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抽象代数讲义第2卷：第2卷线性代数

抽象代数讲义第2卷：第2卷线性代数

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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

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

分类: 自然科学

13人买过

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

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