswouldbeifwecouldbreakthemintoapartinvolving

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SectionIV.JordanForm 391 theirrsteigenvalue 1 whichwerepresentusingitsJordanblock,apartwith 2 ,etc. Thisidealisinfactwhathappens.Foranytransformation t : V V ,we shallbreakthespace V intothedirectsumofapartonwhich t )]TJ/F11 9.9626 Tf 8.572 0 Td [( 1 isnilpotent, plusapartonwhich t )]TJ/F11 9.9626 Tf 9.428 0 Td [( 2 isnilpotent,etc.Moreprecisely,weshalltakethree stepstogettothissection'smajortheoremandthethirdstepshowsthat V = N 1 t )]TJ/F11 9.9626 Tf 9.963 0 Td [( 1 N 1 t )]TJ/F11 9.9626 Tf 9.962 0 Td [( ` where 1 ;:::; ` are t 'seigenvalues. Supposethat t : V V isalineartransformation.Notethattherestriction of t toasubspace M neednotbealineartransformationon M becausetheremay bean ~m 2 M with t ~m 62 M .Toensurethattherestrictionofatransformation toa`part'ofaspaceisatransformationonthepartweneedthenextcondition. 2.6Denition Let t : V V beatransformation.Asubspace M is t invariant ifwhenever ~m 2 M then t ~m 2 M shorter: t M M Twoexamplesarethatthegeneralizednullspace N 1 t andthegeneralized rangespace R 1 t ofanytransformation t areinvariant.Forthegeneralizednull space,if ~v 2 N 1 t then t n ~v = ~ 0where n isthedimensionoftheunderlying spaceandso t ~v 2 N 1 t because t n t ~v iszeroalso.Forthegeneralized rangespace,if ~v 2 R 1 t then ~v = t n ~w forsome ~w andthen t ~v = t n +1 ~w = t n t ~w showsthat t ~v isalsoamemberof R 1 t Thusthespaces N 1 t )]TJ/F11 9.9626 Tf 10.171 0 Td [( i and R 1 t )]TJ/F11 9.9626 Tf 10.17 0 Td [( i are t )]TJ/F11 9.9626 Tf 10.171 0 Td [( i invariant.Observe alsothat t )]TJ/F11 9.9626 Tf 9.053 0 Td [( i isnilpotenton N 1 t )]TJ/F11 9.9626 Tf 9.053 0 Td [( i because,simply,if ~v hastheproperty thatsomepowerof t )]TJ/F11 9.9626 Tf 10.318 0 Td [( i mapsittozero|thatis,ifitisinthegeneralized nullspace|thensomepowerof t )]TJ/F11 9.9626 Tf 10.385 0 Td [( i mapsittozero.Thegeneralizednull space N 1 t )]TJ/F11 9.9626 Tf 10.171 0 Td [( i isa`part'ofthespaceonwhichtheactionof t )]TJ/F11 9.9626 Tf 10.171 0 Td [( i iseasy tounderstand. Thenextresultistherstofourthreesteps.Itestablishesthat t )]TJ/F11 9.9626 Tf 8.897 0 Td [( j leaves t )]TJ/F11 9.9626 Tf 9.962 0 Td [( i 'spartunchanged. 2.7Lemma Asubspaceis t invariantifandonlyifitis t )]TJ/F11 9.9626 Tf 10.444 0 Td [( invariantfor anyscalar .Inparticular,where i isaneigenvalueofalineartransformation t ,thenforanyothereigenvalue j ,thespaces N 1 t )]TJ/F11 9.9626 Tf 10 0 Td [( i and R 1 t )]TJ/F11 9.9626 Tf 10.001 0 Td [( i are t )]TJ/F11 9.9626 Tf 9.962 0 Td [( j invariant. Proof Fortherstsentencewecheckthetwoimplicationsofthe`ifandonly if'separately.Oneofthemiseasy:ifthesubspaceis t )]TJ/F11 9.9626 Tf 10.046 0 Td [( invariantforany thentaking =0showsthatitis t invariant.Fortheotherimplicationsuppose thatthesubspaceis t invariant,sothatif ~m 2 M then t ~m 2 M ,andlet beanyscalar.Thesubspace M isclosedunderlinearcombinationsandsoif t ~m 2 M then t ~m )]TJ/F11 9.9626 Tf 10.477 0 Td [(~m 2 M .Thusif ~m 2 M then t )]TJ/F11 9.9626 Tf 10.477 0 Td [( ~m 2 M ,as required. Thesecondsentencefollowsstraightfromtherst.Becausethetwospaces are t )]TJ/F11 9.9626 Tf 9.604 0 Td [( i invariant,theyaretherefore t invariant.Fromthis,applyingtherst sentenceagain,weconcludethattheyarealso t )]TJ/F11 9.9626 Tf 9.962 0 Td [( j invariant. QED Moreinformationonrestrictionsoffunctionsisintheappendix.

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392 ChapterFive.Similarity Thesecondstepofthethreethatwewilltaketoprovethissection'smajor resultmakesuseofanadditionalpropertyof N 1 t )]TJ/F11 9.9626 Tf 9.729 0 Td [( i and R 1 t )]TJ/F11 9.9626 Tf 9.729 0 Td [( i ,that theyarecomplementary.Recallthatifaspaceisthedirectsumoftwoothers V = N R thenanyvector ~v inthespacebreaksintotwoparts ~v = ~n + ~r where ~n 2 N and ~r 2 R ,andrecallalsothatif B N and B R arebasesfor N and R thentheconcatenation B N B R islinearlyindependentandsothetwo partsof ~v donotoverlap".Thenextresultsaysthatforanysubspaces N and R thatarecomplementaryaswellas t invariant,theactionof t on ~v breaks intothenon-overlapping"actionsof t on ~n andon ~r 2.8Lemma Let t : V V beatransformationandlet N and R be t invariant complementarysubspacesof V .Then t canberepresentedbyamatrixwith blocksofsquaresubmatrices T 1 and T 2 T 1 Z 2 Z 1 T 2 g dim N -manyrows g dim R -manyrows where Z 1 and Z 2 areblocksofzeroes. Proof Sincethetwosubspacesarecomplementary,theconcatenationofabasis for N andabasisfor R makesabasis B = h ~ 1 ;:::;~ p ;~ 1 ;:::;~ q i for V .We shallshowthatthematrix Rep B;B t = 0 B B @ Rep B t ~ 1 Rep B t ~ q 1 C C A hasthedesiredform. Anyvector ~v 2 V isin N ifandonlyifitsnal q componentsarezeroes whenitisrepresentedwithrespectto B .As N is t invariant,eachofthe vectorsRep B t ~ 1 ,...,Rep B t ~ p hasthatform.Hencethelowerleftof Rep B;B t isallzeroes. Theargumentfortheupperrightissimilar. QED Toseethat t hasbeendecomposedintoitsactionontheparts,observe thattherestrictionsof t tothesubspaces N and R arerepresented,with respecttotheobviousbases,bythematrices T 1 and T 2 .So,withsubspaces thatareinvariantandcomplementary,wecansplittheproblemofexamininga lineartransformationintotwolower-dimensionalsubproblems.Thenextresult illustratesthisdecompositionintoblocks. 2.9Lemma If T isamatriceswithsquaresubmatrices T 1 and T 2 T = T 1 Z 2 Z 1 T 2 wherethe Z 'sareblocksofzeroes,then j T j = j T 1 jj T 2 j .

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SectionIV.JordanForm 393 Proof Supposethat T is n n ,that T 1 is p p ,andthat T 2 is q q .Inthe permutationformulaforthedeterminant j T j = X permutations t 1 ; t 2 ; t n; n sgn eachtermcomesfromarearrangementofthecolumnnumbers1 ;:::;n intoa neworder ;:::; n .Theupperrightblock Z 2 isallzeroes,soifa hasat leastoneof p +1 ;:::;n amongitsrst p columnnumbers ;:::; p then thetermarisingfrom iszero,e.g.,if = n then t 1 ; t 2 ; :::t n; n = 0 t 2 ; :::t n; n =0. Sotheaboveformulareducestoasumoverallpermutationswithtwo halves:anysignicant isthecompositionofa 1 thatrearrangesonly1 ;:::;p anda 2 thatrearrangesonly p +1 ;:::;p + q .Now,thedistributivelawand thefactthatthesignumofacompositionistheproductofthesignumsgives thatthis j T 1 jj T 2 j = X perms 1 of1 ;:::;p t 1 ; 1 t p; 1 p sgn 1 X perms 2 of p +1 ;:::;p + q t p +1 ; 2 p +1 t p + q; 2 p + q sgn 2 equals j T j = P signicant t 1 ; t 2 ; t n; n sgn QED 2.10Example 2000 1200 0030 0003 = 20 12 30 03 =36 FromLemma2.9weconcludethatiftwosubspacesarecomplementaryand t invariantthen t isnonsingularifandonlyifitsrestrictionstobothsubspaces arenonsingular. Nowforthepromisedthird,nal,steptothemainresult. 2.11Lemma Ifalineartransformation t : V V hasthecharacteristicpolynomial x )]TJ/F11 9.9626 Tf 10.276 0 Td [( 1 p 1 ::: x )]TJ/F11 9.9626 Tf 10.276 0 Td [( ` p ` then V = N 1 t )]TJ/F11 9.9626 Tf 10.276 0 Td [( 1 N 1 t )]TJ/F11 9.9626 Tf 10.276 0 Td [( ` anddim N 1 t )]TJ/F11 9.9626 Tf 9.962 0 Td [( i = p i Proof Becausedim V isthedegree p 1 + + p ` ofthecharacteristicpolynomial,toestablishstatement1weneedonlyshowthatstatementholds andthat N 1 t )]TJ/F11 9.9626 Tf 9.962 0 Td [( i N 1 t )]TJ/F11 9.9626 Tf 9.963 0 Td [( j istrivialwhenever i 6 = j Forthelatter,byLemma2.7,both N 1 t )]TJ/F11 9.9626 Tf 7.944 0 Td [( i and N 1 t )]TJ/F11 9.9626 Tf 7.944 0 Td [( j are t invariant. Noticethatanintersectionof t invariantsubspacesis t invariantandsothe restrictionof t to N 1 t )]TJ/F11 9.9626 Tf 9.245 0 Td [( i N 1 t )]TJ/F11 9.9626 Tf 9.245 0 Td [( j isalineartransformation.Butboth t )]TJ/F11 9.9626 Tf 9.979 0 Td [( i and t )]TJ/F11 9.9626 Tf 9.98 0 Td [( j arenilpotentonthissubspaceandsoif t hasanyeigenvalues

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394 ChapterFive.Similarity ontheintersectionthenitsonly"eigenvalueisboth i and j .Thatcannot be,sothisrestrictionhasnoeigenvalues: N 1 t )]TJ/F11 9.9626 Tf 10.396 0 Td [( i N 1 t )]TJ/F11 9.9626 Tf 10.395 0 Td [( j istrivial Lemma3.10showsthattheonlytransformationwithoutanyeigenvaluesison thetrivialspace. Toprovestatement,xtheindex i .Decompose V as N 1 t )]TJ/F11 9.9626 Tf 10.73 0 Td [( i R 1 t )]TJ/F11 9.9626 Tf 9.963 0 Td [( i andapplyLemma2.8. T = T 1 Z 2 Z 1 T 2 g dim N 1 t )]TJ/F11 9.9626 Tf 9.962 0 Td [( i -manyrows g dim R 1 t )]TJ/F11 9.9626 Tf 9.962 0 Td [( i -manyrows ByLemma2.9, j T )]TJ/F11 9.9626 Tf 9.545 0 Td [(xI j = j T 1 )]TJ/F11 9.9626 Tf 9.545 0 Td [(xI jj T 2 )]TJ/F11 9.9626 Tf 9.545 0 Td [(xI j .Bytheuniquenessclauseofthe FundamentalTheoremofArithmetic,thedeterminantsoftheblockshavethe samefactorsasthecharacteristicpolynomial j T 1 )]TJ/F11 9.9626 Tf 8.878 0 Td [(xI j = x )]TJ/F11 9.9626 Tf 8.878 0 Td [( 1 q 1 ::: x )]TJ/F11 9.9626 Tf 8.879 0 Td [( ` q ` and j T 2 )]TJ/F11 9.9626 Tf 10.427 0 Td [(xI j = x )]TJ/F11 9.9626 Tf 10.428 0 Td [( 1 r 1 ::: x )]TJ/F11 9.9626 Tf 10.428 0 Td [( ` r ` ,andthesumofthepowersofthese factorsisthepowerofthefactorinthecharacteristicpolynomial: q 1 + r 1 = p 1 ..., q ` + r ` = p ` .Statementwillbeprovedifwewillshowthat q i = p i and that q j =0forall j 6 = i ,becausethenthedegreeofthepolynomial j T 1 )]TJ/F11 9.9626 Tf 9.6 0 Td [(xI j | whichequalsthedimensionofthegeneralizednullspace|isasrequired. Forthat,rst,astherestrictionof t )]TJ/F11 9.9626 Tf 9.596 0 Td [( i to N 1 t )]TJ/F11 9.9626 Tf 9.596 0 Td [( i isnilpotentonthat space,theonlyeigenvalueof t onitis i .Thusthecharacteristicequationof t on N 1 t )]TJ/F11 9.9626 Tf 9.963 0 Td [( i is j T 1 )]TJ/F11 9.9626 Tf 9.963 0 Td [(xI j = x )]TJ/F11 9.9626 Tf 9.963 0 Td [( i q i .Andthus q j =0forall j 6 = i Nowconsidertherestrictionof t to R 1 t )]TJ/F11 9.9626 Tf 10.409 0 Td [( i .ByNoteII.2.2,themap t )]TJ/F11 9.9626 Tf 10.015 0 Td [( i isnonsingularon R 1 t )]TJ/F11 9.9626 Tf 10.016 0 Td [( i andso i isnotaneigenvalueof t onthat subspace.Therefore, x )]TJ/F11 9.9626 Tf 9.963 0 Td [( i isnotafactorof j T 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(xI j ,andso q i = p i QED Ourmajorresultjusttranslatesthosestepsintomatrixterms. 2.12Theorem Anysquarematrixissimilartoonein Jordanform 0 B B B B B @ J 1 {zeroes{ J 2 J ` )]TJ/F6 4.9813 Tf 5.396 0 Td [(1 {zeroes{ J ` 1 C C C C C A whereeach J istheJordanblockassociatedwiththeeigenvalue ofthe originalmatrixthatis,isallzeroesexceptfor 'sdownthediagonaland somesubdiagonalones. Proof Givenan n n matrix T ,considerthelinearmap t : C n C n thatit representswithrespecttothestandardbases.Usethepriorlemmatowrite C n = N 1 t )]TJ/F11 9.9626 Tf 9.917 0 Td [( 1 N 1 t )]TJ/F11 9.9626 Tf 9.917 0 Td [( ` where 1 ;:::; ` aretheeigenvaluesof t Becauseeach N 1 t )]TJ/F11 9.9626 Tf 10.073 0 Td [( i is t invariant,Lemma2.8andthepriorlemmashow that t isrepresentedbyamatrixthatisallzeroesexceptforsquareblocksalong thediagonal.TomakethoseblocksintoJordanblocks,pickeach B i tobea stringbasisfortheactionof t )]TJ/F11 9.9626 Tf 9.963 0 Td [( i on N 1 t )]TJ/F11 9.9626 Tf 9.962 0 Td [( i QED

