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i i “book” — 2011/3/4 — 17:06 — page i — #1 i i A Guide to Advanced Linear Algebra i i i i i i “book” — 2011/3/4 — 17:06 — page ii — #2 i i (cid:13)c 2011by TheMathematicalAssociationofAmerica(Incorporated) LibraryofCongressCatalogCardNumber2011923993 PrintEditionISBN978-0-88385-351-1 ElectronicEditionISBN978-0-88385-967-4 PrintedintheUnitedStatesofAmerica CurrentPrinting(lastdigit): 10987654321 i i i i i i “book” — 2011/3/4 — 17:06 — page iii — #3 i i TheDolcianiMathematicalExpositions NUMBERFORTY-FOUR MAAGuides#6 A Guide to Advanced Linear Algebra Steven H. Weintraub Lehigh University PublishedandDistributedby TheMathematicalAssociationofAmerica i i i i i i “book” — 2011/3/4 — 17:06 — page iv — #4 i i DOLCIANIMATHEMATICALEXPOSITIONS CommitteeonBooks FrankFarris,Chair DolcianiMathematicalExpositionsEditorialBoard UnderwoodDudley,Editor JeremyS.Case RosalieA.Dance TevianDray ThomasM.Halverson PatriciaB.Humphrey MichaelJ.McAsey MichaelJ.Mossinghoff JonathanRogness ThomasQ.Sibley i i i i i i “book” — 2011/3/4 — 17:06 — page v — #5 i i The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical AssociationofAmericawasestablishedthroughagenerousgifttotheAssociation fromMaryP.Dolciani,ProfessorofMathematicsatHunterCollegeoftheCityUni- versityofNewYork.Inmakingthegift,ProfessorDolciani,herselfanexceptionally talentedandsuccessfulexpositorofmathematics,hadthepurposeoffurtheringthe idealofexcellenceinmathematicalexposition. TheAssociation,foritspart,wasdelightedtoacceptthegraciousgestureinitiat- ingtherevolvingfundforthisseriesfromonewhohasservedtheAssociationwith distinction,bothasamemberoftheCommitteeonPublicationsandasamemberof theBoardofGovernors.ItwaswithgenuinepleasurethattheBoardchosetoname theseriesinherhonor. Thebooksintheseriesareselectedfortheirlucidexpositorystyleandstimu- latingmathematicalcontent.Typically,theycontainanamplesupplyofexercises, manywithaccompanyingsolutions.Theyareintendedtobesufficientlyelementary fortheundergraduateandeventhemathematicallyinclinedhigh-schoolstudentto understandandenjoy,butalsoto beinterestingandsometimeschallengingto the moreadvancedmathematician. 1. MathematicalGems,RossHonsberger 2. MathematicalGemsII,RossHonsberger 3. MathematicalMorsels,RossHonsberger 4. MathematicalPlums,RossHonsberger(ed.) 5. GreatMomentsinMathematics(Before1650),HowardEves 6. MaximaandMinimawithoutCalculus,IvanNiven 7. GreatMomentsinMathematics(After1650),HowardEves 8. MapColoring,Polyhedra,andtheFour-ColorProblem,DavidBarnette 9. MathematicalGemsIII,RossHonsberger 10. MoreMathematicalMorsels,RossHonsberger 11. Old and New Unsolved Problems in Plane Geometry and Number Theory, VictorKleeandStanWagon 12. ProblemsforMathematicians,YoungandOld,PaulR.Halmos 13. Excursions in Calculus: An Interplay of the Continuous and the Discrete, RobertM.Young 14. TheWohascumCountyProblemBook,GeorgeT.Gilbert,MarkKrusemeyer, andLorenC.Larson 15. LionHuntingandOtherMathematicalPursuits:ACollectionofMathematics, Verse,andStoriesbyRalphP.Boas,Jr.,editedbyGeraldL.Alexandersonand DaleH.Mugler 16. LinearAlgebraProblemBook,PaulR.Halmos 17. FromErdo˝stoKiev:ProblemsofOlympiadCaliber,RossHonsberger 18. WhichWayDidtheBicycleGo?...andOtherIntriguingMathematicalMys- teries,JosephD.E.Konhauser,DanVelleman,andStanWagon i i i i i i “book” — 2011/3/4 — 17:06 — page vi — #6 i i 19. InPo´lya’sFootsteps:MiscellaneousProblemsandEssays,RossHonsberger 20. DiophantusandDiophantineEquations,I.G.Bashmakova(UpdatedbyJoseph SilvermanandtranslatedbyAbeShenitzer) 21. LogicasAlgebra,PaulHalmosandStevenGivant 22. Euler:TheMasterofUsAll,WilliamDunham 23. TheBeginningsandEvolutionofAlgebra,I.G.BashmakovaandG.S.Smirnova (TranslatedbyAbeShenitzer) 24. MathematicalChestnutsfromAroundtheWorld,RossHonsberger 25. CountingonFrameworks:MathematicstoAidtheDesignofRigidStructures, JackE.Graver 26. MathematicalDiamonds,RossHonsberger 27. ProofsthatReallyCount:TheArtofCombinatorialProof,ArthurT.Benjamin andJenniferJ.Quinn 28. MathematicalDelights,RossHonsberger 29. Conics,KeithKendig 30. Hesiod’s Anvil: