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Linear Algebra Done Right.pdf

Linear Algebra Done Right

zixu19810610
2010-01-07 0人阅读 0 0 0 暂无简介 举报

简介:本文档为《Linear Algebra Done Rightpdf》,可适用于工程科技领域

LinearAlgebraDoneRight,SecondEditionSheldonAxlerSpringerContentsPrefacetotheInstructorixPrefacetotheStudentxiiiAcknowledgmentsxvChapterVectorSpacesComplexNumbersDefinitionofVectorSpacePropertiesofVectorSpacesSubspacesSumsandDirectSumsExercisesChapterFiniteDimensionalVectorSpacesSpanandLinearIndependenceBasesDimensionExercisesChapterLinearMapsDefinitionsandExamplesSpacesandRangesTheMatrixofaLinearMapInvertibilityExercisesvviContentsChapterPolynomialsDegreeComplexCoefficientsRealCoefficientsExercisesChapterEigenvaluesandEigenvectorsInvariantSubspacesPolynomialsAppliedtoOperatorsUpperTriangularMatricesDiagonalMatricesInvariantSubspacesonRealVectorSpacesExercisesChapterInnerProductSpacesInnerProductsNormsOrthonormalBasesOrthogonalProjectionsandMinimizationProblemsLinearFunctionalsandAdjointsExercisesChapterOperatorsonInnerProductSpacesSelfAdjointandNormalOperatorsTheSpectralTheoremNormalOperatorsonRealInnerProductSpacesPositiveOperatorsIsometriesPolarandSingularValueDecompositionsExercisesChapterOperatorsonComplexVectorSpacesGeneralizedEigenvectorsTheCharacteristicPolynomialDecompositionofanOperatorContentsviiSquareRootsTheMinimalPolynomialJordanFormExercisesChapterOperatorsonRealVectorSpacesEigenvaluesofSquareMatricesBlockUpperTriangularMatricesTheCharacteristicPolynomialExercisesChapterTraceandDeterminantChangeofBasisTraceDeterminantofanOperatorDeterminantofaMatrixVolumeExercisesSymbolIndexIndexPrefacetotheInstructorYouareprobablyabouttoteachacoursethatwillgivestudentstheirsecondexposuretolinearalgebraDuringtheirfirstbrushwiththesubject,yourstudentsprobablyworkedwithEuclideanspacesandmatricesIncontrast,thiscoursewillemphasizeabstractvectorspacesandlinearmapsTheaudacioustitleofthisbookdeservesanexplanationAlmostalllinearalgebrabooksusedeterminantstoprovethateverylinearoperatoronafinitedimensionalcomplexvectorspacehasaneigenvalueDeterminantsaredifficult,nonintuitive,andoftendefinedwithoutmotivationToprovethetheoremaboutexistenceofeigenvaluesoncomplexvectorspaces,mostbooksmustdefinedeterminants,provethatalinearmapisnotinvertibleifandonlyifitsdeterminantequals,andthendefinethecharacteristicpolynomialThistortuous(torturous)pathgivesstudentslittlefeelingforwhyeigenvaluesmustexistIncontrast,thesimpledeterminantfreeproofspresentedhereoffermoreinsightOncedeterminantshavebeenbanishedtotheendofthebook,anewrouteopenstothemaingoaloflinearalgebraunderstandingthestructureoflinearoperatorsThisbookstartsatthebeginningofthesubject,withnoprerequisitesotherthantheusualdemandforsuitablemathematicalmaturityEvenifyourstudentshavealreadyseensomeofthematerialinthefirstfewchapters,theymaybeunaccustomedtoworkingexercisesofthetypepresentedhere,mostofwhichrequireanunderstandingofproofs•VectorspacesaredefinedinChapter,andtheirbasicpropertiesaredeveloped•Linearindependence,span,basis,anddimensionaredefinedinChapter,whichpresentsthebasictheoryoffinitedimensionalvectorspacesixxPrefacetotheInstructor•LinearmapsareintroducedinChapterThekeyresulthereisthatforalinearmapT,thedimensionofthespaceofTplusthedimensionoftherangeofTequalsthedimensionofthedomainofT•ThepartofthetheoryofpolynomialsthatwillbeneededtounderstandlinearoperatorsispresentedinChapterIfyoutakeclasstimegoingthroughtheproofsinthischapter(whichcontainsnolinearalgebra),thenyouprobablywillnothavetimetocoversomeimportantaspectsoflinearalgebraYourstudentswillalreadybefamiliarwiththetheoremsaboutpolynomialsinthischapter,soyoucanaskthemtoreadthestatementsoftheresultsbutnottheproofsThecuriousstudentswillreadsomeoftheproofsanyway,whichiswhytheyareincludedinthetext•TheideaofstudyingalinearoperatorbyrestrictingittosmallsubspacesleadsinChaptertoeigenvectorsThehighlightofthechapterisasimpleproofthatoncomplexvectorspaces,eigenvaluesalwaysexistThisresultisthenusedtoshowthateachlinearoperatoronacomplexvectorspacehasanuppertriangularmatrixwithrespecttosomebasisSimilartechniquesareusedtoshowthateverylinearoperatoronarealvectorspacehasaninvariantsubspaceofdimensionorThisresultisusedtoprovethateverylinearoperatoronanodddimensionalrealvectorspacehasaneigenvalueAllthisisdonewithoutdefiningdeterminantsorcharacteristicpolynomials!•InnerproductspacesaredefinedinChapter,andtheirbasicpropertiesaredevelopedalongwithstandardtoolssuchasorthonormalbases,theGramSchmidtprocedure,andadjointsThischapteralsoshowshoworthogonalprojectionscanbeusedtosolvecertainminimizationproblems•Thespectraltheorem,whichcharacterizesthelinearoperatorsforwhichthereexistsanorthonormalbasisconsistingofeigenvectors,isthehighlightofChapterTheworkinearlierchapterspaysoffherewithespeciallysimpleproofsThischapteralsodealswithpositiveoperators,linearisometries,thepolardecomposition,andthesingularvaluedecompositionPrefacetotheInstructorxi•Theminimalpolynomial,characteristicpolynomial,andgeneralizedeigenvectorsareintroducedinChapterThemainachievementofthischapteristhedescriptionofalinearoperatoronacomplexvectorspaceintermsofitsgeneralizedeigenvectorsThisdescriptionenablesonetoprovealmostalltheresultsusuallyprovedusingJordanformForexample,thesetoolsareusedtoprovethateveryinvertiblelinearoperatoronacomplexvectorspacehasasquarerootThechapterconcludeswithaproofthateverylinearoperatoronacomplexvectorspacecanbeputintoJordanform•LinearoperatorsonrealvectorspacesoccupycenterstageinChapterHeretwodimensionalinvariantsubspacesmakeupforthepossiblelackofeigenvalues,leadingtoresultsanalogoustothoseobtainedoncomplexvectorspaces•ThetraceanddeterminantaredefinedinChapterintermsofthecharacteristicpolynomial(definedearlierwithoutdeterminants)Oncomplexvectorspaces,thesedefinitionscanberestated:thetraceisthesumoftheeigenvaluesandthedeterminantistheproductoftheeigenvalues(bothcountingmultiplicity)TheseeasytorememberdefinitionswouldnotbepossiblewiththetraditionalapproachtoeigenvaluesbecausethatmethodusesdeterminantstoprovethateigenvaluesexistThestandardtheoremsaboutdeterminantsnowbecomemuchclearerThepolardecompositionandthecharacterizationofselfadjointoperatorsareusedtoderivethechangeofvariablesformulaformultivariableintegralsinafashionthatmakestheappearanceofthedeterminantthereseemnaturalThisbookusuallydevelopslinearalgebrasimultaneouslyforrealandcomplexvectorspacesbylettingFdenoteeithertherealorthecomplexnumbersAbstractfieldscouldbeusedinstead,buttodosowouldintroduceextraabstractionwithoutleadingtoanynewlinearalgebraAnotherreasonforrestrictingattentiontotherealandcomplexnumbersisthatpolynomialscanthenbethoughtofasgenuinefunctionsinsteadofthemoreformalobjectsneededforpolynomialswithcoefficientsinfinitefieldsFinally,evenifthebeginningpartofthetheoryweredevelopedwitharbitraryfields,innerproductspaceswouldpushconsiderationbacktojustrealandcomplexvectorspacesxiiPrefacetotheInstructorEveninabookasshortasthisone,youcannotexpecttocovereverythingGoingthroughthefirsteightchaptersisanambitiousgoalforaonesemestercourseIfyoumustreachChapter,thenIsuggestcoveringChapters,,andquickly(studentsmayhaveseenthismaterialinearliercourses)andskippingChapter(inwhichcaseyoushoulddiscusstraceanddeterminantsonlyoncomplexvectorspaces)AgoalmoreimportantthanteachinganyparticularsetoftheoremsistodevelopinstudentstheabilitytounderstandandmanipulatetheobjectsoflinearalgebraMathematicscanbelearnedonlybydoingfortunately,linearalgebrahasmanygoodhomeworkproblemsWhenteachingthiscourse,Iusuallyassigntwoorthreeoftheexerciseseachclass,duethenextclassGoingoverthehomeworkmighttakeupathirdorevenhalfofatypicalclassAsolutionsmanualforalltheexercisesisavailable(withoutcharge)onlytoinstructorswhoareusingthisbookasatextbookToobtainthesolutionsmanual,instructorsshouldsendanemailrequesttome(orcontactSpringerifIamnolongeraround)Pleasecheckmywebsiteforalistoferrata(whichIhopewillbeemptyoralmostempty)andotherinformationaboutthisbookIwouldgreatlyappreciatehearingaboutanyerrorsinthisbook,evenminoronesIwelcomeyoursuggestionsforimprovements,eventinyonesPleasefeelfreetocontactmeHavefun!SheldonAxlerMathematicsDepartmentSanFranciscoStateUniversitySanFrancisco,CA,USAemail:axlermathsfsueduwwwhomepage:http:mathsfsueduaxlerPrefacetotheStudentYouareprobablyabouttobeginyoursecondexposuretolinearalgebraUnlikeyourfirstbrushwiththesubject,whichprobablyemphasizedEuclideanspacesandmatrices,wewillfocusonabstractvectorspacesandlinearmapsThesetermswillbedefinedlater,sodon’tworryifyoudon’tknowwhattheymeanThisbookstartsfromthebeginningofthesubject,assumingnoknowledgeoflinearalgebraThekeypointisthatyouareabouttoimmerseyourselfinseriousmathematics,withanemphasisonyourattainingadeepunderstandingofthedefinitions,theorems,andproofsYoucannotexpecttoreadmathematicsthewayyoureadanovelIfyouzipthroughapageinlessthananhour,youareprobablygoingtoofastWhenyouencounterthephrase“asyoushouldverify”,youshouldindeeddotheverification,whichwillusuallyrequiresomewritingonyourpartWhenstepsareleftout,youneedtosupplythemissingpiecesYoushouldponderandinternalizeeachdefinitionForeachtheorem,youshouldseekexamplestoshowwhyeachhypothesisisnecessaryPleasecheckmywebsiteforalistoferrata(whichIhopewillbeemptyoralmostempty)andotherinformationaboutthisbookIwouldgreatlyappreciatehearingaboutanyerrorsinthisbook,evenminoronesIwelcomeyoursuggestionsforimprovements,eventinyonesHavefun!SheldonAxlerMathematicsDepartmentSanFranciscoStateUniversitySanFrancisco,CA,USAemail:axlermathsfsueduwwwhomepage:http:mathsfsueduaxlerxiiiAcknowledgmentsIoweahugeintellectualdebttothemanymathematicianswhocreatedlinearalgebraduringthelasttwocenturiesInwritingthisbookItriedtothinkaboutthebestwaytopresentlinearalgebraandtoproveitstheorems,withoutregardtothestandardmethodsandproofsusedinmosttextbooksThusIdidnotconsultotherbookswhilewritingthisone,thoughthememoryofmanybooksIhadstudiedinthepastsurelyinfluencedmeMostoftheresultsinthisbookbelongtothecommonheritageofmathematicsAspecialcaseofatheoremmayfirsthavebeenprovedinantiquity(whichforlinearalgebrameansthenineteenthcentury),thenslowlysharpenedandimprovedoverdecadesbymanymathematiciansBestowingpropercreditonallthecontributorswouldbeadifficulttaskthatIhavenotundertakenInnocaseshouldthereaderassumethatanytheorempresentedhererepresentsmyoriginalcontributionManypeoplehelpedmakethisabetterbookForusefulsuggestionsandcorrections,IamgratefultoWilliamArveson(forsuggestingtheproofof),MarilynBrouwer,WilliamBrown,RobertBurckel,PaulCohn,JamesDudziak,DavidFeldman(forsuggestingtheproofof),PamelaGorkin,AramHarrow,PanFongHo,DanKalman,RobertKantrowitz,RamanaKappagantu,MizanKhan,MikaelLindstro¨m,JacobPlotkin,ElenaPoletaeva,MihaelaPoplicher,RichardPotter,WadeRamey,MarianRobbins,JonathanRosenberg,JoanStamm,ThomasStarbird,JayValanju,andThomasvonFoersterFinally,IthankSpringerforprovidingmewithhelpwhenIneededitandforallowingmethefreedomtomakethefinaldecisionsaboutthecontentandappearanceofthisbookxvChapterVectorSpacesLinearalgebraisthestudyoflinearmapsonfinitedimensionalvectorspacesEventuallywewilllearnwhatallthesetermsmeanInthischapterwewilldefinevectorspacesanddiscusstheirelementarypropertiesInsomeareasofmathematics,includinglinearalgebra,bettertheoremsandmoreinsightemergeifcomplexnumbersareinvestigatedalongwithrealnumbersThuswebeginbyintroducingthecomplexnumbersandtheirbasicproperties✽ChapterVectorSpacesComplexNumbersYoushouldalreadybefamiliarwiththebasicpropertiesofthesetRofrealnumbersComplexnumberswereinventedsothatwecantakesquarerootsofnegativenumbersThekeyideaistoassumewehaveasquarerootof−,denotedi,andmanipulateitusingtheusualrulesThesymboliwasfirstusedtodenote√−bytheSwissmathematicianLeonhardEulerinofarithmeticFormally,acomplexnumberisanorderedpair(a,b),wherea,b∈R,butwewillwritethisasabiThesetofallcomplexnumbersisdenotedbyC:C={abi:a,b∈R}Ifa∈R,weidentifyaiwiththerealnumberaThuswecanthinkofRasasubsetofCAdditionandmultiplicationonCaredefinedby(abi)(cdi)=(ac)(bd)i,(abi)(cdi)=(ac−bd)(adbc)iherea,b,c,d∈RUsingmultiplicationasdefinedabove,youshouldverifythati=−Donotmemorizetheformulafortheproductoftwocomplexnumbersyoucanalwaysrederiveitbyrecallingthati=−andthenusingtheusualrulesofarithmeticYoushouldverify,usingthefamiliarpropertiesoftherealnumbers,thatadditionandmultiplicationonCsatisfythefollowingproperties:commutativitywz=zwandwz=zwforallw,z∈Cassociativity(zz)z=z(zz)and(zz)z=z(zz)forallz,z,z∈Cidentitiesz=zandz=zforallz∈Cadditiveinverseforeveryz∈C,thereexistsauniquew∈Csuchthatzw=multiplicativeinverseforeveryz∈Cwithz�=,thereexistsauniquew∈Csuchthatzw=ComplexNumbersdistributivepropertyλ(wz)=λwλzforallλ,w,z∈CForz∈C,welet−zdenotetheadditiveinverseofzThus−zistheuniquecomplexnumbersuchthatz(−z)=SubtractiononCisdefinedbyw−z=w(−z)forw,z∈CForz∈Cwithz�=,weletzdenotethemultiplicativeinverseofzThuszistheuniquecomplexnumbersuchthatz(z)=DivisiononCisdefinedbywz=w(z)forw,z∈Cwithz�=Sothatwecanconvenientlymakedefinitionsandprovetheoremsthatapplytobothrealandcomplexnumbers,weadoptthefollowingnotation:TheletterFisusedbecauseRandCareexamplesofwhatarecalledfieldsInthisbookwewillnotneedtodealwithfieldsotherthanRorCManyofthedefinitions,theorems,andproofsinlinearalgebrathatworkforbothRandCalsoworkwithoutchangeifanarbitraryfieldreplacesRorCThroughoutthisbook,FstandsforeitherRorCThusifweproveatheoreminvolvingF,wewillknowthatitholdswhenFisreplacedwithRandwhenFisreplacedwithCElementsofFarecalledscalarsTheword“scalar”,whichmeansnumber,isoftenusedwhenwewanttoemphasizethatanobjectisanumber,asopposedtoavector(vectorswillbedefinedsoon)Forz∈Fandmapositiveinteger,wedefinezmtodenotetheproductofzwithitselfmtimes:zm=z·····z︸︷︷︸mtimesClearly(zm)n=zmnand(wz)m=wmzmforallw,z∈Fandallpositiveintegersm,nChapterVectorSpacesDefinitionofVectorSpaceBeforedefiningwhatavectorspaceis,let’slookattwoimportantexamplesThevectorspaceR,whichyoucanthinkofasaplane,consistsofallorderedpairsofrealnumbers:R={(x,y):x,y∈R}ThevectorspaceR,whichyoucanthinkofasordinaryspace,consistsofallorderedtriplesofrealnumbers:R={(x,y,z):x,y,z∈R}Togeneralize

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