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SectionIV.JordanForm 395 Jordanformisacanonicalformforsimilarityclassesofsquarematrices, providedthatwemakeituniquebyarrangingtheJordanblocksfromleast eigenvaluetogreatestandthenarrangingthesubdiagonal1blocksinsideeach Jordanblockfromlongesttoshortest. 2.13Example Thismatrixhasthecharacteristicpolynomial x )]TJ/F8 9.9626 Tf 9.672 0 Td [(2 2 x )]TJ/F8 9.9626 Tf 9.672 0 Td [(6. T = 0 @ 201 062 002 1 A Wewillhandletheeigenvalues2and6separately. Computationofthepowers,andthenullspacesandnullities,of T )]TJ/F8 9.9626 Tf 10.272 0 Td [(2 I is routine.RecallfromExample2.3theconventionoftaking T torepresenta transformation,here t : C 3 C 3 ,withrespecttothestandardbasis. power p T )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 I p N t )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 p nullity 1 0 B @ 001 042 000 1 C A f 0 B @ x 0 0 1 C A x 2 C g 1 2 0 B @ 000 0168 000 1 C A f 0 B @ x )]TJ/F11 9.9626 Tf 7.749 0 Td [(z= 2 z 1 C A x;z 2 C g 2 3 0 B @ 000 06432 000 1 C A {same{| Sothegeneralizednullspace N 1 t )]TJ/F8 9.9626 Tf 10.096 0 Td [(2hasdimensiontwo.We'venotedthat therestrictionof t )]TJ/F8 9.9626 Tf 10.496 0 Td [(2isnilpotentonthissubspace.Fromthewaythatthe nullitiesgrowweknowthattheactionof t )]TJ/F8 9.9626 Tf 10.076 0 Td [(2onastringbasis ~ 1 7! ~ 2 7! ~ 0. Thustherestrictioncanberepresentedinthecanonicalform N 2 = 00 10 =Rep B;B t )]TJ/F8 9.9626 Tf 9.963 0 Td [(2 B 2 = h 0 @ 1 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 1 A ; 0 @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 0 0 1 A i wheremanychoicesofbasisarepossible.Consequently,theactionoftherestrictionof t to N 1 t )]TJ/F8 9.9626 Tf 9.962 0 Td [(2isrepresentedbythismatrix. J 2 = N 2 +2 I =Rep B 2 ;B 2 t = 20 12 Thesecondeigenvalue'scomputationsareeasier.Becausethepowerof x )]TJ/F8 9.9626 Tf 8.909 0 Td [(6 inthecharacteristicpolynomialisone,therestrictionof t )]TJ/F8 9.9626 Tf 8.756 0 Td [(6to N 1 t )]TJ/F8 9.9626 Tf 8.757 0 Td [(6must benilpotentofindexone.Itsactiononastringbasismustbe ~ 3 7! ~ 0andsince itisthezeromap,itscanonicalform N 6 isthe1 1zeromatrix.Consequently,

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396 ChapterFive.Similarity thecanonicalform J 6 fortheactionof t on N 1 t )]TJ/F8 9.9626 Tf 8.635 0 Td [(6isthe1 1matrixwiththe singleentry6.Forthebasiswecanuseanynonzerovectorfromthegeneralized nullspace. B 6 = h 0 @ 0 1 0 1 A i Takentogether,thesetwogivethattheJordanformof T is Rep B;B t = 0 @ 200 120 006 1 A where B istheconcatenationof B 2 and B 6 2.14Example Contrastthepriorexamplewith T = 0 @ 221 062 002 1 A whichhasthesamecharacteristicpolynomial x )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 2 x )]TJ/F8 9.9626 Tf 9.962 0 Td [(6. Whilethecharacteristicpolynomialisthesame, power p T )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 I p N t )]TJ/F8 9.9626 Tf 9.962 0 Td [(2 p nullity 1 0 B @ 021 042 000 1 C A f 0 B @ x )]TJ/F11 9.9626 Tf 7.748 0 Td [(z= 2 z 1 C A x;z 2 C g 2 2 0 B @ 084 0168 000 1 C A {same{| heretheactionof t )]TJ/F8 9.9626 Tf 8.953 0 Td [(2isstableafteronlyoneapplication|therestrictionofof t )]TJ/F8 9.9626 Tf 8.934 0 Td [(2to N 1 t )]TJ/F8 9.9626 Tf 8.934 0 Td [(2isnilpotentofindexonlyone.Sothecontrastwiththeprior exampleisthatwhilethecharacteristicpolynomialtellsustolookattheaction ofthe t )]TJ/F8 9.9626 Tf 9.469 0 Td [(2onitsgeneralizednullspace,thecharacteristicpolynomialdoesnot describecompletelyitsactionandwemustdosomecomputationstond,in thisexample,thattheminimalpolynomialis x )]TJ/F8 9.9626 Tf 9.383 0 Td [(2 x )]TJ/F8 9.9626 Tf 9.383 0 Td [(6.Therestrictionof t )]TJ/F8 9.9626 Tf 9.765 0 Td [(2tothegeneralizednullspaceactsonastringbasisas ~ 1 7! ~ 0and ~ 2 7! ~ 0, andwegetthisJordanblockassociatedwiththeeigenvalue2. J 2 = 20 02 Fortheothereigenvalue,theargumentsforthesecondeigenvalueofthe priorexampleapplyagain.Therestrictionof t )]TJ/F8 9.9626 Tf 10.307 0 Td [(6to N 1 t )]TJ/F8 9.9626 Tf 10.307 0 Td [(6isnilpotent ofindexoneitcan'tbeofindexlessthanone,andsince x )]TJ/F8 9.9626 Tf 10.296 0 Td [(6isafactorof

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SectionIV.JordanForm 397 thecharacteristicpolynomialtothepoweroneitcan'tbeofindexmorethan oneeither.Thus t )]TJ/F8 9.9626 Tf 10.394 0 Td [(6'scanonicalform N 6 isthe1 1zeromatrix,andthe associatedJordanblock J 6 isthe1 1matrixwithentry6. Therefore, T isdiagonalizable. Rep B;B t = 0 @ 200 020 006 1 A B = B 2 B 6 = h 0 @ 1 0 0 1 A ; 0 @ 0 1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 1 A ; 0 @ 3 4 0 1 A i Checkingthatthethirdvectorin B isinthenullspaceof t )]TJ/F8 9.9626 Tf 9.963 0 Td [(6isroutine. 2.15Example Abitofcomputingwith T = 0 B B B B @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(14000 03000 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 )]TJ/F8 9.9626 Tf 7.748 0 Td [(100 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 )]TJ/F8 9.9626 Tf 7.748 0 Td [(42 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 15414 1 C C C C A showsthatitscharacteristicpolynomialis x )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 3 x +1 2 .Thistable power p T )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 I p N t )]TJ/F8 9.9626 Tf 9.962 0 Td [(3 p nullity 1 0 B B B B B B @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(44000 00000 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 )]TJ/F8 9.9626 Tf 7.749 0 Td [(400 3 )]TJ/F8 9.9626 Tf 7.749 0 Td [(9 )]TJ/F8 9.9626 Tf 7.749 0 Td [(4 )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 15411 1 C C C C C C A f 0 B B B B B B @ )]TJ/F8 9.9626 Tf 7.749 0 Td [( u + v = 2 )]TJ/F8 9.9626 Tf 7.749 0 Td [( u + v = 2 u + v = 2 u v 1 C C C C C C A u;v 2 C g 2 2 0 B B B B B B @ 16 )]TJ/F8 9.9626 Tf 7.749 0 Td [(16000 00000 0161600 )]TJ/F8 9.9626 Tf 7.749 0 Td [(16321600 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(16 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1600 1 C C C C C C A f 0 B B B B B B @ )]TJ/F11 9.9626 Tf 7.749 0 Td [(z )]TJ/F11 9.9626 Tf 7.749 0 Td [(z z u v 1 C C C C C C A z;u;v 2 C g 3 3 0 B B B B B B @ )]TJ/F8 9.9626 Tf 7.749 0 Td [(6464000 00000 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(64 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6400 64 )]TJ/F8 9.9626 Tf 7.748 0 Td [(128 )]TJ/F8 9.9626 Tf 7.749 0 Td [(6400 0646400 1 C C C C C C A {same{| showsthattherestrictionof t )]TJ/F8 9.9626 Tf 10.21 0 Td [(3to N 1 t )]TJ/F8 9.9626 Tf 10.21 0 Td [(3actsonastringbasisviathe twostrings ~ 1 7! ~ 2 7! ~ 0and ~ 3 7! ~ 0. Asimilarcalculationfortheothereigenvalue

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398 ChapterFive.Similarity power p T +1 I p N t +1 p nullity 1 0 B B B B B B @ 04000 04000 0 )]TJ/F8 9.9626 Tf 7.748 0 Td [(4000 3 )]TJ/F8 9.9626 Tf 7.748 0 Td [(9 )]TJ/F8 9.9626 Tf 7.749 0 Td [(43 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 15415 1 C C C C C C A f 0 B B B B B B @ )]TJ/F8 9.9626 Tf 7.748 0 Td [( u + v 0 )]TJ/F11 9.9626 Tf 7.748 0 Td [(v u v 1 C C C C C C A u;v 2 C g 2 2 0 B B B B B B @ 016000 016000 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(16000 8 )]TJ/F8 9.9626 Tf 7.749 0 Td [(40 )]TJ/F8 9.9626 Tf 7.749 0 Td [(168 )]TJ/F8 9.9626 Tf 7.749 0 Td [(8 82416824 1 C C C C C C A {same{| showsthattherestrictionof t +1toitsgeneralizednullspaceactsonastring basisviathetwoseparatestrings ~ 4 7! ~ 0and ~ 5 7! ~ 0. Therefore T issimilartothisJordanformmatrix. 0 B B B B @ )]TJ/F8 9.9626 Tf 7.748 0 Td [(10000 0 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1000 00300 00130 00003 1 C C C C A Weclosewiththestatementthatthesubjectsconsideredearlierinthis Chpaterareindeed,inthissense,exhaustive. 2.16Corollary Everysquarematrixissimilartothesumofadiagonalmatrix andanilpotentmatrix. Exercises 2.17 DothecheckforExample2.3. 2.18 EachmatrixisinJordanform.Stateitscharacteristicpolynomialandits minimalpolynomial. a 30 13 b )]TJ/F29 8.9664 Tf 7.168 0 Td [(10 0 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 c 200 120 00 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 = 2 d 300 130 013 e 0 B @ 3000 1300 0030 0013 1 C A f 0 B @ 4000 1400 00 )]TJ/F29 8.9664 Tf 7.168 0 Td [(40 001 )]TJ/F29 8.9664 Tf 7.167 0 Td [(4 1 C A g 500 020 003 h 0 B @ 5000 0200 0020 0003 1 C A i 0 B @ 5000 0200 0120 0003 1 C A X 2.19 FindtheJordanformfromthegivendata. a Thematrix T is5 5withthesingleeigenvalue3.Thenullitiesofthepowers are: T )]TJ/F29 8.9664 Tf 9.532 0 Td [(3 I hasnullitytwo, T )]TJ/F29 8.9664 Tf 9.532 0 Td [(3 I 2 hasnullitythree, T )]TJ/F29 8.9664 Tf 9.532 0 Td [(3 I 3 hasnullity four,and T )]TJ/F29 8.9664 Tf 9.216 0 Td [(3 I 4 hasnullityve.

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SectionIV.JordanForm 399 b Thematrix S is5 5withtwoeigenvalues.Fortheeigenvalue2thenullities are: S )]TJ/F29 8.9664 Tf 9.396 0 Td [(2 I hasnullitytwo,and S )]TJ/F29 8.9664 Tf 9.397 0 Td [(2 I 2 hasnullityfour.Fortheeigenvalue )]TJ/F29 8.9664 Tf 7.167 0 Td [(1thenullitiesare: S +1 I hasnullityone. 2.20 Findthechangeofbasismatricesforeachexample. a Example2.13 b Example2.14 c Example2.15 X 2.21 FindtheJordanformandaJordanbasisforeachmatrix. a )]TJ/F29 8.9664 Tf 7.168 0 Td [(104 )]TJ/F29 8.9664 Tf 7.168 0 Td [(2510 b 5 )]TJ/F29 8.9664 Tf 7.168 0 Td [(4 9 )]TJ/F29 8.9664 Tf 7.168 0 Td [(7 c 400 213 504 d 543 )]TJ/F29 8.9664 Tf 7.168 0 Td [(10 )]TJ/F29 8.9664 Tf 7.168 0 Td [(3 1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(21 e 973 )]TJ/F29 8.9664 Tf 7.168 0 Td [(9 )]TJ/F29 8.9664 Tf 7.168 0 Td [(7 )]TJ/F29 8.9664 Tf 7.167 0 Td [(4 444 f 22 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(11 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(22 g 0 B @ 7122 14 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 )]TJ/F29 8.9664 Tf 7.168 0 Td [(215 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 1128 1 C A X 2.22 FindallpossibleJordanformsofatransformationwithcharacteristicpolynomial x )]TJ/F29 8.9664 Tf 9.215 0 Td [(1 2 x +2 2 2.23 FindallpossibleJordanformsofatransformationwithcharacteristicpolynomial x )]TJ/F29 8.9664 Tf 9.215 0 Td [(1 3 x +2. X 2.24 FindallpossibleJordanformsofatransformationwithcharacteristicpolynomial x )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 3 x +1andminimalpolynomial x )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 2 x +1. 2.25 FindallpossibleJordanformsofatransformationwithcharacteristicpolynomial x )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 4 x +1andminimalpolynomial x )]TJ/F29 8.9664 Tf 9.215 0 Td [(2 2 x +1. X 2.26 Diagonalizethese. a 11 00 b 01 10 X 2.27 FindtheJordanmatrixrepresentingthedierentiationoperatoron P 3 X 2.28 Decideifthesetwoaresimilar. 1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 4 )]TJ/F29 8.9664 Tf 7.167 0 Td [(3 )]TJ/F29 8.9664 Tf 7.167 0 Td [(10 1 )]TJ/F29 8.9664 Tf 7.167 0 Td [(1 2.29 FindtheJordanformofthismatrix. 0 )]TJ/F29 8.9664 Tf 7.168 0 Td [(1 10 AlsogiveaJordanbasis. 2.30 Howmanysimilarityclassesaretherefor3 3matriceswhoseonlyeigenvalues are )]TJ/F29 8.9664 Tf 7.168 0 Td [(3and4?

