Basic Algebra

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Basic Algebra - Mathematics and Statistics - McGill University

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[PDF] Basic Algebra - Stony Brook Mathematics

DigitalSecondEditions

ByAnthonyW.Knapp

BasicAlgebra

AdvancedAlgebra

BasicRealAnalysis,

AdvancedRealAnalysis

Anthony W. Knapp

Basic Algebra

Along with a Companion VolumeAdvanced Algebra

Digital Second Edition, 2016

Published by the Author

East Setauket, New York

AnthonyW.Knapp

81UpperSheepPastureRoad

EastSetauket,N.Y.11733-1729,U.S.A.

Emailto:aknapp@math.stonybrook.edu

Homepage:www.math.stonybrook.edu/?aknapp

Title:BasicAlgebra

68P30.

FirstEdition,ISBN-13978-0-8176-3248-9

PublishedbyBirkh¨auserBoston

DigitalSecondEdition,nottobesold,noISBN

PublishedbytheAuthor

MediaInc.

ToSusan

ToMyChildren,SarahandWilliam,

ToMyAlgebraTeachers:

RalphFox,JohnFraleigh,RobertGunning,

CONTENTS

ContentsofAdvancedAlgebrax

PrefacetotheSecondEditionxi

PrefacetotheFirstEditionxiii

ListofFiguresxvii

DependenceAmongChaptersxix

StandardNotationxx

GuidefortheReaderxxi

I.PRELIMINARIESABOUTTHEINTEGERS,

POLYNOMIALS,ANDMATRICES1

1.DivisionandEuclideanAlgorithms1

2.UniqueFactorizationofIntegers4

3.UniqueFactorizationofPolynomials9

4.PermutationsandTheirSigns15

5.RowReduction19

6.MatrixOperations24

7.Problems30

1.Spanning,LinearIndependence,andBases33

2.VectorSpacesDefinedbyMatrices38

3.LinearMaps42

4.DualSpaces50

5.QuotientsofVectorSpaces54

7.Determinants65

9.BasesintheInfinite-DimensionalCase78

10.Problems82

III.INNER-PRODUCTSPACES89

1.InnerProductsandOrthonormalSets89

2.Adjoints99

3.SpectralTheorem105

4.Problems112

IV.GROUPSANDGROUPACTIONS117

1.GroupsandSubgroups118

2.QuotientSpacesandHomomorphisms129

3.DirectProductsandDirectSums135

4.RingsandFields141

5.PolynomialsandVectorSpaces148

6.GroupActionsandExamples159

7.SemidirectProducts167

8.SimpleGroupsandCompositionSeries171

10.SylowTheorems185

11.CategoriesandFunctors189

12.Problems200

V.THEORYOFASINGLELINEARTRANSFORMATION211

1.Introduction211

3.CharacteristicandMinimalPolynomials218

4.ProjectionOperators226

5.PrimaryDecomposition228

6.JordanCanonicalForm231

7.ComputationswithJordanForm238

8.Problems241