Linear Algebra: Vector Spaces and Linear Transformations

✍ Scribed by Meighan I. Dillon

• Publisher: American Mathematical Society
• Year: 2022
• Tongue: English
• Leaves: 395
• Series: Pure and Applied Undergraduate Texts, 57
• Category: Library

✦ Synopsis

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 LU-factorization, 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.

✦ 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. LU-Factorization
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. Self-Adjoint 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. Self-Adjoint 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. Well-Definedness
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 3-Space
B.5. The Dot Product
B.6. The Cross-Product
Appendix C. More Set Theory
C.1. Partially Ordered Sets
C.2. Zorn’s Lemma
Appendix D. Infinite Dimension
Bibliography
Index
Back Cover