falling and spinning through heavenand earth, Andrew J. Simoson 31. AGardenofIntegrals,FrankE.Burk 32. AGuidetoComplexVariables(MAAGuides#1),StevenG.Krantz 33. SinkorFloat?ThoughtProblemsinMathandPhysics,KeithKendig 34. BiscuitsofNumberTheory,ArthurT.BenjaminandEzraBrown 35. UncommonMathematical Excursions:Polynomia andRelated Realms,Dan Kalman 36. WhenLessisMore:VisualizingBasicInequalities,ClaudiAlsinaandRoger B.Nelsen 37. AGuidetoAdvancedRealAnalysis(MAAGuides#2),GeraldB.Folland 38. AGuidetoRealVariables(MAAGuides#3),StevenG.Krantz 39. Voltaire’s Riddle: Microme´gas and the measure of all things, Andrew J. Simoson 40. AGuidetoTopology,(MAAGuides#4),StevenG.Krantz 41. AGuidetoElementaryNumberTheory,(MAAGuides#5),UnderwoodDud- ley 42. Charming Proofs: A Journeyinto Elegant Mathematics, Claudi Alsina and RogerB.Nelsen 43. MathematicsandSports,editedbyJosephA.Gallian 44. AGuidetoAdvancedLinearAlgebra,(MAAGuides#6),StevenH.Weintraub MAAServiceCenter P.O.Box91112 Washington,DC20090-1112 1-800-331-1MAA FAX:1-301-206-9789 i i i i i i “book” — 2011/3/4 — 17:06 — page vii — #7 i i Preface Linear algebra is a beautifuland mature field ofmathematics, and mathe- maticianshavedevelopedhighlyeffectivemethodsforsolvingitsproblems. Itisasubjectwellworthstudyingforitsownsake. Morethanthat,linearalgebraoccupiesacentralplaceinmodernmath- ematics. Students in algebra studying Galois theory, students in analysis studyingfunctionspaces, studentsintopologystudyinghomologyandco- homology,orforthatmatterstudentsinjustaboutanyareaofmathematics, studyingjustaboutanything,needtohaveasoundknowledgeoflinearal- gebra. Wehavewrittenabookthatwehopewillbebroadlyuseful.Thecoreof linearalgebraisessentialtoeverymathematician,andwenotonlytreatthis core, butaddmaterialthatisessentialtomathematiciansinspecific fields, evenifnotallofitisessentialtoeverybody. Thisisabookforadvancedstudents.Wepresumeyouarealreadyfamil- iarwithelementarylinearalgebra,andthatyouknowhowtomultiplyma- trices, solvelinearsystems, etc. We donottreatelementary material here, thoughin places we return to elementary material from a more advanced standpointtoshowyouwhatitreallymeans. However,wedonotpresume youarealreadyamaturemathematician,andinplacesweexplainwhat(we feel)isthe“right”waytounderstandthematerial.Theauthorfeelsthatone ofthemaindutiesofateacheristoprovideaviewpointonthesubject,and wetakepainstodothathere. Onethingthatyoushouldlearnaboutlinearalgebranow,ifyouhavenot alreadydoneso,isthefollowing:Linearalgebraisaboutvectorspacesand lineartransformations,notaboutmatrices.Thisisverymuchtheapproach ofthisbook,asyouwillseeuponreadingit. We treat both the finite and infinite dimensional cases in this book, and point out the differences between them, but the bulk of our attention isdevoted tothefinitedimensionalcase. There are tworeasons:First, the vii i i i i i i “book” — 2011/3/4 — 17:06 — page viii — #8 i i viii A GuidetoAdvanced Linear Algebra strongestresultsareavailablehere,andsecond,thisisthecasemostwidely used in mathematics. (Of course, matrices are available only in the finite dimensionalcase,but,evenhere,wealmostalwaysargueintermsoflinear transformationsratherthanmatrices.) We regard linear algebra as part of algebra, and that guides our approach.Butwehavefollowedamiddleground.Oneoftheprincipalgoals of this book is to derive canonical forms for linear transformationson finite dimensional vector spaces, i.e., rational and Jordan canonical forms. The quickestandperhapsmostenlighteningapproachistoderivethemas corollariesofthebasicstructuretheoremsformodulesoveraprincipalideal domain (PID).Doingso wouldrequirea gooddeal of background,which wouldlimit the utilityof this book. Thus our main line of approach does notusethese,thoughweindicatethisapproachinanappendix.Insteadwe adoptamoredirectargument. Wehavewrittenabookthatwefeelisathorough,thoughintentionally not encyclopedic, treatment of linear algebra, one that contains material that is both important and deservedly “well known”. In a few places we have succumbed to temptation and included material that is not