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Linear Algebra Thoroughly Explained Milan Vujicˇic´ Linear Algebra Thoroughly Explained Author Editor MilanVujicˇic´ JeffreySanderson (1931–2005) EmeritusProfessor, SchoolofMathematics&Statistics, UniversityofStAndrews, StAndrews, Scotland ISBN:978-3-540-74637-9 e-ISBN:978-3-540-74639-3 LibraryofCongressControlNumber:2007936399 (cid:2)c 2008Springer-VerlagBerlinHeidelberg Thisworkissubjecttocopyright. Allrightsarereserved,whetherthewholeorpartofthematerial is concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyright LawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liabletoprosecutionundertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. CoverDesign:eStudioCalamarS.L. Printedonacid-freepaper 9 8 7 6 5 4 3 2 1 springer.com Foreword Thereareazillionbooksonlinearalgebra,yetthisonefindsitsownuniqueplace among them. It starts as an introduction at undergraduatelevel, covers the essen- tial results at postgraduatelevel and reachesthe full power of the linear algebraic methodsneededbyresearchers,especiallythoseinvariousfieldsofcontemporary theoreticalphysics,appliedmathematicsandengineering.Atfirstsight,thetitleof thebookmayseemsomewhatpretentiousbutitfaithfullyreflectsitsobjectiveand, indeed,itsachievements. MilanVujicˇic´startedhisscientificcarrierintheoreticalnuclearphysicsinwhich he relied heavily in his research problemson linear algebraic and group theoretic methods. Subsequently,he movedto the field of grouptheory itself and its appli- cationsinvarioustopicsinphysics.Inparticular,heachieved,togetherwithFedor Herbut,importantresultsinthefoundationsofanddistantcorrelationsinquantum mechanics, where his understandingand skill in linear algebra was precedent.He wasknownasanacuteandlearnedmathematicalphysicist. AtfirstVujicˇic´ taughtgrouptheoryatgraduatelevel.However,histeachingca- reerblossomedwhenhemovedtothePhysicsFacultyoftheUniversityofBelgrade, anditcontinued,evenafterretirement,attheUniversityofMalta,wherehetaught linearalgebraatthemostbasicleveltoteachingdiplomastudents.Hecontinuously interestedhimselfintheproblemsofteaching,andwithworthyresults.Indeed,his didactic works were outstanding and he was frequently singled out by students, in their teaching evaluation questionnaires, as a superb teacher of mathematical physics. ThisbookisbasedonlecturesthatVujicˇic´ gavetobothundergraduateandpost- graduate students over a period of several decades. Its guiding principle is to de- velopthesubjectrigorouslybuteconomically,withminimalprerequisitesandwith plentyofgeometricintuition.Thebookoffersapracticalsystemofstudieswithan abundanceofworkedexamplescoordinatedinsuchawayastopermitthediligent studenttoprogresscontinuouslyfromthefirsteasylessonstoarealmasteryofthe subject.Throughoutthisbook,theauthorhassucceededinmaintainingrigourwhile givingthereaderanintuitiveunderstandingofthesubject.Hehasimbuedthebook with the same good sense and helpfulness that characterized his teaching during v vi Foreword hislifetime.Sadly,havingjustcompletedthebook,MilanVujicˇic´ suddenlydiedin December2005. HavingknownMilanwell,asmythesisadvisor,acolleagueandadearfriend,I amcertainthathewouldwishthisbooktobededicatedtohiswifeRadmilaandhis sonsBorisandAndrejfortheirpatience,supportandlove. Belgrade,July2007 DjordjeSˇijacˇki, Acknowledgements ThanksareduetoseveralpeoplewhohavehelpedinvariouswaystobringProfessor Vujicˇic´’s manuscript to publication. Vladislav Pavlovicˇ producedthe initial Latex copy,andsubsequently,Dr.PatriciaHeggieprovidedtimelyandinvaluabletechni- calhelpinthisarea.ProfessorsJohnCornwellandNikolaRuskucoftheUniversity of St. Andrewsread andmade helpfulcommentsuponthe manuscriptin the light of which Professor Milan Damnjanovicˇ of the University of Belgrade made some amendments.Finally,itisapleasuretothankProfessorDjordjeSˇijacˇkioftheUni- versityofBelgradeandtheSerbianAcademyofSciencesforwritingtheForeword. vii Contents 1 VectorSpaces.................................................. 1 1.1 Introduction ............................................... 1 1.2 GeometricalVectorsinaPlane ............................... 2 1.3 VectorsinaCartesian(Analytic)PlaneR2...................... 