Softcover ISBN:  9781470469863 
Product Code:  AMSTEXT/57 
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eBook ISBN:  9781470472009 
Product Code:  AMSTEXT/57.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Softcover ISBN:  9781470469863 
eBook: ISBN:  9781470472009 
Product Code:  AMSTEXT/57.B 
List Price:  $170.00 $127.50 
MAA Member Price:  $153.00 $114.75 
AMS Member Price:  $136.00 $102.00 
Softcover ISBN:  9781470469863 
Product Code:  AMSTEXT/57 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
eBook ISBN:  9781470472009 
Product Code:  AMSTEXT/57.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Softcover ISBN:  9781470469863 
eBook ISBN:  9781470472009 
Product Code:  AMSTEXT/57.B 
List Price:  $170.00 $127.50 
MAA Member Price:  $153.00 $114.75 
AMS Member Price:  $136.00 $102.00 

Book DetailsPure and Applied Undergraduate TextsVolume: 57; 2023; 367 ppMSC: Primary 15
This textbook is directed towards students who are familiar with matrices and their use in solving systems of linear equations. The emphasis is on the algebra supporting the ideas that make linear algebra so important, both in theoretical and practical applications. The narrative is written to bring along students who may be new to the level of abstraction essential to a working understanding of linear algebra. The determinant is used throughout, placed in some historical perspective, and defined several different ways, including in the context of exterior algebras. The text details proof of the existence of a basis for an arbitrary vector space and addresses vector spaces over arbitrary fields. It develops LUfactorization, Jordan canonical form, and real and complex inner product spaces. It includes examples of inner product spaces of continuous complex functions on a real interval, as well as the background material that students may need in order to follow those discussions. Special classes of matrices make an entrance early in the text and subsequently appear throughout. The last chapter of the book introduces the classical groups.
ReadershipUndergraduate and graduate students interested in linear algebra.