quite so wellknown,butthatinouropinionshouldbe. We hopethatyouwillbeenlightenednotonlybythespecificmaterial in the book but by its style of argument–we hope it will help you learn to “think like a mathematician”. We also hope this book will serve as a valuablereferencethroughoutyourmathematicalcareer. Here is a rough outlineof the text. We begin, in Chapter 1, by intro- ducing the basic notionsof linear algebra, vector spaces and linear trans- formations,andestablishsomeoftheirmostimportantproperties.InChap- ter 2 we introduce coordinates for vectors and matrices for linear trans- formations. Inthe first half of Chapter3 we establishthe basic properties of determinants, and in the last halfof that chapter we give some oftheir applications.Chapters 4 and 5 are devotedto the analysis ofthe structure ofa singlelinear transformationfroma finitedimensionalvectorspace to itself. In particular, in these chapters, we develop eigenvalues, eigenvec- tors,andgeneralizedeigenvectors,andderiverationalandJordancanonical forms. In Chapter 6 we introduce additional structure on a vector space, thatofa(bilinear,sesquilinear,orquadratic)form,andanalyzetheseforms. In Chapter 7 we specialize the situationof Chapter 6 to that of a positive definiteinnerproductona real or complex vectorspace, andinparticular derivethespectraltheorem.InChapter8weprovideanintroductiontoLie groups,whicharecentralobjectsinmathematicsandareameetingplacefor i i i i i i “book” — 2011/3/4 — 17:06 — page ix — #9 i i Preface ix algebra,analysis,andtopology.(Forthischapterwerequiretheadditional backgroundknowledgeoftheinversefunctiontheorem.)InAppendixAwe review basic properties of polynomialsand polynomialrings that we use, andinAppendixBwerederivesomeofourresultsoncanonicalformsofa lineartransformationfromthestructuretheoremsformodulesoveraPID. We have provided complete proofs of just about all the results in this book, except that we have often omitted proofs that are routine without comment. Aswehaveremarkedabove, wehave triedtowriteabookthatwillbe widely applicable. This book is written in an algebraic spirit, so the student of algebra will find items of interest and particular applications, too numerous to mention here, throughoutthe book. The student of analysis will appreciate the fact that we not only consider finite dimensional vec- torspaces, butalso infinitedimensionalones, and willalsoappreciate our material on inner product spaces and our particular examples of function spaces. The student of algebraic topology will appreciate our dimension- countingargumentsandourcarefulattentiontoduality,andthestudentof differentialtopologywillappreciate ourmaterial onorientationsofvector spacesandourintroductiontoLiegroups. Nobookcan treateverything.Withtheexceptionofashortsectionon Hilbertmatrices,wedonottreatcomputationalissuesatall.Theydonotfit inwithourtheoreticalapproach.Studentsinnumericalanalysis,forexam- ple,willneedtolookelsewhereforthismaterial. Toclosethispreface,weestablishsomenotationalconventions.Wewill denotebothsets(usuallybutnotalwayssetsofvectors)andlineartransfor- mationsbyscriptlettersA;B;:::;Z.Wewilltendtousescriptlettersnear thefrontofthealphabetforsetsandscriptlettersneartheendofthealpha- betforlineartransformations.T willalwaysdenotealineartransformation andI willalwaysdenotetheidentitylineartransformation.Someparticu- larlineartransformationswillhave particularnotations,ofteninboldface. Capitalletterswilldenoteeithervectorspaces ormatrices.Wewilltendto denotevectorspaces bycapitallettersneartheendofthealphabet,andV willalways denote a vector space. Also, I willalmost always denote the identitymatrix.EandF willdenotearbitraryfieldsandQ, R,andC will denotethefieldsofrational,real,andcomplexnumbersrespectively.Zwill denote theringofintegers. We willuse A B tomean thatA is a sub- (cid:18) set ofB and A B tomean thatA is a propersubset ofB. A .a / ij (cid:26) D will mean that A is the matrix whose entry in the .i;j/ position is a . ij A Œv v v (cid:141)willmean thatAisthematrixwhoseithcolumn 1 2 n D j j (cid:1)(cid:1)(cid:1) j i i i i