5 1.4 ScalarMultiplication(TheProductofaNumberwithaVector) .... 7 1.5 TheDotProductofTwoVectors(ortheEuclideanInnerProduct ofTwoVectorsinR2)....................................... 8 1.6 ApplicationsoftheDotProductandScalarMultiplication ........ 10 1.7 VectorsinThree-DimensionalSpace(SpatialVectors)............ 15 1.8 TheCrossProductinR3..................................... 18 1.9 The Mixed Triple Productin R3. Applicationsof the Cross andMixedProducts ........................................ 21 1.10 EquationsofLinesinThree-DimensionalSpace................. 24 1.11 EquationsofPlanesinThree-DimensionalSpace ................ 26 1.12 RealVectorSpacesandSubspaces ............................ 28 1.13 LinearDependenceandIndependence.SpanningSubsetsandBases 30 1.14 TheThreeMostImportantExamplesofFinite-DimensionalReal VectorSpaces.............................................. 33 1.14.1 TheVectorSpaceRn(NumberColumns) ................ 33 1.14.2 TheVectorSpaceRn×n(Matrices)...................... 35 1.14.3 TheVectorSpaceP (Polynomials) ..................... 37 3 1.15 SomeSpecialTopicsaboutMatrices........................... 39 1.15.1 MatrixMultiplication................................. 39 1.15.2 SomeSpecialMatrices................................ 40 A Determinants.................................................. 45 A.1 DefinitionsofDeterminants.................................. 45 A.2 PropertiesofDeterminants................................... 49 ix x Contents 2 LinearMappingsandLinearSystems ............................ 59 2.1 AShortPlanfortheFirst5SectionsofChapter2................ 59 2.2 SomeGeneralStatementsaboutMapping ...................... 60 2.3 TheDefinitionofLinearMappings(Linmaps) .................. 62 2.4 TheKernelandtheRangeofL ............................... 63 2.5 TheQuotientSpaceV /kerLandtheIsomorphismV /ker ∼=L(cid:2) ran L 65 n n 2.6 RepresentationTheory ...................................... 67 2.6.1 TheVectorSpaceLˆ(V ,W ) ........................... 68 n m 2.6.2 TheLinearMapM:Rn→Rm ......................... 69 2.6.3 TheThreeIsomorphismsv, wandv−w ................. 70 2.6.4 HowtoCalculatetheRepresentingMatrixM............. 72 2.7 AnExample(RepresentationofaLinmapWhichActsbetween VectorSpacesofPolynomials) ............................... 75 2.8 SystemsofLinearEquations(LinearSystems) .................. 79 2.9 TheFourTasks ............................................ 85 2.10 TheColumnSpaceandtheRowSpace ........................ 86 2.11 Two Examples of Linear Dependence of Columns andRowsofaMatrix ....................................... 88 2.12 ElementaryRowOperations(Eros)andElementaryMatrices...... 91 2.12.1 Eros ............................................... 91 2.12.2 ElementaryMatrices ................................. 93 2.13 TheGJFormofaMatrix .................................... 95 2.14 An Example (Preservation of Linear Independence andDependenceinGJForm)................................. 97 2.15 The Existence of the Reduced Row-Echelon (GJ) FormforEveryMatrix ...................................... 99 2.16 TheStandardMethodforSolvingAX¯ =b¯ .....................101 2.16.1 When Does a Consistent System AX¯ =b¯ Have aUniqueSolution?...................................102 2.16.2 WhenaConsistentSystemAX¯ =b¯ HasNo UniqueSolution .....................................108 2.17 TheGJMProcedure–aNewApproachtoSolvingLinearSystems withNonuniqueSolutions ...................................109 2.17.1 DetailedExplanation .................................110 2.18 SummaryofMethodsforSolvingSystemsofLinearEquations....116 3 Inner-ProductVectorSpaces(EuclideanandUnitarySpaces) .......119 3.1 EuclideanSpacesE ........................................119 n 3.2 UnitarySpacesU (orComplexInner-productVectorSpaces) .....126 n 3.3 Orthonormal Bases and the Gram-Schmidt Procedure forOrthonormalizationofBases ..............................131 3.4 DirectandOrthogonalSumsofSubspacesandtheOrthogonal ComplementofaSubspace ..................................139 3.4.1 DirectandOrthogonalSumsofSubspaces ...............139 3.4.2 TheOrthogonalComplementofaSubspace..............141 Contents xi 4 DualSpacesandtheChangeofBasis .............................145 ∗ 