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400 ChapterFive.Similarity X 2.31 Provethatamatrixisdiagonalizableifandonlyifitsminimalpolynomial hasonlylinearfactors. 2.32 Giveanexampleofalineartransformationonavectorspacethathasno non-trivialinvariantsubspaces. 2.33 Showthatasubspaceis t )]TJ/F32 8.9664 Tf 9.215 0 Td [( 1 invariantifandonlyifitis t )]TJ/F32 8.9664 Tf 9.215 0 Td [( 2 invariant. 2.34 Proveordisprove:two n n matricesaresimilarifandonlyiftheyhavethe samecharacteristicandminimalpolynomials. 2.35 The trace ofasquarematrixisthesumofitsdiagonalentries. a Findtheformulaforthecharacteristicpolynomialofa2 2matrix. b Showthattraceisinvariantundersimilarity,andsowecansensiblyspeak ofthe`traceofamap'. Hint: seetheprioritem. c Istraceinvariantundermatrixequivalence? d Showthatthetraceofamapisthesumofitseigenvaluescountingmultiplicities. e Showthatthetraceofanilpotentmapiszero.Doestheconversehold? 2.36 TouseDenition2.6tocheckwhetherasubspaceis t invariant,weseemingly havetocheckalloftheinnitelymanyvectorsinanontrivialsubspacetoseeif theysatisfythecondition.Provethatasubspaceis t invariantifandonlyifits subbasishasthepropertythatforallofitselements, t ~ isinthesubspace. X 2.37 Is t invariancepreservedunderintersection?Underunion?Complementation? Sumsofsubspaces? 2.38 GiveawaytoordertheJordanblocksifsomeoftheeigenvaluesarecomplex numbers.Thatis,suggestareasonableorderingforthecomplexnumbers. 2.39 Let P j R bethevectorspaceovertherealsofdegree j polynomials.Show thatif j k then P j R isaninvariantsubspaceof P k R underthedierentiation operator.In P 7 R ,doesanyof P 0 R ,..., P 6 R haveaninvariantcomplement? 2.40 In P n R ,thevectorspaceovertherealsofdegree n polynomials, E = f p x 2P n R p )]TJ/F32 8.9664 Tf 7.167 0 Td [(x = p x forall x g and O = f p x 2P n R p )]TJ/F32 8.9664 Tf 7.167 0 Td [(x = )]TJ/F32 8.9664 Tf 7.167 0 Td [(p x forall x g arethe even andthe odd polynomials; p x = x 2 isevenwhile p x = x 3 isodd. Showthattheyaresubspaces.Aretheycomplementary?Aretheyinvariantunder thedierentiationtransformation? 2.41 Lemma2.8saysthatif M and N areinvariantcomplementsthen t hasa representationinthegivenblockformwithrespecttothesameendingasstarting basis,ofcourse.Doestheimplicationreverse? 2.42 Amatrix S isthe squareroot ofanother T if S 2 = T .Showthatanynonsingularmatrixhasasquareroot.

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Topic:MethodofPowers 401 Topic:MethodofPowers Inpractice,calculatingeigenvaluesandeigenvectorsisadicultproblem.Finding,andsolving,thecharacteristicpolynomialofthelargematricesoftenencounteredinapplicationsistooslowandtoohard.Othertechniques,indirect onesthatavoidthecharacteristicpolynomial,areused.Hereweshallseesuch amethodthatissuitableforlargematricesthatare`sparse'thegreatmajority oftheentriesarezero. Supposethatthe n n matrix T hasthe n distincteigenvalues 1 2 ,..., n Then R n hasabasisthatiscomposedoftheassociatedeigenvectors h ~ 1 ;:::; ~ n i Forany ~v 2 R n ,where ~v = c 1 ~ 1 + + c n ~ n ,iterating T on ~v givesthese. T~v = c 1 1 ~ 1 + c 2 2 ~ 2 + + c n n ~ n T 2 ~v = c 1 2 1 ~ 1 + c 2 2 2 ~ 2 + + c n 2 n ~ n T 3 ~v = c 1 3 1 ~ 1 + c 2 3 2 ~ 2 + + c n 3 n ~ n T k ~v = c 1 k 1 ~ 1 + c 2 k 2 ~ 2 + + c n k n ~ n Ifoneoftheeigenvaluse,say, 1 ,hasalargerabsolutevaluethananyofthe othereigenvaluesthenitstermwilldominatetheaboveexpression.Putanother way,dividingthroughby k 1 givesthis, T k ~v k 1 = c 1 ~ 1 + c 2 k 2 k 1 ~ 2 + + c n k n k 1 ~ n and,because 1 isassumedtohavethelargestabsolutevalue,as k getslarger thefractionsgotozero.Thus,theentireexpressiongoesto c 1 ~ 1 Thatisaslongas c 1 isnotzero,as k increases,thevectors T k ~v will tendtowardthedirectionoftheeigenvectorsassociatedwiththedominant eigenvalue,and,consequently,theratiosofthelengths k T k ~v k = k T k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ~v k will tendtowardthatdominanteigenvalue. Forexamplesamplecomputercodeforthisfollowstheexercises,because thematrix T = 30 8 )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 istriangular,itseigenvaluesarejusttheentriesonthediagonal,3and )]TJ/F8 9.9626 Tf 7.749 0 Td [(1. Arbitrarilytaking ~v tohavethecomponents1and1gives ~v T~vT 2 ~v T 9 ~vT 10 ~v 1 1 3 7 9 17 19683 39367 59049 118097 andtheratiobetweenthelengthsofthelasttwois2 : 9999. Twoimplementationissuesmustbeaddressed.Therstissueisthat,instead ofndingthepowersof T andapplyingthemto ~v ,wewillcompute ~v 1 as T~v and

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402 ChapterFive.Similarity thencompute ~v 2 as T~v 1 ,etc.i.e.,weneverseparatelycalculate T 2 T 3 ,etc.. Thesematrix-vectorproductscanbedonequicklyevenif T islarge,provided thatitissparse.Thesecondissueisthat,toavoidgeneratingnumbersthatare solargethattheyoverowourcomputer'scapability,wecannormalizethe ~v i 's ateachstep.Forinstance,wecandivideeach ~v i byitslengthotherpossibilities aretodivideitbyitslargestcomponent,orsimplybyitsrstcomponent.We thusimplementthismethodbygenerating ~w 0 = ~v 0 = k ~v 0 k ~v 1 = T~w 0 ~w 1 = ~v 1 = k ~v 1 k ~v 2 = T~w 2 ~w k )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = ~v k )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = k ~v k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 k ~v k = T~w k untilwearesatised.Thenthevector ~v k isanapproximationofaneigenvector, andtheapproximationofthedominanteigenvalueistheratio k ~v k k = k ~w k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 k = k ~v k k Onewaywecouldbe`satised'istoiterateuntilourapproximationofthe eigenvaluesettlesdown.Wecoulddecide,forinstance,tostoptheiteration processnotaftersomexednumberofsteps,butinsteadwhen k ~v k k diers from k ~v k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 k bylessthanonepercent,orwhentheyagreeuptothesecond signicantdigit. Therateofconvergenceisdeterminedbytherateatwhichthepowersof k 2 = 1 k gotozero,where 2 istheeigenvalueofsecondlargestnorm.Ifthat ratioismuchlessthanonethenconvergenceisfast,butifitisonlyslightly lessthanonethenconvergencecanbequiteslow.Consequently,themethodof powersisnotthemostcommonlyusedwayofndingeigenvaluesalthoughit isthesimplestone,whichiswhyitishereastheillustrationofthepossibilityof computingeigenvalueswithoutsolvingthecharacteristicpolynomial.Instead, thereareavarietyofmethodsthatgenerallyworkbyrstreplacingthegiven matrix T withanotherthatissimilartoitandsohasthesameeigenvalues,but isinsomereducedformsuchas tridiagonalform :theonlynonzeroentriesare onthediagonal,orjustaboveorbelowit.Thenspecialtechniquescanbeused tondtheeigenvalues.Oncetheeigenvaluesareknown,theeigenvectorsof T canbeeasilycomputed.Theseothermethodsareoutsideofourscope.Agood referenceis[Goult, etal. ] Exercises 1 Useteniterationstoestimatethelargesteigenvalueofthesematrices,starting fromthevectorwithcomponents1and2.Comparetheanswerwiththeone obtainedbysolvingthecharacteristicequation.

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Topic:MethodofPowers 403 a 15 04 b 32 )]TJ/F29 8.9664 Tf 7.168 0 Td [(10 2 Redothepriorexercisebyiteratinguntil k ~v k k)-255(k ~v k )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 k hasabsolutevalueless than0 : 01Ateachstep,normalizebydividingeachvectorbyitslength.Howmany iterationsarerequired?Aretheanswerssignicantlydierent? 3 Useteniterationstoestimatethelargesteigenvalueofthesematrices,starting fromthevectorwithcomponents1,2,and3.Comparetheanswerwiththeone obtainedbysolvingthecharacteristicequation. a 401 )]TJ/F29 8.9664 Tf 7.168 0 Td [(210 )]TJ/F29 8.9664 Tf 7.168 0 Td [(201 b )]TJ/F29 8.9664 Tf 7.168 0 Td [(122 222 )]TJ/F29 8.9664 Tf 7.168 0 Td [(3 )]TJ/F29 8.9664 Tf 7.168 0 Td [(6 )]TJ/F29 8.9664 Tf 7.168 0 Td [(6 4 Redothepriorexercisebyiteratinguntil k ~v k k)-255(k ~v k )]TJ/F31 5.9776 Tf 5.756 0 Td [(1 k hasabsolutevalueless than0 : 01.Ateachstep,normalizebydividingeachvectorbyitslength.How manyiterationsdoesittake?Aretheanswerssignicantlydierent? 5 Whathappensif c 1 =0?Thatis,whathappensiftheinitialvectordoesnotto haveanycomponentinthedirectionoftherelevanteigenvector? 6 Howcanthemethodofpowersbeadoptedtondthesmallesteigenvalue? ComputerCode ThisisthecodeforthecomputeralgebrasystemOctavethatwasusedto dothecalculationabove.Ithasbeenlightlyeditedtoremoveblanklines,etc. >T=[3,0; 8,-1] T= 30 8-1 >v0=[1;2] v0= 1 1 >v1=T*v0 v1= 3 7 >v2=T*v1 v2= 9 17 >T9=T**9 T9= 196830 39368-1 >T10=T**10 T10= 590490 1180961 >v9=T9*v0 v9=

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404 ChapterFive.Similarity 19683 39367 >v10=T10*v0 v10= 59049 118096 >normv10/normv9 ans=2.9999 Remark:weareignoringthepowerofOctavehere;therearebuilt-infunctionstoautomaticallyapplyquitesophisticatedmethodstondeigenvaluesand eigenvectors.Instead,weareusingjustthesystemasacalculator.

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Topic:StablePopulations 405 Topic:StablePopulations Imagineareserveparkwithanimalsfromaspeciesthatwearetryingtoprotect. Theparkdoesn'thaveafenceandsoanimalscrosstheboundary,bothfrom theinsideoutandintheotherdirection.Everyyear,10%oftheanimalsfrom insideoftheparkleave,and1%oftheanimalsfromtheoutsidendtheirway in.Wecanaskifwecanndastablelevelofpopulationforthispark:istherea populationthat,onceestablished,willstayconstantovertime,withthenumber ofanimalsleavingequaltothenumberofanimalsentering? Toanswerthatquestion,wemustrstestablishtheequations.Lettheyear n populationintheparkbe p n andintherestoftheworldbe r n p n +1 = : 90 p n + : 01 r n r n +1 = : 10 p n + : 99 r n WecansetthissystemupasamatrixequationseetheMarkovChaintopic. p n +1 r n +1 = : 90 : 01 : 10 : 99 p n r n Now,stablelevel"meansthat p n +1 = p n and r n +1 = r n ,sothatthematrix equation ~v n +1 = T~v n becomes ~v = T~v .Wearethereforelookingforeigenvectors for T thatareassociatedwiththeeigenvalue1.Theequation I )]TJ/F11 9.9626 Tf 9.963 0 Td [(T ~v = ~ 0is : 10 : 01 : 10 : 01 p r = 0 0 whichgivestheeigenspace:vectorswiththerestrictionthat p = : 1 r .Coupled withadditionalinformation,thatthetotalworldpopulationofthisspeciesisis p + r =110000,wendthatthestablestateis p =10 ; 000and r =100 ; 000. Ifwestartwithaparkpopulationoftenthousandanimals,sothattherestof theworldhasonehundredthousand,theneveryyeartenpercentathousand animalsofthoseinsidewillleavethepark,andeveryyearonepercenta thousandofthosefromtherestoftheworldwillenterthepark.Itisstable, self-sustaining. Nowimaginethatwearetryingtograduallybuildupthetotalworldpopulationofthisspecies.Wecantry,forinstance,tohavetheworldpopulation growatarateof1%peryear.Inthiscase,wecantakeastable"statefor thepark'spopulationtobethatitalsogrowsat1%peryear.Theequation ~v n +1 =1 : 01 ~v n = T~v n leadsto : 01 I )]TJ/F11 9.9626 Tf 9.963 0 Td [(T ~v = ~ 0,whichgivesthissystem. : 11 : 01 : 10 : 02 p r = 0 0 Thematrixisnonsingular,andsotheonlysolutionis p =0and r =0.Thus, thereisnousableinitialpopulationthatwecanestablishattheparkand expectthatitwillgrowatthesamerateastherestoftheworld.

