线性代数

章节摘录

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前言

This textbook giyes 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 instanc~。 the jntrinsic treatment of the classicaJ adjoint of a linear transformation in Chapter IV,as welI as the discussion of quaternions and the classifica. tion of associative division algebras in Chapter VII.Chapters XII and XlIl 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.I should like to thank Mr.M.L.Johnson who made many useful suggestions for the problems in the third edition.I am also grateful to my colleague S.Halperin who assisted in the revision of Chapters XII and XIII and to Mr.F.Gomez who helped to prepare the SUbject index.Finally，I have to express my deep gratitude to my colleague J.R.Van- stone who worked closely with me in the preparation of alI the revisions and additions and who generously helped with the proof reading.

内容概要

作者：(美国) 哥汝布 (Greub.W.)

书籍目录

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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作者简介

《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. ...

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