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Linear Algebra Done Right(Axler).pdf

Linear Algebra Done Right(Axler…

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简介:本文档为《Linear Algebra Done Right(Axler)pdf》,可适用于高等教育领域,主题内容包含ContentsPrefacetotheInstructorixPrefacetotheStudentxiiiAcknowledgmentsxvCh符等。

ContentsPrefacetotheInstructorixPrefacetotheStudentxiiiAcknowledgmentsxvChapterVectorSpacesComplexNumbersDefinitionofVectorSpacePropertiesofVectorSpacesSubspacesSumsandDirectSumsExercisesChapterFiniteDimensionalVectorSpacesSpanandLinearIndependenceBasesDimensionExercisesChapterLinearMapsDefinitionsandExamplesSpacesandRangesTheMatrixofaLinearMapInvertibilityExercisesvviContentsChapterPolynomialsDegreeComplexCoefficientsRealCoefficientsExercisesChapterEigenvaluesandEigenvectorsInvariantSubspacesPolynomialsAppliedtoOperatorsUpperTriangularMatricesDiagonalMatricesInvariantSubspacesonRealVectorSpacesExercisesChapterInnerProductSpacesInnerProductsNormsOrthonormalBasesOrthogonalProjectionsandMinimizationProblemsLinearFunctionalsandAdjointsExercisesChapterOperatorsonInnerProductSpacesSelfAdjointandNormalOperatorsTheSpectralTheoremNormalOperatorsonRealInnerProductSpacesPositiveOperatorsIsometriesPolarandSingularValueDecompositionsExercisesChapterOperatorsonComplexVectorSpacesGeneralizedEigenvectorsTheCharacteristicPolynomialDecompositionofanOperatorContentsviiSquareRootsTheMinimalPolynomialJordanFormExercisesChapterOperatorsonRealVectorSpacesEigenvaluesofSquareMatricesBlockUpperTriangularMatricesTheCharacteristicPolynomialExercisesChapterTraceandDeterminantChangeofBasisTraceDeterminantofanOperatorDeterminantofaMatrixVolumeExercisesSymbolIndexIndexPrefacetotheInstructorYouareprobablyabouttoteachacoursethatwillgivestudentstheirsecondexposuretolinearalgebraDuringtheirfirstbrushwiththesubject,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,MikaelLindstrom,JacobPlotkin,ElenaPoletaeva,MihaelaPoplicher,RichardPotter,WadeRamey,MarianRobbins,JonathanRosenberg,JoanStamm,ThomasStarbird,JayValanju,andThomasvonFoersterFinally,IthankSpringerforprovidingmewithhelpwhenIneededitandforallowingmethefreedomtomakethefinaldecisionsaboutthecontentandappearanceofthisbookxvChapterVectorSpacesLinearalgebraisthestudyoflinearmapsonfinitedimensionalvectorspacesEventuallywewilllearnwhatallthesetermsmeanInthischapterwewilldefinevectorspacesanddiscusstheirelementarypropertiesInsomeareasofmathematics,includinglinearalgebra,bettertheoremsandmoreinsightemergeifcomplexnumbersareinvestigatedalongwithrealnumbersThuswebeginbyintroducingthecomplexnumbersandtheirbasicpropertiesChapterVectorSpacesComplexNumbersYoushouldalreadybefamiliarwiththebasicpropertiesofthesetRofrealnumbersComplexnumberswereinventedsothatwecantakesquarerootsofnegativenumbersThekeyideaistoassumewehaveasquarerootof,denotedi,andmanipulateitusingtheusualrulesThesymboliwasfirstusedtodenotebytheSwissmathematicianLeonhardEulerinofarithmeticFormally,acomplexnumberisanorderedpair(a,b),wherea,bR,butwewillwritethisasabiThesetofallcomplexnumbersisdenotedbyC:C={abi:a,bR}IfaR,weidentifyaiwiththerealnumberaThuswecanthinkofRasasubsetofCAdditionandmultiplicationonCaredefinedby(abi)(cdi)=(ac)(bd)i,(abi)(cdi)=(acbd)(adbc)iherea,b,c,dRUsingmultiplicationasdefinedabove,youshouldverifythati=Donotmemorizetheformulafortheproductoftwocomplexnumbersyoucanalwaysrederiveitbyrecallingthati=andthenusingtheusualrulesofarithmeticYoushouldverify,usingthefamiliarpropertiesoftherealnumbers,thatadditionandmultiplicationonCsatisfythefollowingproperties:commutativitywz=zwandwz=zwforallw,zCassociativity(zz)z=z(zz)and(zz)z=z(zz)forallz,z,zCidentitiesz=zandz=zforallzCadditiveinverseforeveryzC,thereexistsauniquewCsuchthatzw=multiplicativeinverseforeveryzCwithz=,thereexistsauniquewCsuchthatzw=ComplexNumbersdistributivepropertyλ(wz)=λwλzforallλ,w,zCForzC,weletzdenotetheadditiveinverseofzThuszistheuniquecomplexnumbersuchthatz(z)=SubtractiononCisdefinedbywz=w(z)forw,zCForzCwithz=,weletzdenotethemultiplicativeinverseofzThuszistheuniquecomplexnumbersuchthatz(z)=DivisiononCisdefinedbywz=w(z)forw,zCwithz=Sothatwecanconvenientlymakedefinitionsandprovetheoremsthatapplytobothrealandcomplexnumbers,weadoptthefollowingnotation:TheletterFisusedbecauseRandCareexamplesofwhatarecalledfieldsInthisbookwewillnotneedtodealwithfieldsotherthanRorCManyofthedefinitions,theorems,andproofsinlinearalgebrathatworkforbothRandCalsoworkwithoutchangeifanarbitraryfieldreplacesRorCThroughoutthisbook,FstandsforeitherRorCThusifweproveatheoreminvolvingF,wewillknowthatitholdswhenFisreplacedwithRandwhenFisreplacedwithCElementsofFarecalledscalarsTheword“scalar”,whichmeansnumber,isoftenusedwhenwewanttoemphasizethatanobjectisanumber,asopposedtoavector(vectorswillbedefinedsoon)ForzFandmapositiveinteger,wedefinezmtodenotetheproductofzwithitselfmtimes:zm=zz︸︷︷︸mtimesClearly(zm)n=zmnand(wz)m=wmzmforallw,zFandallpositiveintegersm,nChapterVectorSpacesDefinitionofVectorSpaceBeforedefiningwhatavectorspaceis,let’slookattwoimportantexamplesThevectorspaceR,whichyoucanthinkofasaplane,consistsofallorderedpairsofrealnumbers:R={(x,y):x,yR}ThevectorspaceR,whichyoucanthinkofasordinaryspace,consistsofallorderedtriplesofrealnumbers:R={(x,y,z):x,y,zR}TogeneralizeRandRtohigherdimensions,wefirstneedtodiscusstheconceptof

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