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406 ChapterFive.Similarity Knowingthatanannualworldpopulationgrowthrateof1%forcesanunstableparkpopulation,wecanaskwhichgrowthratestherearethatwould allowaninitialpopulationfortheparkthatwillbeself-sustaining.Weconsider ~v = T~v andsolvefor 0= )]TJ/F11 9.9626 Tf 9.963 0 Td [(: 9 : 01 : 10 )]TJ/F11 9.9626 Tf 9.963 0 Td [(: 99 = )]TJ/F11 9.9626 Tf 9.963 0 Td [(: 9 )]TJ/F11 9.9626 Tf 9.962 0 Td [(: 99 )]TJ/F8 9.9626 Tf 9.962 0 Td [( : 10 : 01= 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 : 89 + : 89 Ashortcuttofactoringthatquadraticisourknowledgethat =1isaneigenvalueof T ,sotheothereigenvalueis : 89.Thustherearetwowaystohavea stableparkpopulationapopulationthatgrowsatthesamerateasthepopulationoftherestoftheworld,despitetheleakyparkboundaries:haveaworld populationthatisdoesnotgroworshrink,andhaveaworldpopulationthat shrinksby11%everyyear. Sothisisonemeaningofeigenvaluesandeigenvectors|theygiveastablestateforasystem.Iftheeigenvalueis1thenthesystemisstatic.If theeigenvalueisn't1thenthesystemiseithergrowingorshrinking,butina dynamically-stableway. Exercises 1 Whatinitialpopulationfortheparkdiscussedaboveshouldbesetupinthecase whereworldpopulationsareallowedtodeclineby11%everyyear? 2 Whatwillhappentothepopulationoftheparkintheeventofagrowthinworld populationof1%peryear?Willitlagtheworldgrowth,orleadit?Assume thattheinitalparkpopulationistenthousand,andtheworldpopulationisone hunderdthousand,andcalculateoveratenyearspan. 3 Theparkdiscussedaboveispartiallyfencedsothatnow,everyyear,only5%of theanimalsfrominsideoftheparkleavestill,about1%oftheanimalsfromthe outsidendtheirwayin.Underwhatconditionscantheparkmaintainastable populationnow? 4 SupposethataspeciesofbirdonlylivesinCanada,theUnitedStates,orin Mexico.Everyyear,4%oftheCanadianbirdstraveltotheUS,and1%ofthem traveltoMexico.Everyyear,6%oftheUSbirdstraveltoCanada,and4% gotoMexico.FromMexico,everyyear10%traveltotheUS,and0%goto Canada. a Givethetransitionmatrix. b Isthereawayforthethreecountriestohaveconstantpopulations? c Findallstablesituations.

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Topic:LinearRecurrences 407 Topic:LinearRecurrences In1202LeonardoofPisa,alsoknownasFibonacci,posedthisproblem. Acertainmanputapairofrabbitsinaplacesurroundedonall sidesbyawall.Howmanypairsofrabbitscanbeproducedfrom thatpairinayearifitissupposedthateverymontheachpairbegets anewpairwhichfromthesecondmonthonbecomesproductive? Thismovespastanelementaryexponentialgrowthmodelforpopulationincreasetoincludethefactthatthereisaninitialperiodwherenewbornsarenot fertile.However,itretainsothersimplyngassumptions,suchasthatthereis nogestationperiodandnomortality. Thenumberofnewbornpairsthatwillappearintheupcomingmonthis simplythenumberofpairsthatwerealivelastmonth,sincethosewillallbe fertile,havingbeenalivefortwomonths.Thenumberofpairsalivenextmonth isthesumofthenumberalivelastmonthandthenumberofnewborns. f n +1= f n + f n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1where f =1, f =1 Theisanexampleofa recurrencerelation itiscalledthatbecausethevalues of f arecalculatedbylookingatother,prior,valuesof f .Fromit,wecan easilyanswerFibonacci'stwelve-monthquestion. month 0123456789101112 pairs 1123581321345589144233 Thesequenceofnumbersdenedbytheaboveequationofwhichtherstfew arelistedisthe Fibonaccisequence .Thematerialofthischaptercanbeused togiveaformulawithwhichwecancancalculate f n +1withouthavingto rstnd f n f n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1,etc. Forthat,observethattherecurrenceisalinearrelationshipandsowecan giveasuitablematrixformulationofit. 11 10 f n f n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1 = f n +1 f n where f f = 1 1 Then,wherewewrite T forthematrixand ~v n forthevectorwithcomponents f n +1and f n ,wehavethat ~v n = T n ~v 0 .Theadvantageofthismatrixformulationisthatbydiagonalizing T wegetafastwaytocomputeitspowers:where T = PDP )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 wehave T n = PD n P )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 ,andthe n -thpowerofthediagonal matrix D isthediagonalmatrixwhoseentriesthatarethe n -thpowersofthe entriesof D Thecharacteristicequationof T is 2 )]TJ/F11 9.9626 Tf 9.86 0 Td [( )]TJ/F8 9.9626 Tf 9.861 0 Td [(1.Thequadraticformulagives itsrootsas+ p 5 = 2and )]TJ 9.963 8.241 Td [(p 5 = 2.Diagonalizinggivesthis. 11 10 = 1+ p 5 2 1 )]TJ 6.227 5.814 Td [(p 5 2 11 1+ p 5 2 0 0 1 )]TJ 6.227 5.813 Td [(p 5 2 1 p 5 )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(1 )]TJ 6.226 5.814 Td [(p 5 2 p 5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 p 5 1+ p 5 2 p 5 !

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408 ChapterFive.Similarity Introducingthevectorsandtakingthe n -thpower,wehave f n +1 f n = 11 10 n f f = 1+ p 5 2 1 )]TJ 6.227 5.813 Td [(p 5 2 11 1+ p 5 2 n 0 0 1 )]TJ 6.227 5.814 Td [(p 5 2 n 1 p 5 )]TJ/F7 6.9738 Tf 8.944 3.922 Td [(1 )]TJ 6.226 5.814 Td [(p 5 2 p 5 )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 p 5 1+ p 5 2 p 5 f f Wecancompute f n fromthesecondcomponentofthatequation. f n = 1 p 5 1+ p 5 2 n )]TJ/F1 9.9626 Tf 9.963 17.036 Td [( 1 )]TJ 9.962 8.242 Td [(p 5 2 n # Noticethat f isdominatedbyitsrsttermbecause )]TJ 10.592 8.242 Td [(p 5 = 2islessthan one,soitspowersgotozero.Althoughwehaveextendedtheelementarymodel ofpopulationgrowthbyaddingadelayperiodbeforetheonsetoffertility,we nonethelessstillgetanasmyptoticallyexponentialfunction. Ingeneral,a linearrecurrencerelation hastheform f n +1= a n f n + a n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 f n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1+ + a n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k f n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k itisalsocalleda dierenceequation .Thisrecurrencerelationis homogeneous becausethereisnoconstantterm;i.e,itcanbeputintotheform0= )]TJ/F11 9.9626 Tf 7.749 0 Td [(f n +1+ a n f n + a n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 f n )]TJ/F8 9.9626 Tf 8.736 0 Td [(1+ + a n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k f n )]TJ/F11 9.9626 Tf 8.736 0 Td [(k .Thisissaidtobearelationof order k .Therelation,alongwiththe initialconditions f ,..., f k completely determineasequence.Forinstance,theFibonaccirelationisoforder2and it,alongwiththetwoinitialconditions f =1and f =1,determinesthe Fibonaccisequencesimplybecausewecancomputeany f n byrstcomputing f f ,etc.InthisTopic,weshallseehowlinearalgebracanbeusedto solvelinearrecurrencerelations. First,wedenethevectorspaceinwhichweareworking.Let V betheset offunctions f fromthenaturalnumbers N = f 0 ; 1 ; 2 ;::: g totherealnumbers. Belowweshallhavefunctionswithdomain f 1 ; 2 ;::: g ,thatis,without0,but itisnotanimportantdistinction. Puttingtheinitialconditionsasideforamoment,foranyrecurrence,wecan considerthesubset S of V ofsolutions.Forexample,withoutinitialconditions, inadditiontothefunction f givenabove,theFibonaccirelationisalsosolvedby thefunction g whoserstfewvaluesare g =1, g =1, g =3, g =4, and g =7. Thesubset S isasubspaceof V .Itisnonemptybecausethezerofunction isasolution.Itisclosedunderadditionsinceif f 1 and f 2 aresolutions,then a n +1 f 1 + f 2 n +1+ + a n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k f 1 + f 2 n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k = a n +1 f 1 n +1+ + a n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k f 1 n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k + a n +1 f 2 n +1+ + a n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k f 2 n )]TJ/F11 9.9626 Tf 9.962 0 Td [(k =0 :

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Topic:LinearRecurrences 409 And,itisclosedunderscalarmultiplicationsince a n +1 rf 1 n +1+ + a n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k rf 1 n )]TJ/F11 9.9626 Tf 9.962 0 Td [(k = r a n +1 f 1 n +1+ + a n )]TJ/F10 6.9738 Tf 6.226 0 Td [(k f 1 n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k = r 0 =0 : Wecangivethedimensionof S .Considerthismapfromthesetoffunctions S tothesetofvectors R k f 7! 0 B B B @ f f f k 1 C C C A Exercise3showsthatthismapislinear.Because,asnotedabove,anysolution oftherecurrenceisuniquelydeterminedbythe k initialconditions,thismapis one-to-oneandonto.Thusitisanisomorphism,andthus S hasdimension k theorderoftherecurrence. Soagain,withoutanyinitialconditions,wecandescribethesetofsolutionsofanylinearhomogeneousrecurrencerelationofdegree k bytakinglinear combinationsofonly k linearlyindependentfunctions.Itremainstoproduce thosefunctions. Forthat,weexpresstherecurrence f n +1= a n f n + + a n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k f n )]TJ/F11 9.9626 Tf 9.652 0 Td [(k withamatrixequation. 0 B B B B B B B @ a n a n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 a n )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 :::a n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k +1 a n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k 100 ::: 00 010 001 . 000 ::: 10 1 C C C C C C C A 0 B B B @ f n f n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1 f n )]TJ/F11 9.9626 Tf 9.962 0 Td [(k 1 C C C A = 0 B B B @ f n +1 f n f n )]TJ/F11 9.9626 Tf 9.963 0 Td [(k +1 1 C C C A Intryingtondthecharacteristicfunctionofthematrix,wecanseethepattern inthe2 2case a n )]TJ/F11 9.9626 Tf 9.962 0 Td [(a n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 1 )]TJ/F11 9.9626 Tf 7.749 0 Td [( = 2 )]TJ/F11 9.9626 Tf 9.963 0 Td [(a n )]TJ/F11 9.9626 Tf 9.962 0 Td [(a n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 and3 3case. 0 @ a n )]TJ/F11 9.9626 Tf 9.963 0 Td [(a n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 a n )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 1 )]TJ/F11 9.9626 Tf 7.749 0 Td [( 0 01 )]TJ/F11 9.9626 Tf 7.749 0 Td [( 1 A = )]TJ/F11 9.9626 Tf 7.748 0 Td [( 3 + a n 2 + a n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 + a n )]TJ/F7 6.9738 Tf 6.227 0 Td [(2

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410 ChapterFive.Similarity Exercise4showsthatthecharacteristicequationisthis. a n )]TJ/F11 9.9626 Tf 9.963 0 Td [(a n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 a n )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 :::a n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k +1 a n )]TJ/F10 6.9738 Tf 6.226 0 Td [(k 1 )]TJ/F11 9.9626 Tf 7.748 0 Td [( 0 ::: 00 01 )]TJ/F11 9.9626 Tf 7.749 0 Td [( 001 . 000 ::: 1 )]TJ/F11 9.9626 Tf 7.749 0 Td [( = )]TJ/F11 9.9626 Tf 7.748 0 Td [( k + a n k )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 + a n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 k )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 + + a n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k +1 + a n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k Wecallthatthepolynomial`associated'withtherecurrencerelation.Wewill bendingtherootsofthispolynomialandsowecandropthe asirrelevant. If )]TJ/F11 9.9626 Tf 7.749 0 Td [( k + a n k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 + a n )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 k )]TJ/F7 6.9738 Tf 6.226 0 Td [(2 + + a n )]TJ/F10 6.9738 Tf 6.226 0 Td [(k +1 + a n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k hasnorepeatedroots thenthematrixisdiagonalizableandwecan,intheory,getaformulafor f n asintheFibonaccicase.But,becauseweknowthatthesubspaceofsolutions hasdimension k ,wedonotneedtodothediagonalizationcalculation,provided thatwecanexhibit k linearlyindependentfunctionssatisfyingtherelation. Where r 1 r 2 ,..., r k arethedistinctroots,considerthefunctions f r 1 n = r n 1 through f r k n = r n k ofpowersofthoseroots.Exercise5showsthateachisa solutionoftherecurrenceandthatthe k ofthemformalinearlyindependent set.So,giventhehomogeneouslinearrecurrence f n +1= a n f n + + a n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k f n )]TJ/F11 9.9626 Tf 8.994 0 Td [(k thatis,0= )]TJ/F11 9.9626 Tf 7.749 0 Td [(f n +1+ a n f n + + a n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k f n )]TJ/F11 9.9626 Tf 8.994 0 Td [(k weconsider theassociatedequation0= )]TJ/F11 9.9626 Tf 7.749 0 Td [( k + a n k )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 + + a n )]TJ/F10 6.9738 Tf 6.226 0 Td [(k +1 + a n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k .Wendits roots r 1 ,..., r k ,andifthoserootsaredistinctthenanysolutionoftherelation hastheform f n = c 1 r n 1 + c 2 r n 2 + + c k r n k for c 1 ;:::;c n 2 R .Thecaseof repeatedrootsisalsoeasilydone,butwewon'tcoverithere|seeanytexton DiscreteMathematics. Now,givensomeinitialconditions,sothatweareinterestedinaparticular solution,wecansolvefor c 1 ,..., c n .Forinstance,thepolynomialassociated withtheFibonaccirelationis )]TJ/F11 9.9626 Tf 7.749 0 Td [( 2 + +1,whoserootsare p 5 = 2andso anysolutionoftheFibonacciequationhastheform f n = c 1 + p 5 = 2 n + c 2 )]TJ 9.97 8.242 Td [(p 5 = 2 n .Includingtheinitialconditionsforthecases n =0and n =1 gives c 1 + c 2 =1 + p 5 = 2 c 1 + )]TJ 9.963 8.242 Td [(p 5 = 2 c 2 =1 whichyields c 1 =1 = p 5and c 2 = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 = p 5,aswascalculatedabove. Weclosebyconsideringthenonhomogeneouscase,wheretherelationhasthe form f n +1= a n f n + a n )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 f n )]TJ/F8 9.9626 Tf 8.797 0 Td [(1+ + a n )]TJ/F10 6.9738 Tf 6.227 0 Td [(k f n )]TJ/F11 9.9626 Tf 8.797 0 Td [(k + b forsomenonzero b .Asintherstchapterofthisbook,onlyasmalladjustmentisneededtomake thetransitionfromthehomogeneouscase.Thisclassicexampleillustrates. In1883,EdouardLucasposedthefollowingproblem. InthegreattempleatBenares,beneaththedomewhichmarks thecenteroftheworld,restsabrassplateinwhicharexedthree diamondneedles,eachacubithighandasthickasthebodyofa

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Topic:LinearRecurrences 411 bee.Ononeoftheseneedles,atthecreation,Godplacedsixtyfour disksofpuregold,thelargestdiskrestingonthebrassplate,and theothersgettingsmallerandsmalleruptothetopone.Thisisthe TowerofBramah.Dayandnightunceasinglytheprieststransfer thedisksfromonediamondneedletoanotheraccordingtothexed andimmutablelawsofBramah,whichrequirethattheprieston dutymustnotmovemorethanonediskatatimeandthathemust placethisdiskonaneedlesothatthereisnosmallerdiskbelow it.Whenthesixty-fourdisksshallhavebeenthustransferredfrom theneedleonwhichatthecreationGodplacedthemtooneofthe otherneedles,tower,temple,andBrahminsalikewillcrumbleinto dusk,andwithathunderclaptheworldwillvanish.Translationof [DeParville]from[Ball&Coxeter]. Howmanydiskmoveswillittake?Insteadoftacklingthesixtyfourdisk problemrightaway,wewillconsidertheproblemforsmallernumbersofdisks, startingwiththree. Tobegin,allthreedisksareonthesameneedle. Aftermovingthesmalldisktothefarneedle,themid-sizeddisktothemiddle needle,andthenmovingthesmalldisktothemiddleneedlewehavethis. Nowwecanmovethebigdiskover.Then,tonish,werepeattheprocessof movingthesmallerdisks,thistimesothattheyenduponthethirdneedle,on topofthebigdisk. Sothethingtoseeisthattomovetheverylargestdisk,thebottomdisk, ataminimumwemust:rstmovethesmallerdiskstothemiddleneedle,then movethebigone,andthenmoveallthesmalleronesfromthemiddleneedleto theendingneedle.Thosethreestepsgiveusthisrecurence. T n +1= T n +1+ T n =2 T n +1where T =1 Wecaneasilygettherstfewvaluesof T .

