线性代数：第4版

ChapterO.Prerequisites

ChapterⅠ.Vectorspaces

§1.Vectorspaces

§2.Linearmappings

§3.Subspacesandfactorspaces

§4.Dhneasion.

§5.Thetopologyofarealfinitedimensionalvectorspace

ChapterⅡ.Linearmappings

§1.Baskproperties.

§2.Operationswithlinearmappings

§3.Linearisomorphisms

§4.Directsumofvectorspaces

§5.Dualvectorspaces

§6.Finitedimensionalvectorspaces

ChapterⅢMatrices

§1.Matricesandsystemsoflinearequations

§2.Multiplicationofmatrices

§3.Basistransformation

§4.Elementarytransformations

ChapterⅣ.Determinants

§1.Determinantfunctions

§2.Thedeterminantofalineartransformation

§3.Thedeterminantofamatrix

§4.Dualdeterminantfunctions

§5.Theadjointmatrix

§6.Thecharacteristicpolynomial

§7.Thetrace

§8.Orientedvectorspaces

ChapterⅤ.Algebras

§1.Basicproperties

§2.Ideals

§3.Changeofcoeffientfieldofavectorspace

ChapterⅥ.Gradationsandhomology

§1.G-gradedvectorspaces

§2.G-gradedalgebras

§3.Differentialspacesanddifferentialalsebras

ChapterⅦ.Innerproductspaces

§1.Theinnerproduct

§2.Orthonormalbases

§3.Normeddeterminantfunctions

§4.Dualityinaninnerproductspace

§5.Normedvectorspaces

§6.Thealgebraofquaternions

ChapterⅧ.Linearmappingsofinnerproductspaces

§1.Theadjointmapping

§2.Selfadjointmappings

§3.Orthogonalprojections

§4.Skewmappings

§5.Isometricmappings

§6.RotationsofEuclideanspacesofdimension2,3and4

§7.Different~ablefamiliesoflinearautomorphisms

ChapterⅨ.Symmetricbilinaarfunctions

§1.Bilinearandquadraticfunctions

§2.ThedecompositionofE

§3.Pairsofsymmetricbilinearfunctions

§4.Pseudo-Euclideanspaces

§5.LinearmappingsofPseudo-Euclideanspaces

ChapterⅩ.Quadries

§1.Arlenespaces

§2.Quadricsintheaffmespace

§3.Affineequivalenceofquadrics

§4.QuadricsintheEuclideanspace

ChapterⅪ.Unitaryspaces

§1.Hermitianfunctions

§2.Unitaryspaces

§3.Linearmappingsofunitaryspaces

§4.UnitarymappingsofthecomplexDiane

§5.ApplicationtoLorentz-transformations

ChapterⅫ.Polynomialalgebra

§1.Basicproperties

§2.Idealsanddivisibility

§3.Factoralgebras

§4.Thestructureoffactoralgebras

ChapterⅩⅢ.Theoryofalineartransformation

§1.Polynomialsinalineartransformation

§2.Generalizedeigenspaces

§3.Cyclicspaces

§4.Irreduciblespaces

§5.Applicationofcyclicspaces

§6.Nilpotentandsere~simpletransformations

§7.Applicationstoinnerproductspaces

Bibliography

SubjectIndex

　　Thistextbookgivesadetailedandcomprehensivepresentationoflinearalgebrabasedonanaxiomatictreatmentoflinearspaces.Forthisfourtheditionsomenewmaterialhasbeenaddedtothetext,forinstance,theintrinsictreatmentoftheclassicaladjointofalineartransformationinChapterIV,aswellasthediscussionofquaternionsandtheclassificationofassociativedivisionalgebrasinChapterVII.ChaptersXIIandXIIIhavebeensubstantiallyrewrittenforthesakeofclarity,butthecontentsremainbasicallythesameasbefore.Finally,anumberofproblemscoveringnewtopics-e.g.complexstructures,Caylaynumbersandsymplecticspaces-havebeenadded.