4.1 TheDualSpaceU ofaUnitarySpaceU ......................145 n n 4.2 TheAdjointOperator .......................................153 4.3 TheChangeofBasesinV (F)................................157 n 4.3.1 TheChangeoftheMatrix-ColumnξThatRepresents aVectorx¯∈V (F)(ContravariantVectors)...............158 n 4.3.2 The Change of then×nMatrix A ThatRepresents an Operator A∈Lˆ(V (F),V (F)) (Mixed Tensor n n oftheSecondOrder) .................................159 4.4 TheChangeofBasesinEuclidean(E )andUnitary(U )Vector n n Spaces....................................................162 4.5 The Change of Biorthogonal Bases in V∗(F) n (CovariantVectors) .........................................164 4.6 The Relation between V (F) and V∗(F) is Symmetric n n (TheInvariantIsomorphismbetweenV (F)andV∗∗(F))..........167 n n 4.7 Isodualism—TheInvariantIsomorphismbetweentheSuperspaces Lˆ(V (F),V (F))andLˆ(V∗(F),V∗(F)) .........................168 n n n n 5 TheEigenProblemorDiagonalFormofRepresentingMatrices.....173 5.1 Eigenvalues,Eigenvectors,andEigenspaces ....................173 5.2 DiagonalizationofSquareMatrices ...........................180 5.3 DiagonalizationofanOperatorinU ..........................183 n 5.3.1 TwoExamplesofNormalMatrices .....................188 5.4 TheActualMethodforDiagonalizationofaNormalOperator .....191 5.5 TheMostImportantSubsetsofNormalOperatorsinU ..........194 n 5.5.1 TheUnitaryOperatorsA†=A−1 .......................194 5.5.2 TheHermitianOperatorsA†=A .......................198 5.5.3 TheProjectionOperatorsP†=P=P2 ..................200 5.5.4 OperationswithProjectionOperators ...................203 5.5.5 TheSpectralFormofaNormalOperatorA...............207 5.6 DiagonalizationofaSymmetricOperatorinE .................208 3 5.6.1 TheActualProcedureforOrthogonalDiagonalization ofaSymmetricOperatorinE .........................214 3 5.6.2 DiagonalizationofQuadraticForms ....................218 5.6.3 ConicSectionsinR2 .................................220 5.7 CanonicalFormofOrthogonalMatrices .......................228 5.7.1 OrthogonalMatricesinRn ............................228 5.7.2 OrthogonalMatricesinR2 (RotationsandReflections).....229 5.7.3 The CanonicalFormsof OrthogonalMatricesin R3 (RotationsandRotationswithInversions)................240 6 TensorProductofUnitarySpaces................................243 6.1 KroneckerProductofMatrices ...............................243 6.2 AxiomsfortheTensorProductofUnitarySpaces................247 6.2.1 TheTensorproductofUnitarySpacesCmandCn .........247 xii Contents 6.2.2 Definition of the Tensor Productof Unitary Spaces, inAnalogywiththePreviousExample ..................249 6.3 MatrixRepresentationoftheTensorProductofUnitarySpaces ....250 6.4 MultipleTensorProductsofaUnitarySpaceU andofitsDual n ∗ SpaceU asthePrincipalExamplesoftheNotionofUnitary n Tensors ...................................................252 6.5 UnitarySpaceofAntilinearOperatorsLˆ (U ,U )astheMain a m n RealizationofU ⊗U ......................................254 m n 6.6 ComparativeTreatmentofMatrixRepresentationsof Linear Operatorsfrom Lˆ(U ,U ) and Antimatrix Representations m n ofAntilinearOperatorsfromLˆ (U ,U )=U ⊗U .............257 a m n m n 7 The Dirac Notation in Quantum Mechanics: Dualism between Unitary Spaces (Sect. 4.1) and Isodualism betweenTheirSuperspaces(Sect.4.7) ............................263 7.1 RepeatingtheStatementsabouttheDualismD..................263 7.2 Invariant Linear and Antilinear Bijections between theSuperspacesLˆ(U ,U )andLˆ(U∗,U∗) ......................266 n n n n 7.2.1 DualismbetweentheSuperspaces ......................266 7.2.2 IsodualismbetweenUnitarySuperspaces ................267 7.3 SuperspacesLˆ(U ,U ) Lˆ(U∗,U∗)astheTensorProductofU n n n n n andU∗,i.e.,U ⊗U∗........................................270 n n n ∗ 7.3.1 TheTensorProductofU andU .......................270 n n 7.3.2 RepresentationandtheTensorNatureofDiads ...........271 7.3.3 TheProofofTensorProductProperties..................272 7.3.4 DiadRepresentationsofOperators......................274 Bibliography.......................................................279 Index .............................................................281