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412 ChapterFive.Similarity n 12345678910 T n 1371531631272555111023 Werecognizethoseasbeingsimplyonelessthanapoweroftwo. Toderivethisequationinsteadofjustguessingatit,wewritetheoriginal relationas )]TJ/F8 9.9626 Tf 7.749 0 Td [(1= )]TJ/F11 9.9626 Tf 7.749 0 Td [(T n +1+2 T n ,considerthehomogeneousrelation0= )]TJ/F11 9.9626 Tf 7.749 0 Td [(T n +2 T n )]TJ/F8 9.9626 Tf 9.987 0 Td [(1,getitsassociatedpolynomial )]TJ/F11 9.9626 Tf 7.748 0 Td [( +2,whichobviouslyhas thesingle,unique,rootof r 1 =2,andconcludethatfunctionssatisfyingthe homogeneousrelationtaketheform T n = c 1 2 n That'sthehomogeneoussolution.Nowweneedaparticularsolution. Becausethenonhomogeneousrelation )]TJ/F8 9.9626 Tf 7.749 0 Td [(1= )]TJ/F11 9.9626 Tf 7.749 0 Td [(T n +1+2 T n issosimple, inafewminutesorbyrememberingthetablewecanspottheparticular solution T n = )]TJ/F8 9.9626 Tf 7.749 0 Td [(1thereareotherparticularsolutions,butthisoneiseasily spotted.Sowehavethat|withoutyetconsideringtheinitialcondition|any solutionof T n +1=2 T n +1isthesumofthehomogeneoussolutionand thisparticularsolution: T n = c 1 2 n )]TJ/F8 9.9626 Tf 9.962 0 Td [(1. Theinitialcondition T =1nowgivesthat c 1 =1,andwe'vegottenthe formulathatgeneratesthetable:the n -diskTowerofHanoiproblemrequiresa minimumof2 n )]TJ/F8 9.9626 Tf 9.963 0 Td [(1moves. Findingaparticularsolutioninmorecomplicatedcasesis,naturally,more complicated.Adelightfulandrewarding,butchallenging,sourceonrecurrencerelationsis[Graham,Knuth,Patashnik].,FormoreontheTowerofHanoi, [Ball&Coxeter]or[Gardner1957]aregoodstartingpoints.Sois[Hofstadter]. Somecomputercodefortryingsomerecurrencerelationsfollowstheexercises. Exercises 1 Solveeachhomogeneouslinearrecurrencerelations. a f n +1=5 f n )]TJ/F29 8.9664 Tf 9.215 0 Td [(6 f n )]TJ/F29 8.9664 Tf 9.215 0 Td [(1 b f n +1=4 f n )]TJ/F29 8.9664 Tf 9.215 0 Td [(1 c f n +1=6 f n +7 f n )]TJ/F29 8.9664 Tf 9.215 0 Td [(1+6 f n )]TJ/F29 8.9664 Tf 9.216 0 Td [(2 2 Giveaformulafortherelationsofthepriorexercise,withtheseinitialconditions. a f =1, f =1 b f =0, f =1 c f =1, f =1, f =3. 3 Checkthattheisomorphismgivenbetwween S and R k isalinearmap.Itis arguedabovethatthismapisone-to-one.Whatisitsinverse? 4 Showthatthecharacteristicequationofthematrixisasstated,thatis,isthe polynomialassociatedwiththerelation.Hint:expandingdownthenalcolumn, andusinginductionwillwork. 5 Givenahomogeneouslinearrecurrencerelation f n +1= a n f n + + a n )]TJ/F33 5.9776 Tf 5.756 0 Td [(k f n )]TJ/F32 8.9664 Tf 8.516 0 Td [(k ,let r 1 ,..., r k betherootsoftheassociatedpolynomial. a Provethateachfunction f r i n = r n k satisestherecurrencewithoutinitial conditions. b Provethatno r i is0. c Provethattheset f f r 1 ;:::;f r k g islinearlyindependent.

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Topic:LinearRecurrences 413 6 Thisreferstothevalue T =18 ; 446 ; 744 ; 073 ; 709 ; 551 ; 615giveninthecomputercodebelow.Transferringonediskpersecond,howmanyyearswouldit takethepriestsattheTowerofHanoitonishthejob? ComputerCode Thiscodeallowsthegenerationoftherstfewvaluesofafunctiondened byarecurrenceandinitialconditions.ItisintheSchemedialectofLISP specically,itwaswrittenforA.Jaer'sfreeschemeinterpreterSCM,although itshouldruninanySchemeimplementation. First,theTowerofHanoicodeisastraightforwardimplementationofthe recurrence. definetower-of-hanoi-movesn if=n1 1 +*tower-of-hanoi-moves-n1 2 1 Noteforreadersunusedtorecursivecode:tocompute T ,thecomputeris toldtocompute2 T )]TJ/F8 9.9626 Tf 9.507 0 Td [(1,whichrequires,ofcourse,computing T .The computerputsthe`times2'andthe`plus1'asideforamomenttodothat.It computes T byusingthissamepieceofcodethat'swhat`recursive'means, andtodothatistoldtocompute2 T )]TJ/F8 9.9626 Tf 10.142 0 Td [(1.Thiskeepsupthenextstep istotrytodo T whiletheotherarithmeticisheldinwaiting,until,after 63steps,thecomputertriestocompute T .Itthenreturns T =1,which nowmeansthatthecomputationof T canproceed,etc.,upuntiltheoriginal computationof T nishes. Thenextroutinecalculatesatableoftherstfewvalues.Somelanguage notes: istheemptylist,thatis,theemptysequence,and cons pushes somethingontothestartofalist.Notethat,inthelastline,theprocedure proc iscalledonargument n definefirst-few-outputsprocn first-few-outputs-helperprocn' ; definefirst-few-outputs-auxprocnlst iffirst-few-outputstower-of-hanoi-moves64 Evaluationtook120mSec 3715316312725551110232047409581911638332767 655351310712621435242871048575209715141943038388607 167772153355443167108863134217727268435455536870911 107374182321474836474294967295858993459117179869183 3435973836768719476735137438953471274877906943549755813887

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414 ChapterFive.Similarity 1099511627775219902325555143980465111038796093022207 175921860444153518437208883170368744177663140737488355327 2814749767106555629499534213111125899906842623 225179981368524745035996273704959007199254740991 180143985094819833602879701896396772057594037927935 144115188075855871288230376151711743576460752303423487 115292150460684697523058430092136939514611686018427387903 922337203685477580718446744073709551615 Thisisalistof T through T .The120mSeccameona50mHz'486 runninginanXTermofXWindowunderLinux.Thesessionwaseditedtoput linebreaksbetweennumbers.

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Appendix Mathematicsismadeofargumentsreasoneddiscoursethatis,notcrockerythrowing.Thissectionisareferencetothemostusedtechniques.Areader havingtroublewith,say,proofbycontradiction,canturnhereforanoutlineof thatmethod. Butthissectiongivesonlyasketch.Formore,theseareclassics: Methods ofLogic byQuine, InductionandAnalogyinMathematics byPolya,and Naive SetTheory byHalmos. IV.3Propositions Thepointatissueinanargumentisthe proposition .Mathematiciansusually writethepointinfullbeforetheproofandlabeliteither Theorem formajor points, Corollary forpointsthatfollowimmediatelyfromapriorone,or Lemma forresultschieyusedtoproveotherresults. Thestatementsexpressingpropositionscanbecomplex,withmanysubparts. Thetruthorfalsityoftheentirepropositiondependsbothonthetruthvalue oftheparts,andonthewordsusedtoassemblethestatementfromitsparts. Not. Forexample,where P isaproposition,`itisnotthecasethat P 'is trueprovidedthat P isfalse.Thus,` n isnotprime'istrueonlywhen n isthe productofsmallerintegers. Wecanpicturethe`not'operationwitha Venndiagram P Wheretheboxenclosesallnaturalnumbers,andinsidethecirclearetheprimes, theshadedareaholdsnumberssatisfying`not P '. Toprovethata`not P 'statementholds,showthat P isfalse. A-1

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A-2 And. Considerthestatementform` P and Q '.Forthestatementtobetrue bothhalvesmusthold:`7isprimeandsois3'istrue,while`7isprimeand3 isnot'isfalse. HereistheVenndiagramfor` P and Q '. P Q Toprove` P and Q ',provethateachhalfholds. Or. A` P or Q 'istruewheneitherhalfholds:`7isprimeor4isprime'is true,while`7isnotprimeor4isprime'isfalse.Wetake`or'inclusivelysothat ifbothhalvesaretrue`7isprimeor4isnot'thenthestatementasawholeis true.Ineverydayspeech,sometimes`or'ismeantinanexclusiveway|Eat yourvegetablesornodessert"doesnotintendbothhalvestohold|butwe willnotuse`or'inthatway. TheVenndiagramfor`or'includesallofbothcircles. P Q Toprove` P or Q ',showthatinallcasesatleastonehalfholdsperhaps sometimesonehalfandsometimestheother,butalwaysatleastone. If-then. An`if P then Q 'statementsometimeswritten` P materiallyimplies Q 'orjust` P implies Q 'or` P = Q 'istrueunless P istruewhile Q isfalse. Thus`if7isprimethen4isnot'istruewhile`if7isprimethen4isalsoprime' isfalse.Contrarytoitsuseincasualspeech,inmathematics`if P then Q 'does notconnotethat P precedes Q orcauses Q Moresubtly,inmathematics`if P then Q 'istruewhen P isfalse:`if4is primethen7isprime'and`if4isprimethen7isnot'arebothtruestatements, sometimessaidtobe vacuouslytrue .Weadoptthisconventionbecausewewant statementslike`ifanumberisaperfectsquarethenitisnotprime'tobetrue, forinstancewhenthenumberis5orwhenthenumberis6. Thediagram P Q

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A-3 showsthat Q holdswhenever P doesanotherphrasingis` P issucienttogive Q '.Noticeagainthatif P doesnothold, Q mayormaynotbeinforce. Therearetwomainwaystoestablishanimplication.Therstwayisdirect: assumethat P istrueand,usingthatassumption,prove Q .Forinstance,to show`ifanumberisdivisibleby5thentwicethatnumberisdivisibleby10', assumethatthenumberis5 n anddeducethat2 n =10 n .Thesecondway isindirect:provethe contrapositive statement:`if Q isfalsethen P isfalse' rephrased,` Q canonlybefalsewhen P isalsofalse'.Asanexample,toshow `ifanumberisprimethenitisnotaperfectsquare',arguethatifitwerea square p = n 2 thenitcouldbefactored p = n n where n


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A-4 Forall. The`forall'prexisthe universalquantier ,symbolized 8 Venndiagramsaren'tveryhelpfulwithquantiers,butinasensethebox wedrawtoborderthediagramshowstheuniversalquantiersinceitdilineates theuniverseofpossiblemembers. Toprovethatastatementholdsinallcases,wemustshowthatitholdsin eachcase.Thus,toprove`everynumberdivisibleby p hasitssquaredivisible by p 2 ',takeasinglenumberoftheform pn andsquareit pn 2 = p 2 n 2 .Thisis atypicalelement"orgenericelement"proof. Thiskindofargumentrequiresthatwearecarefultonotassumeproperties forthatelementotherthanthoseinthehypothesis|forinstance,thistypeof wrongargumentisacommonmistake:if n isdivisiblebyaprime,say2,so that n =2 k then n 2 = k 2 =4 k 2 andthesquareofthenumberisdivisible bythesquareoftheprime".Thatisanargumentaboutthecase p =2,butit isn'taproofforgeneral p Thereexists. Wewillalsousethe existentialquantier ,symbolized 9 and read`thereexists'. Asnotedabove,Venndiagramsarenotmuchhelpwithquantiers,buta pictureof`thereisanumbersuchthat P 'wouldshowboththattherecanbe morethanoneandthatnotallnumbersneedsatisfy P P Anexistencepropositioncanbeprovedbyproducingsomethingsatisfying theproperty:once,tosettlethequestionofprimalityof2 2 5 +1,Eulerproduced itsdivisor641.Butthereareproofsshowingthatsomethingexistswithoutsayinghowtondit;Euclid'sargumentgiveninthenextsubsectionshowsthere areinnitelymanyprimeswithoutnamingthem.Ingeneral,whiledemonstratingexistenceisbetterthannothing,givinganexampleisbetter,andan exhaustivelistofallinstancesisgreat.Still,mathematicianstakewhatthey canget. Finally,alongwithArethereany?"weoftenaskHowmany?"That iswhytheissueofuniquenessoftenarisesinconjunctionwithquestionsof existence.Manytimesthetwoargumentsaresimplerifseparated,sonotethat justasprovingsomethingexistsdoesnotshowitisunique,neitherdoesproving somethingisuniqueshowthatitexists.Obviously`thenaturalnumberwith

