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首页 Linear Algebra Throughly Explained(Milan).pdf

Linear Algebra Throughly Explained(Milan).pdf

Linear Algebra Throughly Explai…

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简介:本文档为《Linear Algebra Throughly Explained(Milan)pdf》,可适用于高等教育领域,主题内容包含LinearAlgebraThoroughlyExplainedMilanVujicˇicLinearAlgebraThoroughlyExplai符等。

LinearAlgebraThoroughlyExplainedMilanVujicˇicLinearAlgebraThoroughlyExplainedAuthorMilanVujicˇic(–)EditorJeffreySandersonEmeritusProfessor,SchoolofMathematicsStatistics,UniversityofStAndrews,StAndrews,ScotlandISBN:eISBN:LibraryofCongressControlNumber:cSpringerVerlagBerlinHeidelbergThisworkissubjecttocopyrightAllrightsarereserved,whetherthewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,reproductiononmicrofilmorinanyotherway,andstorageindatabanksDuplicationofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember,,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringerViolationsareliabletoprosecutionundertheGermanCopyrightLawTheuseofgeneraldescriptivenames,registerednames,trademarks,etcinthispublicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluseCoverDesign:eStudioCalamarSLPrintedonacidfreepaperspringercomForewordThereareazillionbooksonlinearalgebra,yetthisonefindsitsownuniqueplaceamongthemItstartsasanintroductionatundergraduatelevel,coverstheessentialresultsatpostgraduatelevelandreachesthefullpowerofthelinearalgebraicmethodsneededbyresearchers,especiallythoseinvariousfieldsofcontemporarytheoreticalphysics,appliedmathematicsandengineeringAtfirstsight,thetitleofthebookmayseemsomewhatpretentiousbutitfaithfullyreflectsitsobjectiveand,indeed,itsachievementsMilanVujicˇicstartedhisscientificcarrierintheoreticalnuclearphysicsinwhichhereliedheavilyinhisresearchproblemsonlinearalgebraicandgrouptheoreticmethodsSubsequently,hemovedtothefieldofgrouptheoryitselfanditsapplicationsinvarioustopicsinphysicsInparticular,heachieved,togetherwithFedorHerbut,importantresultsinthefoundationsofanddistantcorrelationsinquantummechanics,wherehisunderstandingandskillinlinearalgebrawasprecedentHewasknownasanacuteandlearnedmathematicalphysicistAtfirstVujicˇictaughtgrouptheoryatgraduatelevelHowever,histeachingcareerblossomedwhenhemovedtothePhysicsFacultyoftheUniversityofBelgrade,anditcontinued,evenafterretirement,attheUniversityofMalta,wherehetaughtlinearalgebraatthemostbasicleveltoteachingdiplomastudentsHecontinuouslyinterestedhimselfintheproblemsofteaching,andwithworthyresultsIndeed,hisdidacticworkswereoutstandingandhewasfrequentlysingledoutbystudents,intheirteachingevaluationquestionnaires,asasuperbteacherofmathematicalphysicsThisbookisbasedonlecturesthatVujicˇicgavetobothundergraduateandpostgraduatestudentsoveraperiodofseveraldecadesItsguidingprincipleistodevelopthesubjectrigorouslybuteconomically,withminimalprerequisitesandwithplentyofgeometricintuitionThebookoffersapracticalsystemofstudieswithanabundanceofworkedexamplescoordinatedinsuchawayastopermitthediligentstudenttoprogresscontinuouslyfromthefirsteasylessonstoarealmasteryofthesubjectThroughoutthisbook,theauthorhassucceededinmaintainingrigourwhilegivingthereaderanintuitiveunderstandingofthesubjectHehasimbuedthebookwiththesamegoodsenseandhelpfulnessthatcharacterizedhisteachingduringvviForewordhislifetimeSadly,havingjustcompletedthebook,MilanVujicˇicsuddenlydiedinDecemberHavingknownMilanwell,asmythesisadvisor,acolleagueandadearfriend,IamcertainthathewouldwishthisbooktobededicatedtohiswifeRadmilaandhissonsBorisandAndrejfortheirpatience,supportandloveBelgrade,JulyDjordjeˇSijacˇki,AcknowledgementsThanksareduetoseveralpeoplewhohavehelpedinvariouswaystobringProfessorVujicˇic’smanuscripttopublicationVladislavPavlovicˇproducedtheinitialLatexcopy,andsubsequently