Table of Contents

Cover

Title page

Contents

List of Figures

Preface

How To Use This Book

Notation and Terminology

To the Student

Introduction

Chapter 1. Vector Spaces

1.1. Fields

1.2. Vector Spaces

1.3. Spanning and Linear Independence

1.4. Bases

1.5. Polynomials

1.6. ℝ and ℂ in Linear Algebra

Chapter 2. Linear Transformations and Subspaces

2.1. Linear Transformations

2.2. Cosets and Quotient Spaces

2.3. Affine Sets and Mappings

2.4. Isomorphism and the Rank Theorem

2.5. Sums, Products, and Projections

Chapter 3. Matrices and Coordinates

3.1. Matrices

3.2. Coordinate Vectors

3.3. Change of Basis

3.4. Vector Spaces of Linear Transformations

3.5. Equivalences

Chapter 4. Systems of Linear Equations

Introduction

4.1. The Solution Set

4.2. Elementary Matrices

4.3. Reduced Row Echelon Form

4.4. Row Equivalence

4.5. An Early Use of the Determinant

4.6. LUFactorization

Chapter 5. Introductions

5.1. Dual Spaces

5.2. Transposition and Duality

5.3. Bilinear Forms, Their Matrices, and Duality

5.4. Linear Operators and Direct Sums

5.5. Groups of Matrices

5.6. SelfAdjoint and Unitary Matrices

Chapter 6. The Determinant Is a Multilinear Mapping

6.1. Multilinear Mappings

6.2. Alternating Multilinear Mappings

6.3. Permutations, Part I

6.4. Permutations, Part II

6.5. The Determinant

6.6. Properties of the Determinant

Chapter 7. Inner Product Spaces

7.1. The Dot Product: Under the Hood

7.2. Inner Products

7.3. Length and Angle

7.4. Orthonormal Sets

7.5. Orthogonal Complements

7.6. Inner Product Spaces of Functions

7.7. Unitary Transformations

7.8. The Adjoint of an Operator

7.9. A Fundamental Theorem

Chapter 8. The Life of a Linear Operator

8.1. Factoring Polynomials

8.2. The Minimal Polynomial

8.3. Eigenvalues

8.4. The Characteristic Polynomial

8.5. Diagonalizability

8.6. SelfAdjoint Matrices Are Diagonalizable

8.7. Rotations and Translations

Chapter 9. Similarity

9.1. Triangularization

9.2. The Primary Decomposition

9.3. Nilpotent Operators, Part I

9.4. Nilpotent Operators, Part II

9.5. Jordan Canonical Form

Chapter 10. 𝐺𝐿_{𝑛}(𝔽) and Friends

10.1. More about Groups

10.2. Homomorphisms and Normal Subgroups

10.3. The Quaternions

10.4. The Special Linear Group

10.5. The Projective Group

10.6. The Orthogonal Group

10.7. The Unitary Group

10.8. The Symplectic Group

Appendix A. Background Review

A.1. Logic and Proof

A.2. Sets

A.3. WellDefinedness

A.4. Counting

A.5. Equivalence Relations

A.6. Mappings

A.7. Binary Operations

Appendix B. ℝ² and ℝ³

B.1. Vectors

B.2. The Real Plane

B.3. The Complex Numbers and ℝ²

B.4. Real 3Space

B.5. The Dot Product

B.6. The CrossProduct

Appendix C. More Set Theory

C.1. Partially Ordered Sets

C.2. Zorn’s Lemma

Appendix D. Infinite Dimension

Bibliography

Index

Back Cover


Additional Material

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This textbook is directed towards students who are familiar with matrices and their use in solving systems of linear equations. The emphasis is on the algebra supporting the ideas that make linear algebra so important, both in theoretical and practical applications. The narrative is written to bring along students who may be new to the level of abstraction essential to a working understanding of linear algebra. The determinant is used throughout, placed in some historical perspective, and defined several different ways, including in the context of exterior algebras. The text details proof of the existence of a basis for an arbitrary vector space and addresses vector spaces over arbitrary fields. It develops LUfactorization, Jordan canonical form, and real and complex inner product spaces. It includes examples of inner product spaces of continuous complex functions on a real interval, as well as the background material that students may need in order to follow those discussions. Special classes of matrices make an entrance early in the text and subsequently appear throughout. The last chapter of the book introduces the classical groups.
Undergraduate and graduate students interested in linear algebra.

Cover

Title page

Contents

List of Figures

Preface

How To Use This Book

Notation and Terminology

To the Student

Introduction

Chapter 1. Vector Spaces

1.1. Fields

1.2. Vector Spaces

1.3. Spanning and Linear Independence

1.4. Bases

1.5. Polynomials

1.6. ℝ and ℂ in Linear Algebra

Chapter 2. Linear Transformations and Subspaces

2.1. Linear Transformations

2.2. Cosets and Quotient Spaces

2.3. Affine Sets and Mappings

2.4. Isomorphism and the Rank Theorem

2.5. Sums, Products, and Projections

Chapter 3. Matrices and Coordinates

3.1. Matrices

3.2. Coordinate Vectors

3.3. Change of Basis

3.4. Vector Spaces of Linear Transformations

3.5. Equivalences

Chapter 4. Systems of Linear Equations

Introduction

4.1. The Solution Set

4.2. Elementary Matrices

4.3. Reduced Row Echelon Form

4.4. Row Equivalence

4.5. An Early Use of the Determinant

4.6. LUFactorization

Chapter 5. Introductions

5.1. Dual Spaces

5.2. Transposition and Duality

5.3. Bilinear Forms, Their Matrices, and Duality

5.4. Linear Operators and Direct Sums

5.5. Groups of Matrices

5.6. SelfAdjoint and Unitary Matrices

Chapter 6. The Determinant Is a Multilinear Mapping

6.1. Multilinear Mappings

6.2. Alternating Multilinear Mappings

6.3. Permutations, Part I

6.4. Permutations, Part II

6.5. The Determinant

6.6. Properties of the Determinant

Chapter 7. Inner Product Spaces

7.1. The Dot Product: Under the Hood

7.2. Inner Products

7.3. Length and Angle

7.4. Orthonormal Sets

7.5. Orthogonal Complements

7.6. Inner Product Spaces of Functions

7.7. Unitary Transformations

7.8. The Adjoint of an Operator

7.9. A Fundamental Theorem

Chapter 8. The Life of a Linear Operator

8.1. Factoring Polynomials

8.2. The Minimal Polynomial

8.3. Eigenvalues

8.4. The Characteristic Polynomial

8.5. Diagonalizability

8.6. SelfAdjoint Matrices Are Diagonalizable

8.7. Rotations and Translations

Chapter 9. Similarity

9.1. Triangularization

9.2. The Primary Decomposition

9.3. Nilpotent Operators, Part I

9.4. Nilpotent Operators, Part II

9.5. Jordan Canonical Form

Chapter 10. 𝐺𝐿_{𝑛}(𝔽) and Friends

10.1. More about Groups

10.2. Homomorphisms and Normal Subgroups

10.3. The Quaternions

10.4. The Special Linear Group

10.5. The Projective Group

10.6. The Orthogonal Group

10.7. The Unitary Group

10.8. The Symplectic Group

Appendix A. Background Review

A.1. Logic and Proof

A.2. Sets

A.3. WellDefinedness

A.4. Counting

A.5. Equivalence Relations

A.6. Mappings

A.7. Binary Operations

Appendix B. ℝ² and ℝ³

B.1. Vectors

B.2. The Real Plane

B.3. The Complex Numbers and ℝ²

B.4. Real 3Space

B.5. The Dot Product

B.6. The CrossProduct

Appendix C. More Set Theory

C.1. Partially Ordered Sets

C.2. Zorn’s Lemma

Appendix D. Infinite Dimension

Bibliography

Index

Back Cover