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A-5 morefactorsthananyother'wouldbeunique,butinfactnosuchnumber exists. IV.5TechniquesofProof Induction. Manyproofsareiterative,Here'swhythestatementistruefor forthecaseofthenumber1,itthenfollowsfor2,andfromthereto3,andso on...".Thesearecalledproofsby induction .Suchaproofhastwosteps.In the basestep thepropositionisestablishedforsomerstnumber,often0or1. Theninthe inductivestep weassumethatthepropositionholdsfornumbers uptosome k anddeducethatitthenholdsforthenextnumber k +1. Hereisanexample. Wewillprovethat1+2+3+ + n = n n +1 = 2. Forthebasestepwemustshowthattheformulaholdswhen n =1. That'seasy,thesumoftherst1numberdoesindeedequal1+1 = 2. Fortheinductivestep,assumethattheformulaholdsforthenumbers 1 ; 2 ;:::;k .Thatis,assumealloftheseinstancesoftheformula. 1=1+1 = 2 and1+2=2+1 = 2 and1+2+3=3+1 = 2 and1+ + k = k k +1 = 2 Fromthisassumptionwewilldeducethattheformulathereforealsoholds inthe k +1nextcase.Thedeductionisstraightforwardalgebra. 1+2+ + k + k +1= k k +1 2 + k +1= k +1 k +2 2 We'veshowninthebasecasethattheabovepropositionholdsfor1.We've shownintheinductivestepthatifitholdsforthecaseof1thenitalsoholds for2;thereforeitdoesholdfor2.We'vealsoshownintheinductivestepthat ifthestatementholdsforthecasesof1and2thenitalsoholdsforthenext case3,etc.Thusitholdsforanynaturalnumbergreaterthanorequalto1. Hereisanotherexample. Wewillprovethateveryintegergreaterthan1isaproductofprimes. Thebasestepiseasy:2istheproductofasingleprime. Fortheinductivestepassumethateachof2 ; 3 ;:::;k isaproductof primes,aimingtoshow k +1isalsoaproductofprimes.Therearetwo

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A-6 possibilities:iif k +1isnotdivisiblebyanumbersmallerthanitself thenitisaprimeandsoistheproductofprimes,andiiif k +1is divisiblethenitsfactorscanbewrittenasaproductofprimesbythe inductivehypothesisandso k +1canberewrittenasaproductofprimes. Thatendstheproof. Remark. ThePrimeFactorizationTheoremofNumberTheorysaysthat notonlydoesafactorizationexist,butthatitisunique.We'veshownthe easyhalf. Therearetwothingstonoteaboutthe`nextnumber'inaninductionargument. Foronething,whileinductionworksontheintegers,it'snogoodonthe reals.Thereisno`next'real. Theotherthingisthatwesometimesuseinductiontogodown,say,from10 to9to8,etc.,downto0.So`nextnumber'couldmean`nextlowestnumber'. Ofcourse,attheendwehavenotshownthefactforallnaturalnumbers,only forthoselessthanorequalto10. Contradiction. Anothertechniqueofproofistoshowsomethingistrueby showingitcan'tbefalse. TheclassicexampleisEuclid's,thatthereareinnitelymanyprimes. Supposethereareonlynitelymanyprimes p 1 ;:::;p k .Consider p 1 p 2 :::p k +1.Noneoftheprimesonthissupposedlyexhaustivelistdivides thatnumberevenly,eachleavesaremainderof1.Buteverynumberis aproductofprimessothiscan'tbe.Thustherecannotbeonlynitely manyprimes. Everyproofbycontradictionhasthesameform:assumethattheproposition isfalseandderivesomecontradictiontoknownfacts. Anotherexampleisthisproofthat p 2isnotarationalnumber. Supposethat p 2= m=n 2 n 2 = m 2 Factoroutthe2's: n =2 k n ^ n and m =2 k m ^ m andrewrite. 2 k n ^ n 2 = k m ^ m 2 ThePrimeFactorizationTheoremsaysthattheremustbethesamenumberoffactorsof2onbothsides,butthereareanoddnumber1+2 k n on theleftandanevennumber2 k m ontheright.That'sacontradiction,so arationalwithasquareof2cannotbe. Bothoftheseexamplesaimedtoprovesomethingdoesn'texist.Anegative propositionoftensuggestsaproofbycontradiction.

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A-7 IV.6Sets,Functions,andRelations Sets. Mathematiciansworkwithcollectionscalled sets .Asetcanbegiven asalistingbetweencurlybracesasin f 1 ; 4 ; 9 ; 16 g ,or,ifthat'sunwieldy,by usingset-buildernotationasin f x x 5 )]TJ/F8 9.9626 Tf 9.963 0 Td [(3 x 3 +2=0 g readthesetofall x suchthat...".Wenamesetswithcapitalromanlettersaswiththeprimes P = f 2 ; 3 ; 5 ; 7 ; 11 ;::: g ,exceptforafewspecialsetssuchastherealnumbers R ,andthecomplexnumbers C .Todenotethatsomethingisan element or member ofasetweuse` 2 ',sothat7 2f 3 ; 5 ; 7 g while8 62f 3 ; 5 ; 7 g WhatdistinguishesasetfromanyothertypeofcollectionisthePrinciple ofExtensionality,thattwosetswiththesameelementsareequal.Becauseof thisprinciple,inasetrepeatscollapse f 7 ; 7 g = f 7 g andorderdoesn'tmatter f 2 ; g = f ; 2 g Weuse` 'forthesubsetrelationship: f 2 ; gf 2 ;; 7 g and` 'forsubset orequalityif A isasubsetof B but A 6 = B then A isa propersubset of B Thesesymbolsmaybeipped,forinstance f 2 ;; 5 gf 2 ; 5 g BecauseofExtensionality,toprovethattwosetsareequal A = B ,justshow thattheyhavethesamemembers.Usuallyweshowmutualinclusion,thatboth A B and A B Setoperations. Venndiagramsarehandyhere.Forinstance, x 2 P canbe pictured P x and` P Q 'lookslikethis. P Q Notethatthisisarepeatofthediagramfor`if...then...'propositions.That's because` P Q 'means`if x 2 P then x 2 Q '. Ingeneral,foreverypropositionallogicoperatorthereisanassociatedset operator.Forinstance,the complement of P is P comp = f x not x 2 P g P

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A-8 the union is P [ Q = f x x 2 P or x 2 Q g P Q andthe intersection is P Q = f x x 2 P and x 2 Q g : P Q Whentwosetssharenomemberstheirintersectionisthe emptyset fg symbolized ? .Anysethastheemptysetforasubset,bythe`vacuouslytrue' propertyofthedenitionofimplication. Sequences. Weshallalsousecollectionswhereorderdoesmatterandwhere repeatsdonotcollapse.Theseare sequences ,denotedwithanglebrackets: h 2 ; 3 ; 7 i6 = h 2 ; 7 ; 3 i .Asequenceoflength2issometimescalledan orderedpair andwrittenwithparentheses: ; 3.Wealsosometimessay`orderedtriple', `ordered4-tuple',etc.Thesetofordered n -tuplesofelementsofaset A is denoted A n .Thusthesetofpairsofrealsis R 2 Functions. WerstseefunctionsinelementaryAlgebra,wheretheyarepresentedasformulase.g., f x =16 x 2 )]TJ/F8 9.9626 Tf 9.602 0 Td [(100,butprogressingtomoreadvanced Mathematicsrevealsmoregeneralfunctions|trigonometricones,exponential andlogarithmicones,andevenconstructslikeabsolutevaluethatinvolvepiecingtogetherparts|andweseethatfunctionsaren'tformulas,insteadthekey ideaisthatafunctionassociateswithitsinput x asingleoutput f x Consequently,a function or map isdenedtobeasetoforderedpairs x;f x suchthat x sucestodetermine f x ,thatis:if x 1 = x 2 then f x 1 = f x 2 thisrequirementisreferredtobysayingafunctionis well-dened Eachinput x isoneofthefunction's arguments andeachoutput f x isa value .Thesetofallargumentsis f 's domain andthesetofoutputvaluesis its range .Usuallywedon'tneedknowwhatisandisnotintherangeandwe insteadworkwithasupersetoftherange,the codomain .Thenotationfora function f withdomain X andcodomain Y is f : X Y Moreonthisisinthesectiononisomorphisms

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A-9 Wesometimesinsteadusethenotation x f 7)167(! 16 x 2 )]TJ/F8 9.9626 Tf 10.147 0 Td [(100,read` x mapsunder f to16 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(100',or`16 x 2 )]TJ/F8 9.9626 Tf 9.963 0 Td [(100isthe image of x '. Somemaps,like x 7! sin =x ,canbethoughtofascombinationsofsimple maps,here, g y =sin y appliedtotheimageof f x =1 =x .The composition of g : Y Z with f : X Y ,isthemapsending x 2 X to g f x 2 Z .Itis denoted g f : X Z .Thisdenitiononlymakessenseiftherangeof f isa subsetofthedomainof g Observethatthe identitymap id: Y Y denedbyid y = y hasthe propertythatforany f : X Y ,thecompositionid f isequalto f .Soan identitymapplaysthesamerolewithrespecttofunctioncompositionthat thenumber0playsinrealnumberaddition,orthatthenumber1playsin multiplication. Inlinewiththatanalogy,denea leftinverse ofamap f : X Y tobea function g :range f X suchthat g f istheidentitymapon X .Ofcourse, a rightinverse of f isa h : Y X suchthat f h istheidentity. Amapthatisbothaleftandrightinverseof f iscalledsimplyan inverse Aninverse,ifoneexists,isuniquebecauseifboth g 1 and g 2 areinversesof f then g 1 x = g 1 f g 2 x = g 1 f g 2 x = g 2 x themiddleequality comesfromtheassociativityoffunctioncomposition,soweoftencallitthe" inverse,written f )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 .Forinstance,theinverseofthefunction f : R R given by f x =2 x )]TJ/F8 9.9626 Tf 9.963 0 Td [(3isthefunction f )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 : R R givenby f )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 x = x +3 = 2. Thesuperscript` f )]TJ/F7 6.9738 Tf 6.227 0 Td [(1 'notationforfunctioninversecanbeconfusing|it doesn'tmean1 =f x .Itisusedbecauseittsintoalargerscheme.Functionsthathavethesamecodomainasdomaincanbeiterated,sothatwhere f : X X ,wecanconsiderthecompositionof f withitself: f f ,and f f f etc.Naturallyenough,wewrite f f as f 2 and f f f as f 3 ,etc.Note thatthefamiliarexponentrulesforrealnumbersobviouslyhold: f i f j = f i + j and f i j = f i j .Therelationshipwiththepriorparagraphisthat,where f is invertible,writing f )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 fortheinverseand f )]TJ/F7 6.9738 Tf 6.227 0 Td [(2 fortheinverseof f 2 ,etc.,gives thatthesefamiliarexponentrulescontinuetohold,once f 0 isdenedtobethe identitymap. Ifthecodomain Y equalstherangeof f thenwesaythatthefunctionis onto Afunctionhasarightinverseifandonlyifitisontothisisnothardtocheck. Ifnotwoargumentsshareanimage,if x 1 6 = x 2 impliesthat f x 1 6 = f x 2 thenthefunctionis one-to-one .Afunctionhasaleftinverseifandonlyifitis one-to-onethisisalsonothardtocheck. Bythepriorparagraph,amaphasaninverseifandonlyifitisbothonto andone-to-one;suchafunctionisa correspondence .Itassociatesoneandonly oneelementofthedomainwitheachelementoftherangeforexample,nite

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A-10 setsmusthavethesamenumberofelementstobematchedupinthisway. Becauseacompositionofone-to-onemapsisone-to-one,andacompositionof ontomapsisonto,acompositionofcorrespondencesisacorrespondence. Wesometimeswanttoshrinkthedomainofafunction.Forinstance,we maytakethefunction f : R R givenby f x = x 2 and,inordertohavean inverse,limitinputargumentstononnegativereals ^ f : R + R .Technically, ^ f isadierentfunctionthan f ;wecallitthe restriction of f tothesmaller domain. Analpointonfunctions:neither x nor f x needbeanumber.Asan example,wecanthinkof f x;y = x + y asafunctionthattakestheordered pair x;y asitsargument. Relations. Somefamiliaroperationsareobviouslyfunctions:additionmaps ; 3to8.Butwhatof` < 'or`='?Weheretaketheapproachofrephrasing `3 < 5'to` ; 5isintherelation < '.Thatis,denea binaryrelation onaset A tobeasetoforderedpairsofelementsof A .Forexample,the < relationis theset f a;b a
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A-11 ... S 0 S 1 S 2 S 3 Thus,therstparagraphsays`samesign'partitionstheintegersintothepositivesandthenegatives.Similarly,theequivalencerelation`='partitionsthe integersintoone-elementsets. Anotherexampleisthefractions.Ofcourse,2 = 3and4 = 6areequivalent fractions.Thatis,fortheset S = f n=d n;d 2 Z and d 6 =0 g ,wedenetwo elements n 1 =d 1 and n 2 =d 2 tobeequivalentif n 1 d 2 = n 2 d 1 .Wecancheckthat thisisanequivalencerelation,thatis,thatitsatisestheabovethreeconditions. Withthat, S isdividedupintoparts. ... : 0 = 1 : 0 = 3 : 1 = 1 : 2 = 2 : 2 = 4 : )]TJ/F6 4.9813 Tf 5.396 0 Td [(2 = )]TJ/F6 4.9813 Tf 5.397 0 Td [(4 : 4 = 3 : 8 = 6 Beforeweshowthatequivalencerelationsalwaysgiverisetopartitions, werstillustratetheargument.Considertherelationshipbetweentwointegersof`sameparity',theset f )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; 3 ; ; 4 ; ; 0 ;::: g i.e.,`givethesame remainderwhendividedby2'.Wewanttosaythatthenaturalnumbers splitintotwopieces,theevensandtheodds,andinsideapieceeachmemberhasthesameparityaseachother.Soforeach x wedenethesetof numbersassociatedwithit: S x = f y x;y 2 `sameparity' g .Someexamplesare S 1 = f :::; )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ; )]TJ/F8 9.9626 Tf 7.748 0 Td [(1 ; 1 ; 3 ;::: g ,and S 4 = f :::; )]TJ/F8 9.9626 Tf 7.749 0 Td [(2 ; 0 ; 2 ; 4 ;::: g ,and S )]TJ/F7 6.9738 Tf 6.226 0 Td [(1 = f :::; )]TJ/F8 9.9626 Tf 7.749 0 Td [(3 ; )]TJ/F8 9.9626 Tf 7.749 0 Td [(1 ; 1 ; 3 ;::: g .Thesearetheparts,e.g., S 1 istheodds. Theorem. Anequivalencerelationinducesapartitionontheunderlyingset. Proof Calltheset S andtherelation R .Inlinewiththeillustrationinthe paragraphabove,foreach x 2 S dene S x = f y x;y 2 R g Observethat,as x isamemberif S x ,theunionofallthesesetsis S .So wewillbedoneifweshowthatdistinctpartsaredisjoint:if S x 6 = S y then S x S y = ? .Wewillverifythisthroughthecontrapositive,thatis,wewlll assumethat S x S y 6 = ? inordertodeducethat S x = S y Let p beanelementoftheintersection.Thenbydenitionof S x and S y ,the two x;p and y;p aremembersof R ,andbysymmetryofthisrelation p;x and p;y arealsomembersof R .Toshowthat S x = S y wewillshoweachisa subsetoftheother. Assumethat q 2 S x sothat q;x 2 R .Usetransitivityalongwith x;p 2 R toconcludethat q;p isalsoanelementof R .But p;y 2 R soanotheruse oftransitivitygivesthat q;y 2 R .Thus q 2 S y .Therefore q 2 S x implies q 2 S y ,andso S x S y Thesameargumentintheotherdirectiongivestheotherinclusion,andso thetwosetsareequal,completingthecontrapositiveargument. QED

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A-12 Wecalleachpartofapartitionan equivalenceclass orinformally,`part'. Wesomtimespickasingleelementofeachequivalenceclasstobethe class representative ... ? ? ? ? Usuallywhenwepickrepresentativeswehavesomenaturalschemeinmind.In thatcasewecallthemthe canonical representatives. Anexampleisthesimplestformofafraction.We'vedened3 = 5and9 = 15 tobeequivalentfractions.Ineverydayworkweoftenusethe`simplestform'or `reducedform'fractionastheclassrepresentatives. ... ? 0 = 1 ? 1 = 1 ? 1 = 2 ? 4 = 3

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Bibliography [Abbot]EdwinAbbott, Flatland ,Dover. [Ackerson]R.H.Ackerson, ANoteonVectorSpaces ,AmericanMathematical Monthly,volume62number10Dec.1955,p.721. [Agnew]JeanneAgnew, ExplorationsinNumberTheory ,Brooks/Cole,1972. [Am.Math.Mon.,Jun.1931]C.A.Ruppproposer,H.T.R.Audesolver,problem 3468,AmericanMathematicalMonthly,volume37number6June-July1931, p.355. [Am.Math.Mon.,Feb.1933]V.F.Ivanoproposer,T.C.Estysolver,problem 3529,AmericanMathematicalMonthly,volume39number2Feb.1933, p.118. [Am.Math.Mon.,Jan.1935]W.R.Ransomproposer,HansrajGuptasolver,Elementaryproblem105,AmericanMathematicalMonthly,volume42number 1Jan.1935,p.47. [Am.Math.Mon.,Jan.1949]C.W.Triggproposer,R.J.Walkersolver,Elementaryproblem813,AmericanMathematicalMonthly,volume56number1Jan. 1949,p.33. [Am.Math.Mon.,Jun.1949]DonWalterproposer,AlexTytunsolver,Elementaryproblem834,AmericanMathematicalMonthly,volume56number6 June-July1949,p.409. [Am.Math.Mon.,Nov.1951]AlbertWilansky, TheRow-SumsoftheInverseMatrix AmericanMathematicalMonthly,volume58number9Nov.1951,p.614. [Am.Math.Mon.,Feb.1953]NormanAnningproposer,C.W.Triggsolver,Elementaryproblem1016,AmericanMathematicalMonthly,volume60number 2Feb.1953,p.115. [Am.Math.Mon.,Apr.1955]VernHaggettproposer,F.W.Saunderssolver,Elementaryproblem1135,AmericanMathematicalMonthly,volume62number 4Apr.1955,p.257. [Am.Math.Mon.,Jan.1963]UnderwoodDudley,ArnoldLebowproposers,David Rothmansolver,Elemantaryproblem1151,AmericanMathematical Monthly,volume70number1Jan.1963,p.93. [Am.Math.Mon.,Dec.1966]HansLiebeck, AProofoftheEqualityofColumnRank andRowRankofaMatrix AmericanMathematicalMonthly,volume73number10Dec.1966,p.1114.

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[Anton]HowardAnton, ElementaryLinearAlgebra ,JohnWiley&Sons,1987. [Arrow]KennethJ.Arrow, SocialChoiceandIndividualValues ,Wiley,1963. [Ball&Coxeter]W.W.RouseBall, MathematicalRecreationsandEssays ,revisedby H.S.M.Coxeter,MacMillan,1962. [Bennett]WilliamJ.Bennett, QuantifyingAmerica'sDecline ,in WallStreetJournal 15March,1993. [Birkho&MacLane]GarrettBirkho,SaundersMacLane, SurveyofModernAlgebra ,thirdedition,Macmillan,1965. [Blass1984]A.Blass, ExistenceofBasesImpliestheAxiomofChoice ,pp.31{33,AxiomaticSetTheory,J.E.Baumgartner,ed.,AmericanMathematicalSociety, ProvidenceRI,1984. [Bridgman]P.W.Bridgman, DimensionalAnalysis ,YaleUniversityPress,1931. [Casey]JohnCasey, TheElementsofEuclid,BooksItoVIandXI ,ninthedition, Hodges,Figgis,andCo.,Dublin,1890. [Clark&Coupe]DavidH.Clark,JohnD.Coupe, TheBangorAreaEconomyIts PresentandFuture ,reporttothecityofBangorME,Mar.1967. [Clarke]ArthurC.Clarke, TechnicalError ,Fantasy,December1946,reprintedin GreatSFStories8,DAWBooks,1982. [Con.Prob.1955] TheContestProblemBook ,1955number38. [Coxeter]H.S.M.Coxeter, ProjectiveGeometry ,secondedition,Springer-Verlag,1974. [Courant&Robbins]RichardCourant,HerbertRobbins, WhatisMathematics? ,OxfordUniversityPress,1978. [Cullen]CharlesG.Cullen, MatricesandLinearTransformations ,secondedition, Dover,1990. [Dalal,et.al.]SiddharthaR.Dalal,EdwardB.Fowlkes,&BruceHoadley, Lesson LearnedfromChallenger:AStatisticalPerspective ,Stats:theMagazinefor StudentsofStatistics,Fall1989,p.3. [Davies]ThomasD.Davies, NewEvidencePlacesPearyatthePole ,NationalGeographicMagazine,vol.177no.1Jan.1990,p.44. [deMestre]NevilledeMestre, TheMathematicsofProjectilesinSport ,Cambridge UniversityPress,1990. [DeParville]DeParville, LaNature ,Paris,1884,partI,p.285{286citationfrom [Ball&Coxeter]. [Duncan]W.J.Duncan, MethodofDimensions ,in EncyclopaedicDictionaryof Physics Macmillan,1962. [Ebbing]DarrellD.Ebbing, GeneralChemistry ,fourthedition,HoughtonMiin, 1993. [Ebbinghaus]H.D.Ebbinghaus, Numbers ,Springer-Verlag,1990. [Einstein]A.Einstein,AnnalsofPhysics,v.35,1911,p.686. [Eggar]M.H.Eggar, PinholeCameras,Perspecitve,andProjectiveGeometry ,AmericanMathematicslMonthly,August-September1998,p.618{630.

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[Feller]WilliamFeller, AnIntroductiontoProbabilityTheoryandItsApplications vol.1,3rded.,JohnWiley,1968. [Finkbeiner]DanielT.FinkbeinerIII, IntroductiontoMatricesandLinearTransformations ,thirdedition,W.H.FreemanandCompany,1978. [Fraleigh&Beauregard]Fraleigh&Beauregard, LinearAlgebra [Gardner]MartinGardner, TheNewAmbidextrousUniverse ,thirdrevisededition, W.H.FreemanandCompany,1990. [Gardner1957]MartinGardner, MathematicalGames:AbouttheremarkablesimilaritybetweentheIcosianGameandtheTowerofHanoi ,ScienticAmerican, May1957,p.150{154. [Gardner,1970]MartinGardner, MathematicalGames,Somemathematicalcuriositiesembeddedinthesolarsystem ,ScienticAmerican,April1970,p.108{ 112. [Gardner,1980]MartinGardner, MathematicalGames,Fromcountingvotestomakingvotescount:themathematicsofelections ,ScienticAmerican,October 1980. [Gardner,1974]MartinGardner, MathematicalGames,Ontheparadoxicalsituations thatarisefromnontransitiverelations ,ScienticAmerican,October1974. [Giordano,Wells,Wilde]FrankR.Giordano,MichaelE.Wells,CarrollO.Wilde, DimensionalAnalysis ,UMAPUnit526,in UMAPModules,1987 ,COMAP,1987. [Giordano,Jaye,Weir]FrankR.Giordano,MichaelJ.Jaye,MauriceD.Weir, The UseofDimensionalAnalysisinMathematicalModeling ,UMAPUnit632,in UMAPModules,1986 ,COMAP,1986. [Glazer]A.M.Glazer, TheStructureofCrystals ,AdamHilger,1987. [Goult, etal. ]R.J.Goult,R.F.Hoskins,J.A.Milner,M.J.Pratt, Computational MethodsinLinearAlgebra ,Wiley,1975. [Graham,Knuth,Patashnik]RonaldL.Graham,DonaldE.Knuth,OrenPatashnik, ConcreteMathematics ,Addison-Wesley,1988. [Halmos]PaulP.Halmos, FiniteDimensionalVectorSpaces ,secondedition,VanNostrand,1958. [Hamming]RichardW.Hamming, IntroductiontoAppliedNumericalAnalysis ,HemispherePublishing,1971. [Hanes]KitHanes, AnalyticProjectiveGeometryanditsApplications ,UMAPUnit 710,UMAPModules,1990,p.111. [Heath]T.Heath, Euclid'sElements ,volume1,Dover,1956. [Homan&Kunze]KennethHoman,RayKunze, LinearAlgebra ,secondedition, Prentice-Hall,1971. [Hofstadter]DouglasR.Hofstadter, MetamagicalThemas:QuestingfortheEssence ofMindandPattern ,BasicBooks,1985. [Iosifescu]MariusIofescu, FiniteMarkovProcessesandTheirApplications ,JohnWiley,1980. [Kelton]ChristinaM.L.Kelton, TrendsontheRelocationofU.S.Manufacturing ,UMI ResearchPress,1983.

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[Kemeny&Snell]JohnG.Kemeny,J.LaurieSnell, FiniteMarkovChains ,D.Van Nostrand,1960. [Kemp]FranklinKemp LinearEquations ,AmericanMathematicalMonthly,volume 89number8Oct.1982,p.608. [Knuth]DonaldE.Knuth, TheArtofComputerProgramming ,AddisonWesley,1988. [Lang]SergeLang, LinearAlgebra ,Addison-Wesley,1966. [Leontief1951]WassilyW.Leontief, Input-OutputEconomics ,ScienticAmerican, volume185number4Oct.1951,p.15. [Leontief1965]WassilyW.Leontief, TheStructureoftheU.S.Economy ,Scientic American,volume212number4Apr.1965,p.25. [Lieberman]DavidLieberman, TheBigBucksAdBattlesOverTV'sMostExpensive Minutes ,TVGuide,Jan.261991,p.11. [Macdonald&Ridge]KennethMacdonald,JohnRidge, SocialMobility ,in British SocialTrendsSince1900 ,A.H.Halsey,Macmillian,1988. [MacmillanDictionary]WilliamD.Halsey,Macmillan,1979. [Math.Mag.,Sept.1952]DeweyDuncanproposer,W.H.Quelchsolver,MathematicsMagazine,volume26number1Sept-Oct.1952,p.48. [Math.Mag.,Jan.1957]M.S.Klamkinproposer,TrickieT-27,MathematicsMagazine,volume30number3Jan-Feb.1957,p.173. [Math.Mag.,Jan.1963,Q237]D.L.Silvermanproposer,C.W.Triggsolver, Quickie237,MathematicsMagazine,volume36number1Jan.1963. [Math.Mag.,Jan.1963,Q307]C.W.Triggproposer.Quickie307,Mathematics Magazine,volume36number1Jan.1963,p.77. [Math.Mag.,Nov.1967]ClarenceC.Morrisonproposer,Quickie,Mathematics Magazine,volume40number4Nov.1967,p.232. [Math.Mag.,Jan.1973]MarvinBittingerproposer,Quickie578,Mathematics Magazine,volume46number5Jan.1973,p.286,296. [Munkres]JamesR.Munkres, ElementaryLinearAlgebra ,Addison-Wesley,1964. [Neimi&Riker]RichardG.Neimi,WilliamH.Riker, TheChoiceofVotingSystems ScienticAmerican,June1976,p.21{27. [Nering]EvarD.Nering, LinearAlgebraandMatrixTheory ,secondedition,John Wiley,1970. [Niven&Zuckerman]I.Niven,H.Zuckerman, AnIntroductiontotheTheoryofNumbers ,thirdedition,JohnWiley,1972. [Oakley&Baker]CletusO.Oakley,JustineC.Baker, LeastSquaresandthe 3:40 Mile ,MathematicsTeacher,Apr.1977. [Ohanian]HansO'Hanian, Physics ,volumeone,W.W.Norton,1985. [Onan]Onan, LinearAlgebra [Petersen]G.M.Petersen, AreaofaTriangle ,AmericanMathematicalMonthly,volume62number4Apr.1955,p.249. [Polya]G.Polya, PatternsofPlausibleInference ,

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[Poundstone]W.Poundstone, GamingtheVote ,HillandWang,2008.ISBN-13:9780-8090-4893-9 [Programmer'sRef.] MicrosoftProgrammersReference MicrosoftPress. [Putnam,1988,A-6]WilliamLowellPutnamMathematicalCompetition,ProblemA6,1988,solution:AmericanMathematicalMonthly,volume96number8Oct. 1989,p.688. [Putnam,1990,A-5]WilliamLowellPutnamMathematicalCompetition,ProblemA5,1990. [Rice]JohnR.Rice, NumericalMathods,Software,andAnalysis ,secondedition, AcademicPress,1993. [Quine]W.V.Quine, MethodsofLogic ,fourthedition,HarvardUniversityPress, 1982. [Rucker]RudyRucker, InnityandtheMind ,Birkhauser,1982. [Ryan]PatrickJ.Ryan, EuclideanandNon-EuclideanGeometry:anAnalyticApproach ,CambridgeUniversityPress,1986. [Seidenberg]A.Seidenberg, LecturesinProjectiveGeometry ,VanNostrand,1962. [Smith]LarrySmith, LinearAlgebra ,secondedition,Springer-Verlag,1984. [Strang93]GilbertStrang TheFundamentalTheoremofLinearAlgebra ,American MathematicalMonthly,Nov.1993,p.848{855. [Strang80]GilbertStrang, LinearAlgebraanditsApplications ,secondedition,HarcourtBraceJovanovich,1980. [Taylor]AlanD.Taylor, MathematicsandPolitics:Strategy,Voting,Power,and Proof ,Springer-Verlag,1995. [Tilley]BurtTilley,privatecommunication,1996. [Trono]TonyTrono,compiler, UniversityofVermontMathematicsDepartmentHigh SchoolPrizeExaminations1958-1991 ,mimeographedprinting,1991. [USSROlympiadno.174] TheUSSRMathematicsOlympiad ,number174. [Weston]J.D.Weston, VolumeinVectorSpaces ,AmericanMathematicalMonthly, volume66number7Aug./Sept.1959,p.575{577. [Weyl]HermannWeyl, Symmetry ,PrincetonUniversityPress,1952. [Wickens]ThomasD.Wickens, ModelsforBehavior ,W.H.Freeman,1982. [Wilkinson1965]Wilkinson,1965 [Wohascumno.2] TheWohascumCountyProblemBook problemnumber2. [Wohascumno.47] TheWohascumCountyProblemBook problemnumber47. [Yaglom]I.M.Yaglom, FelixKleinandSophusLie:EvolutionoftheIdeaofSymmetry intheNineteenthCentury ,translatedbySergeiSossinsky,Birkhauser,1988. [Zwicker]WilliamS.Zwicker, TheVoters'Paradox,Spin,andtheBordaCount ,MathematicalSocialSciences,vol.2291,p.187{227.

