Linear Algebra

Vector Spaces and Linear Transformations

Meighan I. Dillon

• Publisher: American Mathematical Society
• Year: 2022
• Print ISBN: 9781470469863
• eBook ISBN: 9781470472009
• Topic: Matemáticas
• Subtopic: Matemáticas general

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
• Chapter 2. Linear Transformations and Subspaces
• Chapter 3. Matrices and Coordinates
• Chapter 4. Systems of Linear Equations
• Chapter 5. Introductions
• Chapter 6. The Determinant Is a Multilinear Mapping
• Chapter 7. Inner Product Spaces
• Chapter 8. The Life of a Linear Operator
• Chapter 9. Similarity
• Chapter 10. 𝐺𝐿_{𝑛}(𝔽) and Friends
• Appendix A. Background Review
• Appendix B. ℝ² and ℝ³
• Appendix C. More Set Theory
• Appendix D. Infinite Dimension
• Bibliography
• Index
• Back Cover