,DrPatriciaHeggieprovidedtimelyandinvaluabletechnicalhelpinthisareaProfessorsJohnCornwellandNikolaRuskucoftheUniversityofStAndrewsreadandmadehelpfulcommentsuponthemanuscriptinthelightofwhichProfessorMilanDamnjanovicˇoftheUniversityofBelgrademadesomeamendmentsFinally,itisapleasuretothankProfessorDjordjeˇSijacˇkioftheUniversityofBelgradeandtheSerbianAcademyofSciencesforwritingtheForewordviiContentsVectorSpacesIntroductionGeometricalVectorsinaPlaneVectorsinaCartesian(Analytic)PlaneRScalarMultiplication(TheProductofaNumberwithaVector)TheDotProductofTwoVectors(ortheEuclideanInnerProductofTwoVectorsinR)ApplicationsoftheDotProductandScalarMultiplicationVectorsinThreeDimensionalSpace(SpatialVectors)TheCrossProductinRTheMixedTripleProductinRApplicationsoftheCrossandMixedProductsEquationsofLinesinThreeDimensionalSpaceEquationsofPlanesinThreeDimensionalSpaceRealVectorSpacesandSubspacesLinearDependenceandIndependenceSpanningSubsetsandBasesTheThreeMostImportantExamplesofFiniteDimensionalRealVectorSpacesTheVectorSpaceRn(NumberColumns)TheVectorSpaceRnn(Matrices)TheVectorSpaceP(Polynomials)SomeSpecialTopicsaboutMatricesMatrixMultiplicationSomeSpecialMatricesADeterminantsADefinitionsofDeterminantsAPropertiesofDeterminantsixxContentsLinearMappingsandLinearSystemsAShortPlanfortheFirstSectionsofChapterSomeGeneralStatementsaboutMappingTheDefinitionofLinearMappings(Linmaps)TheKernelandtheRangeofLTheQuotientSpaceVnkerLandtheIsomorphismVnkerL′=ranLRepresentationTheoryTheVectorSpaceˆL(Vn,Wm)TheLinearMapM:RnRmTheThreeIsomorphismsv,wandvwHowtoCalculatetheRepresentingMatrixMAnExample(RepresentationofaLinmapWhichActsbetweenVectorSpacesofPolynomials)SystemsofLinearEquations(LinearSystems)TheFourTasksTheColumnSpaceandtheRowSpaceTwoExamplesofLinearDependenceofColumnsandRowsofaMatrixElementaryRowOperations(Eros)andElementaryMatricesErosElementaryMatricesTheGJFormofaMatrixAnExample(PreservationofLinearIndependenceandDependenceinGJForm)TheExistenceoftheReducedRowEchelon(GJ)FormforEveryMatrixTheStandardMethodforSolvingAX=bWhenDoesaConsistentSystemAX=bHaveaUniqueSolutionWhenaConsistentSystemAX=bHasNoUniqueSolutionTheGJMProcedure–aNewApproachtoSolvingLinearSystemswithNonuniqueSolutionsDetailedExplanationSummaryofMethodsforSolvingSystemsofLinearEquationsInnerProductVectorSpaces(EuclideanandUnitarySpaces)EuclideanSpacesEnUnitarySpacesUn(orComplexInnerproductVectorSpaces)OrthonormalBasesandtheGramSchmidtProcedureforOrthonormalizationofBasesDirectandOrthogonalSumsofSubspacesandtheOrthogonalComplementofaSubspaceDirectandOrthogonalSumsofSubspacesTheOrthogonalComplementofaSubspaceContentsxiDualSpacesandtheChangeofBasisTheDualSpaceUnofaUnitarySpaceUnTheAdjointOperatorTheChangeofBasesinVn(F)TheChangeoftheMatrixColumnξThatRepresentsaVectorxVn(F)(ContravariantVectors)TheChangeofthennMatrixAThatRepresentsanOperatorAˆL(Vn(F),Vn(F))(MixedTensoroftheSecondOrder)TheChangeofBasesinEuclidean(En)andUnitary(Un)VectorSpacesTheChangeofBiorthogonalBasesinVn(F)(CovariantVectors)TheRelationbetweenVn(F)andVn(F)isSymmetric(TheInvariantIsomorphismbetweenVn(F)andVn(F))IsodualismTheInvariantIsomorphismbetweentheSuperspacesˆL(Vn(F),Vn(F))andˆL(Vn(F),Vn(F))TheEigenProblemorDiagonalFormofRepresentingMatricesEigenvalues,Eigenvectors,andEigenspacesDiagonalizationofSquareMatricesDiagonalizationofanOperatorinUnTwoExamplesofNormalMatricesTheActualMethodforDiagonalizationofaNormalOperatorTheMostImportantSubsetsofNormalOperatorsinUnTheUnitaryOperatorsA†=ATheHermitianOperatorsA†=ATheProjectionOperatorsP†=P=POperationswithProjectionOperatorsTheSpectralFormofaNormalOperatorADiagonalizationofaSymmetricOperatorinETheActualProcedureforOrthogonalDiagonalizationofaSymmetricOperatorinEDiagonalizationofQuadraticFormsConicSectionsinRCanonicalFormofOrthogonalMatricesOrthogonalMatricesinRnOrthogonalMatricesinR(RotationsandReflections)TheCanonicalFormsofOrthogonalMatricesinR(RotationsandRotationswithInversions)TensorProductofUnitarySpacesKroneckerProductofMatricesAxiomsfortheTensorProductofUnitarySpacesTheTensorproductofUnitarySpacesCmandCnxiiContentsDefinitionoftheTensorProductofUnitarySpaces,inAnalogywiththePreviousExampleMatrixRepresentationoftheTensorProductofUnitarySpacesMultipleTensorProductsofaUnitarySpaceUnandofitsDualSpaceUnasthePrincipalExamplesoftheNotionofUnitaryTensorsUnitarySpaceofAntilinearOperatorsˆLa(Um,Un)astheMainRealizationofUmUnComparativeTreatmentofMatrixRepresentationsofLinearOperatorsfromˆL(Um,Un)andAntimatrixRepresentationsofAntilinearOperatorsfromˆLa(Um,Un)=UmUnTheDiracNotationinQuantumMechanics:DualismbetweenUnitarySpaces(Sect)andIsodualismbetweenTheirSuperspaces(Sect)RepeatingtheStatementsabouttheDualismDInvariantLinearandAntilinearBijectionsbetweentheSuperspacesˆL(Un,Un)andˆL(Un,Un)DualismbetweentheSuperspacesIsodualismbetweenUnitarySuperspacesSuperspacesˆL(Un,Un)ˆL(Un,Un)astheTensorProductofUnandUn,ie,UnUnTheTensorProductofUnandUnRepresentationandtheTensorNatureofDiadsTheProofofTensorProductPropertiesDiadRepresentationsofOperatorsBibliographyIndexChapterVectorSpacesIntroductionTheideaofavectorisoneofthegreatestcontributionstomathematics,whichcamedirectlyfromphysicsNamely,vectorsarebasicmathematicalobjectsofclassicalphysicssincetheydescribephysicalquantitiesthathavebothmagnitudeanddirection(displacement,velocity,acceleration,forces,egmechanical,electrical,magnetic,gravitational,etc)Geometricalvectors(arrows)intwodimensionalplanesandinthethreedimensionalspace(inwhichwelive)formrealvectorspacesdefinedbytheadditionofvectorsandthemultiplicationofnumberswithvectorsTobeabletodescribelengthsandangles(whichareessentialforphysicalapplications),realvectorspacesareprovidedwiththedotproductoftwovectorsSuchvectorspacesarethencalledEuclideanspacesThetheoryofrealvectorspacescanbegeneralizedtoincludeothersetsofobjects:thesetofallrealmatriceswithmrowsandncolumns,thesetofallrealpolynomials,thesetofallrealpolynomialswhoseorderissmallerthannN,thesetofallrealfunctionswhichhavethesamedomainofdefinition,thesetsofallcontinuous,differentiableorintegrablefunctions,thesetofallsolutionsofagivenhomogeneoussystemoflinearequations,etcMostofthesegeneralizedvectorspacesaremanydimensionalThemosttypicalandveryusefularethevectorspacesofmatrixcolumnsx=xxxnTofnrowsandonecolumn,wheren=,,,,andthecomponentsxi,i=,,,n,arerealnumbersWedenotethesespacesbyRn,whichistheusualnotationforthesetsoforderedntuplesofrealnumbers(Theorderedntuplescan,ofcourse,berepresentedbymatrixrowsx,xxnaswell,butthematrixcolumnsaremoreappropriatewhenwedealwithmatrixtransformationsinRn,whichareappliedtotheleftAx,whereAisanmnrealmatrix)WeshallcalltheelementsofRnnvectorsVectorSpacesThevectorspacesRnforn=,,playanimportantroleingeometry,describinglinesandplanes,aswellastheareaoftrianglesandparallelogramsandthevolumeofaparallelepipedThevectorspacesRnforn>havenogeometricalinterpretationNevertheless,theyareessentialformanyproblemsinmathematics(egforsystemsoflinearequations),inphysics(n=,space–timeeventsinthespecialtheoryofrelativity),aswellasineconomics(lineareconomicmodels)Modernphysics,inparticularQuantumMechanics,aswellasthetheoryofelementaryparticles,usescomplexvectorspacesAsfarasQuantumMechanicsisconcerned,therewereatfirsttwoapproaches:thewavemechanicsofSchrodingerandthematrixmechanicsofHeisenbergVonNeumannprovedthatbothareisomorphictotheinfinitedimensionalunitary(complex)vectorspace(calledHilbertspace)ThegeometryofHilbertspaceisnowuniversallyacceptedasthemathematicalmodelforQuantumMechanicsTheStandardModelofelementaryparticlestreatssev

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