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Index accuracy ofGauss'method,68{71 roundingerror,69 addition vector,80 additiveinverse,80 adjointmatrix,328 angle,42 antipodal,340 antisymmetricmatrix,138 argument,A-8 arrowdiagram,217,233,238,242,353 augmentedmatrix,14 automorphism,163 dilation,163 reection,163 rotation,163 back-substitution,5 basestep ofinduction,A-5 basis,112{123 changeof,238 denition,112 orthogonal,256 orthogonalization,257 orthonormal,258 standard,113,352 standardoverthecomplexnumbers, 352 string,373 besttline,269 blockmatrix,311 box,321 orientation,321 sense,321 volume,321 Clanguage,68 canonicalform formatrixequivalence,245 fornilpotentmatrices,376 forrowequivalence,58 forsimilarity,394 canonicalrepresentative,A-12 Cauchy-SchwartzInequality,41 Cayley-Hamiltontheorem,384 centralprojection,337 changeofbasis,238{249 characteristic vectors,values,359 characteristicequation,362 characteristicpolynomial,362 characterized,172 characterizes,246 Chemistryproblem,1,9 chemistryproblem,22 circuits parallel,73 series,73 series-parallel,74 class equivalence,A-12 closure,95 ofnullspace,369 ofrangespace,369 codomain,A-8 cofactor,327 column,13 rank,125 vector,15 columnrank full,130 columnspace,125 complement,A-7 complementarysubspaces,135 orthogonal,263 complexnumbers vectorspaceover,90 component,15

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composition,A-9 self,367 computeralgebrasystems,62{63 concatenation,133 conditioningnumber,71 congruentgures,287 congruentplanegures,287 contradiction,A-6 contrapositive,A-3 convexset,183 coordinates homogeneous,340 withrespecttoabasis,115 corollary,A-1 correspondence,161,A-9 coset,193 Cramer'srule,331{333 crossproduct,298 crystals,142{145 diamond,143 graphite,143 salt,142 unitcell,143 daVinci,Leonardo,337 determinant,294,299{318 cofactor,327 Cramer'srule,332 denition,299 exists,309,315 Laplaceexpansion,327 minor,327 permutationexpansion,308,312,334 diagonalmatrix,210,225 diagonalizable,356{359 dierenceequation,408 homogeneous,408 dilation,163,276 representing,203 dimension,120 physical,152 directmap,290 directsum,131 ,139 denition,135 external,168 internal,168 oftwosubspaces,135 directionvector,35 distance-preserving,287 divisiontheorem,350 domain,A-8 dotproduct,40 doubleprecision,69 dualspace,194 echelonform,5 freevariable,12 leadingvariable,5 reduced,47 eigenspace,362 eigenvalue,eigenvector ofamatrix,360 ofatransformation,359 element,A-7 elementary matrix,227,275 elementaryreductionoperations,4 pivoting,4 rescaling,4 swapping,4 elementaryrowoperations,4 emptyset,A-8 entry,13 equivalence class,A-12 canonicalrepresentative,A-12 relation,A-10 representative,A-12 equivalencerelation,A-10,A-11 isomorphism,169 matrixequivalence,244 matrixsimilarity,353 rowequivalence,50 equivalentstatements,A-3 ErlangerProgram,287 Euclid,287 evenfunctions,99,137 evenpolynomials,400 externaldirectsum,168 Fibonaccisequence,407 eld,140{141 denition,140 nite-dimensionalvectorspace,119 at,36 form,56 freevariable,12 fullcolumnrank,130 fullrowrank,130

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function,A-8 inverseimage,185 argument,A-8 codomain,A-8 composition,216,A-9 correspondence,A-9 domain,A-8 even,99 identity,A-9 inverse,232,A-9 leftinverse,231 multilinear,304 odd,99 one-to-one,A-9 onto,A-9 range,A-8 restriction,A-10 rightinverse,231 structurepreserving,161,165 seehomomorphism,176 two-sidedinverse,232 value,A-8 well-dened,A-8 zero,177 FundamentalTheorem ofLinearAlgebra,268 Gauss'method,2 accuracy,68{71 back-substitution,5 elementaryoperations,4 Gauss-Jordan,47 Gauss-Jordan,47 generalizednullspace,369 generalizedrangespace,369 GeometryofLinearMaps,274{280 Gram-Schmidtprocess,255{260 historyless Markovchain,281 homogeneouscoordinatevector,340 homogeneouscoordinates,292 homogeneousequation,21 homomorphism,176 composition,216 matrixrepresenting,195{205 nonsingular,190,208 nullity,188 nullspace,188 rangespace,184 rank,207 zero,177 idealline,343 idealpoint,343 identity function,A-9 matrix,225 if-thenstatement,A-2 ill-conditioned,69 image underafunction,A-9 impropersubspace,92 incidencematrix,229 index ofnilpotency,372 induction,23,A-5 inductivestep ofinduction,A-5 inheritedoperations,82 innerproduct,40 Input-OutputAnalysis,64{67 internaldirectsum,135,168 intersection,A-8 invariant subspace,379 invariantsubspace denition,391 inverse,232,A-9 additive,80 exists,232 left,232,A-9 matrix,329 right,232,A-9 two-sided,A-9 inversefunction,232 inverseimage,185 inversion,313 isometry,287 isomorphism,159{175 characterizedbydimension,172 denition,161 ofaspacewithitself,163 Jordanblock,390 Jordanform,381{400 representssimilarityclasses,394 kernel,188 Kirchho'sLaws,73

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Klein,F.,287 Laplaceexpansion,326{330 computesdeterminant,327 leadingvariable,5 leastsquares,269{273 lemma,A-1 length,39 Leontief,W.,64 line bestt,269 inprojectiveplane,341 lineatinnity,343 lineofbestt,269{273 linear transposeoperation,130 linearcombination,52 LinearCombinationLemma,53 linearequation,2 coecients,2 constant,2 homogeneous,21 inconsistentsystems,269 satisedbyavector,15 solutionof,2 Gauss'method,3 Gauss-Jordan,47 solutionsof Cramer'srule,332 systemof,2 linearmap dilation,276 reection,290 rotation,274,290 seehomomorphism,176 skew,277 trace,400 linearrecurrence,408 linearrecurrences,407{414 linearrelationship,102 linearsurface,36 lineartransformation seetransformation,179 linearlydependent,102 linearlyindependent,102 LINPACK,62 map,A-8 distance-preserving,287 extendedlinearly,173 selfcomposition,367 Maple,62 Markovchain,281 historyless,281 Markovchains,281{286 Markovmatrix,285 materialimplication,A-2 Mathematica,62 mathematicalinduction,23,A-5 MATLAB,62 matrix,13 adjoint,328 antisymmetric,138 augmented,14 block,245,311 changeofbasis,238 characteristicpolynomial,362 cofactor,327 column,13 columnspace,125 conditioningnumber,71 determinant,294,299 diagonal,210,225 diagonalizable,356 diagonalized,244 elementaryreduction,227,275 entry,13 equivalent,244 identity,221,225 incidence,229 inverse,329 maindiagonal,225 Markov,230,285 matrix-vectorproduct,198 minimalpolynomial,221,382 minor,327 multiplication,216 nilpotent,372 nonsingular,27,208 orthogonal,289 orthonormal,287{292 permutation,226 rank,207 representation,197 row,13 rowequivalence,50 rowrank,124 rowspace,124 scalarmultiple,213 similar,324

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similarity,353 singular,27 skew-symmetric,311 submatrix,303 sum,213 symmetric,118,138,214,221,229, 268 trace,214,230,400 transition,281 transpose,19,126,214 triangular,204,230,330 unit,223 Vandermonde,311 matrixequivalence,242{249 canonicalform,245 denition,244 matrix:form,56 mean arithmetic,44 geometric,44 member,A-7 methodofpowers,401{404 minimalpolynomial,221,382 minor,327 morphism,161 multilinear,304 multiplication matrix-matrix,216 matrix-vector,198 MuPAD,62 mutualinclusion,A-7 naturalrepresentative,A-12 networks,72{77 Kirchho'sLaws,73 nilpotent,370{380 canonicalformfor,376 denition,372 matrix,372 transformation,372 nilpotentcy index,372 nonsingular,208,232 homomorphism,190 matrix,27 normalize,258 nullity,188 nullspace,188 closureof,369 generalized,369 Octave,62 oddfunctions,99,137 one-to-onefunction,A-9 ontofunction,A-9 oppositemap,290 order ofarecurrence,408 orderedpair,A-8 orientation,321,324 orthogonal,42 basis,256 complement,263 mutually,255 projection,263 orthogonalmatrix,289 orthogonalization,257 orthonormalbasis,258 orthonormalmatrix,287{292 pair ordered,A-8 parallelepiped,321 parallelogramrule,34 parameter,13 partialpivoting,70 partition,A-10{A-12 matrixequivalenceclasses,244,246 rowequivalenceclasses,51 partitions intoisomorphismclasses,170 permutation,307 inversions,313 matrix,226 signum,314 permutationexpansion,308,312,334 perp,263 perpendicular,42 perspective triangles,343 Physicsproblem,1 pivoting full,70 partial scaled,70 pivotingonrows,4 planegure,287 congruence,287 point atinnity,343 inprojectiveplane,340

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polynomial even,400 minimal,382 ofmap,matrix,381 polynomials divisiontheorem,350 populations,stable,405{406 potential,72 powers,methodof,401{404 preservesstructure,176 probabilityvector,281 projection,176,185,250,268,387 alongasubspace,260 central,337 vanishingpoint,337 intoaline,251 intoasubspace,260 orthogonal,251,263 ProjectiveGeometry,337{347 projectivegeometry DualityPrinciple,342 projectiveplane idealline,343 idealpoint,343 lines,341 prooftechniques induction,23 proper subset,A-7 propersubspace,92 proposition,A-1 propositions equivalent,A-3 quantier,A-3 existential,A-4 universal,A-4 quantiers,A-3 range,A-8 rangespace,184 closureof,369 generalized,369 rank,127,207 column,125 ofahomomorphism,184,189 recurrence,327,408 homogeneous,408 initialconditions,408 reducedechelonform,47 reection,290 glide,290 reectionoripaboutaline,163 reexivity,A-10 relation,A-10 equivalence,A-10 reexive,A-10 symmetric,A-10 transitive,A-10 relationship linear,102 representation ofamatrix,197 ofavector,115 representative,A-12 canonical,A-12 forrowequivalenceclasses,58 ofmatrixequivalenceclasses,245 ofsimilarityclasses,395 rescalingrows,4 resistance,72 resistance:equivalent,76 resistor,72 restriction,A-10 rigidmotion,287 rotation,274,290 rotationorturning,163 represented,200 row,13 rank,124 vector,15 rowequivalence,50 rowrank full,130 rowspace,124 scalar,80 scalarmultiple matrix,213 vector,15,34,80 scalarproduct,40 scaledpartialpivoting,70 SchwartzInequality,41 SciLab,62 selfcomposition ofmaps,367 sense,321 sequence,A-8 concatenation,133 set,A-7

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complement,A-7 element,A-7 empty,A-8 intersection,A-8 member,A-7 union,A-8 sets,A-7 dependent,independent,102 empty,104 mutualinclusion,A-7 propersubset,A-7 spanof,95 subset,A-7 sgn seesignum,314 signum,314 similar,298,324 canonicalform,394 similarmatrices,353 similartriangles,290 similarity,353{366 similaritytransformation,366 singleprecision,68 singular matrix,27 size,319,321 skew,277 skew-symmetric,311 span,95 ofasingleton,99 spin,148 squareroot,400 stablepopulations,405{406 standardbasis,113 state,281 absorbtive,281 Staticsproblem,5 string,373 basis,373 ofbasisvectors,371 structure preservation,176 submatrix,303 subspace,91{100 closed,93 complementary,135 denition,91 directsum,135 improper,92 independence,135 invariant,391 proper,92 sum,131 sum ofmatrices,213 ofsubspaces,131 vector,15,34,80 summationnotation forpermutationexpansion,308 swappingrows,4 symmetricmatrix,118,138,214,221 symmetry,A-10 systemoflinearequations,2 Gauss'method,2 solving,2 theorem,A-1 trace,214,230,400 transformation characteristicpolynomial,362 composedwithitself,367 diagonalizable,356 eigenspace,362 eigenvalue,eigenvector,359 Jordanformfor,394 minimalpolynomial,382 nilpotent,372 canonicalrepresentative,376 projection,387 sizechange,321 transitionmatrix,281 transitivity,A-10 translation,287 transpose,19,126 determinant,309,317 interactionwithsumandscalarmultiplication,214 TriangleInequality,40 triangles similar,290 triangularmatrix,230 Triangularization,204 trivialspace,83,113 turningmap,163 union,A-8 unitmatrix,223 vacuouslytrue,A-2 value,A-8

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Vandermondematrix,311 vanishingpoint,337 vector,15,33 angle,42 canonicalposition,34 column,15 component,15 crossproduct,298 direction,35 dotproduct,40 free,33 homogeneouscoordinate,340 length,39 orthogonal,42 probability,281 representationof,115,238 row,15 satisesanequation,15 scalarmultiple,15,34,80 sum,15,34,80 unit,44 zero,22,80 vectorspace,80{100 basis,112 closure,80 complexscalars,90 denition,80 dimension,120 dual,194 nitedimensional,119 homomorphism,176 isomorphism,161 map,176 overcomplexnumbers,349 subspace,91 trivial,83,113 Venndiagram,A-1 voltagedrop,73 volume,321 votingparadox,146 majoritycycle,146 rationalpreferenceorder,146 votingparadoxes,146{151 spin,148 well-dened,A-8 Wheatstonebridge,74 zero divisor,221 zerodivison,237 zerodivisor,221 zerohomomorphism,177 zerovector,22,80