Linear Algebra: Vector Spaces and Linear Transformations

Description

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. 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