抽象代数讲义 第2卷：第2卷 线性代数

图书标准信息

• 作者 Nathan Jacobson 著
• 出版社 世界图书出版公司
• 出版时间 2000-12
• 版次 1
• ISBN 9787506200615
• 定价 45.00元
• 装帧 平装
• 开本 24开
• 纸张 胶版纸
• 页数 280页
• 正文语种 英语
• 原版书名 Lectures in Abstract Algebra 2 Linear Algebra

【内容简介】

Thepresentvolumeisthesecondintheauthor'sseriesofthreedealingwithabstractalgebra.ForanunderstandingofthisvolumeacertainfamiliaritywiththebasicconceptstreatedinVolumeI：groups,rings,fields,homomorphisms,ispresupposed.However,wehavetriedtomakethisaccountoflinearalgebraindependentofadetailedknowledgeofourfirstvolume.Referencestospecificresultsaregivenoccasionallybutsomeofthefundamentalconceptsneededhavebeentreatedagain.Inshort,itishopedthatthisvolumecanbereadwithcompleteunderstandingbyany'studentwhoismathematicallysufficientlymatureandwhohasafamiliaritywiththestandardnotionsofmodernalgebra.

【目录】

CHAPTERI：FINITEDIMENSIONAVECTORSPACES
1.Abstractvectorspaces
2.Rightvectorspaces
3.o-modules
4.Lineardependence
5.Invarianceofdimensionality
6.Basesandmatrices
7.Applicationstomatrixtheory
8.Rankofasetofvectors
9.Factorspaces
10.Algebraofsubspaces
11.Independentsubspaces，directsums

CHAPTERII：LINEARTRANSFORMATIONS
1.Definitionandexamples
2.Compositionsoflineartransformations
3.Thematrixofalineartransformation
4.Compositionsofmatrices
5.Changeofbasis.Equivalenceandsimilarityofmatrices
6.Rankspaceandnullspaceofalineartransformation
7.Systemsoflinearequations
8.Lineartransformationsinrightvectorspaces
9.Linearfunctions
10.Dualitybetweenafinitedimensionalspaceanditsconjugatespace
11.Transposeofalineartransformation
12.Matricesofthetranspose
13.Projections

CHAPTERIII：THETHEORYOFASINGLELINEARTRANSFORMATION
1.Theminimumpolynomialofalineartransformation
2.Cyclicsubspaces
3.Existenceofavectorwhoseorderistheminimumpolynomial
4.Cycliclineartransformations
5.The[]-moduledeterminedbyalineartransformation
6.Finitelygeneratedo-modules，o，aprincipalidealdomain
7.NormalizationofthegeneratorsofFandof
8.Equivalenceofmatriceswithelementsinaprincipalidealdomain
9.Structureoffinitelygenerateda-modules
10.Invariancetheorems
11.Decompositionofavectorspacerelativetoalineartransformation
12.Thecharacteristicandminimumpolynomials
13.DirectproofofTheorem13
14.Formalpropertiesofthetraceandthecharacteristicpolynomial
15.Theringofa-endomorphismsofacyclico-module
16.Determinationoftheringofa-endomorphismsofafinitelygeneratedomodule，oprincipal
17.Thelineartransformationswhichcommutewithagivenlineartransformation
18.Thecenterofthering

CHAPTERIV：SETSOFLINEARTRANSFORMATIONS
1.Invariantsubspaces
2.Inducedlineartransformations
3.Compositionseries
4.Decomposability
5.Completereducibility
6.Relationtothetheoryofoperatorgroupsandthetheoryofmodules
7.Reducibility，decomposability，completereducibilityforasinglelineartransformation
8.Theprimarycomponentsofaspacerelativetoalineartransformation
9.Setsofcommutativelineartransformations

CHAPTERV：BILINEARFORMS
1.Bilinearforms
2.Matricesofabilinearform
3.Non-degenerateforms
4.Transposeofalineartransformationrelativetoapairofbilinearforms
5.Anotherrelationbetweenlineartransformationsandbilinearforms
6.Scalarproducts
7.Hermitianscalarproducts
8.Matricesofhermitianscalarproducts
9.Symmetricandhermitianscalarproductsoverspecialdivisionrings
10.Alternatescalarproducts
11.Witt''stheorem
12.Non-alternateskew-symmetricforms

CHAPTERVI：EUCLIDEANANDUNITARYSPACES
1.Cartesianbases
2.Lineartransformationsandscalarproducts
3.Orthogonalcompletereducibility
4.Symmetric，skewandorthogonallineartransformations
5.Canonicalmatricesforsymmetricandskewlineartransformations
6.Commutativesymmetricandskewlineartransformations
7.Normalandorthogonallineartransformations
8.Semi-definitetransformations
9.Polarfactorizationofanarbitrarylineartransformation
10.Unitarygeometry
11.Analyticfunctionsoflineartransformations

CHAPTERVII：PRODUCTSOFVECTORSPACES
1.Productgroupsofvectorspaces
2.Directproductsoflineartransformations
3.Two-sidedvectorspaces
4.TheKroneckerproduct
5.Kroneckerproductsoflineartransformationsandofmatrices
6.Tensorspaces
7.Symmetryclassesoftensors
8.Extensionofthefieldofavectorspace
9.Atheoremonsimilarityofsetsofmatrices
10.Alternativedefinitionofanalgebra.Kroneckerproductofalgebras

CHAPTERviii：THERINGOFLINEARTRANSFORMATIONS
1.Simplicityof
2.Operatormethods
3.Theleftidealsof
4.Rightideals
5.Isomorphismsofringsoflineartransformations

CHAPTERIX：INFINITEDIMENSIONALVECTORSPACES
1.Existenceofabasis
2.Invarianceofdimensionality
3.Subspaces
4.Lineartransformationsandmatrices
5.Dimensionalityoftheconjugatespace
6.Finitetopologyforlineartransformations
7.TotalsubspacesofR*
8.Dualspaces.Kroneckerproducts
9.Two-sidedidealsintheringoflineartransformations
10.Denseringsoflineartransformations
11.Isomorphismtheorems
12.Anti-automorphismsandscalarproducts
13.Schur''slemma.Ageneraldensitytheorem
14.Irreduciblealgebrasoflineartransformations
Index