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《线性代数(第4版)》讲述了：This textbook gives a detailed and comprehensive presentation of linear algebra based on an axiomatic treatment of linear spaces. For this fourth edition some new material has been added to the text, for instance, the intrinsic treatment of the classical adjoint of a linear transformation in Chapter IV, as well as the discussion of quaternions and the classification of associative division algebras in Chapter VII. Chapters XII and XIII have been substantially rewritten for the sake of clarity, but the contents remain basically the same as before. Finally, a number of problems covering new topics- e.g. complex structures, Caylay numbers and symplectic spaces- have been added. ...

Chapter0.Prerequisites
ChapterI.Vector spaces
1.Vector spaces
2.Linear mappings
3.Subspaces and factor spaces
4.Dimension
5.Thetopology of arealfinitedimensional vector space
ChapterII.Linearmappings
1.Basicproperties
2.Operationswithlinearmappings
3.Linearisomorphisms
4.Directsumofvectorspaces
5.Dualvectorspaces
6.Finitedimensionalvectorspaces
ChapterIII.Matrices
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.Changeofcoefficientfieldofavectorspace
ChapterVI.Gradationsandhomology
1.G-gradedvectorspaces
2.G-gradedalgebras
3.Differentialspacesanddifferentialalgebras
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.Differentiablefamiliesoflinearautomorphisms
ChapterIX.Symmetricbilinearfunctions
1.Bilinearandquadraticfunctions
2.ThedecompositionofE
3.Pairsofsymmetricbilinearfunctions
4.Pseudo-Euclideanspaces
5.LinearmappingsofPseudo-Euclideanspaces
ChapterX.Quadrics
1.Affinespaces
2.Quadricsintheaffinespace
3.Affmeequivalenceofquadrics
4.QuadricsintheEuclideanspace
ChapterXI.Unitaryspaces
1.Hermitianfunctions
2.Unitaryspaces
3.Linearmappingsofunitaryspaces
4.Unitarymappingsofthecomplexplane
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.Nilpotentandsemisimpletransformations
7.Applicationstoinnerproductspaces
Bibliography
SubjectIndex
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