Linear Algebra Done Right [4 ed.] 3031410254, 9783031410253

Table of contents :
Cover
Title Page
Copyright
About the Author
Contents
Preface for Students
Preface for Instructors
The author’s top ten
Major improvements and additions for the fourth edition
Acknowledgments
Chapter 1. Vector Spaces
1A 𝐑ⁿ and 𝐂ⁿ
Complex Numbers
Lists
𝐅ⁿ
Digression on Fields
Exercises 1A
1B Definition of Vector Space
Exercises 1B
1C Subspaces
Sums of Subspaces
Direct Sums
Exercises 1C
Chapter 2. Finite-Dimensional Vector Spaces
2A Span and Linear Independence
Linear Combinations and Span
Linear Independence
Exercises 2A
2B Bases
Exercises 2B
2C Dimension
Exercises 2C
Chapter 3. Linear Maps
3A Vector Space of Linear Maps
Definition and Examples of Linear Maps
Algebraic Operations on ℒ(𝑉, 𝑊)
Exercises 3A
3B Null Spaces and Ranges
Null Space and Injectivity
Range and Surjectivity
Fundamental Theorem of Linear Maps
Exercises 3B
3C Matrices
Representing a Linear Map by a Matrix
Addition and Scalar Multiplication of Matrices
Matrix Multiplication
Column–Row Factorization and Rank of a Matrix
Exercises 3C
3D Invertibility and Isomorphisms
Invertible Linear Maps
Isomorphic Vector Spaces
Linear Maps Thought of as Matrix Multiplication
Change of Basis
Exercises 3D
3E Products and Quotients of Vector Spaces
Products of Vector Spaces
Quotient Spaces
Exercises 3E
3F Duality
Dual Space and Dual Map
Null Space and Range of Dual of Linear Map
Matrix of Dual of Linear Map
Exercises 3F
Chapter 4. Polynomials
Zeros of Polynomials
Division Algorithm for Polynomials
Factorization of Polynomials over 𝐂
Factorization of Polynomials over 𝐑
Exercises 4
Chapter 5. Eigenvalues and Eigenvectors
5A Invariant Subspaces
Eigenvalues
Polynomials Applied to Operators
Exercises 5A
5B The Minimal Polynomial
Existence of Eigenvalues on Complex Vector Spaces
Eigenvalues and the Minimal Polynomial
Eigenvalues on Odd-Dimensional Real Vector Spaces
Exercises 5B
5C Upper-Triangular Matrices
Exercises 5C
5D Diagonalizable Operators
Diagonal Matrices
Conditions for Diagonalizability
Gershgorin Disk Theorem
Exercises 5D
5E Commuting Operators
Exercises 5E
Chapter 6. Inner Product Spaces
6A Inner Products and Norms
Inner Products
Norms
Exercises 6A
6B Orthonormal Bases
Orthonormal Lists and the Gram–Schmidt Procedure
Linear Functionals on Inner Product Spaces
Exercises 6B
6C Orthogonal Complements and Minimization Problems
Orthogonal Complements
Minimization Problems
Pseudoinverse
Exercises 6C
Chapter 7. Operators on Inner Product Spaces
7A Self-Adjoint and Normal Operators
Adjoints
Self-Adjoint Operators
Normal Operators
Exercises 7A
7B Spectral Theorem
Real Spectral Theorem
Complex Spectral Theorem
Exercises 7B
7C Positive Operators
Exercises 7C
7D Isometries, Unitary Operators, and Matrix Factorization
Isometries
Unitary Operators
QR Factorization
Cholesky Factorization
Exercises 7D
7E Singular Value Decomposition
Singular Values
SVD for Linear Maps and for Matrices
Exercises 7E
7F Consequences of Singular Value Decomposition
Norms of Linear Maps
Approximation by Linear Maps with Lower-Dimensional Range
Polar Decomposition
Operators Applied to Ellipsoids and Parallelepipeds
Volume via Singular Values
Properties of an Operator as Determined by Its Eigenvalues
Exercises 7F
Chapter 8. Operators on Complex Vector Spaces
8A Generalized Eigenvectors and Nilpotent Operators
Null Spaces of Powers of an Operator
Generalized Eigenvectors
Nilpotent Operators
Exercises 8A
8B Generalized Eigenspace Decomposition
Generalized Eigenspaces
Multiplicity of an Eigenvalue
Block Diagonal Matrices
Exercises 8B
8C Consequences of Generalized Eigenspace Decomposition
Square Roots of Operators
Jordan Form
Exercises 8C
8D Trace: A Connection Between Matrices and Operators
Exercises 8D
Chapter 9. Multilinear Algebra and Determinants
9A Bilinear Forms and Quadratic Forms
Bilinear Forms
Symmetric Bilinear Forms
Quadratic Forms
Exercises 9A
9B Alternating Multilinear Forms
Multilinear Forms
Alternating Multilinear Forms and Permutations
Exercises 9B
9C Determinants
Defining the Determinant
Properties of Determinants
Exercises 9C
9D Tensor Products
Tensor Product of Two Vector Spaces
Tensor Product of Inner Product Spaces
Tensor Product of Multiple Vector Spaces
Exercises 9D
Photo Credits
Symbol Index
Index
Colophon: Notes on Typesetting

Citation preview

Undergraduate Texts in Mathematics

Sheldon Axler

Linear Algebra Done Right Fourth Edition

Undergraduate Texts in Mathematics

Undergraduate Texts in Mathematics

Series Editors Pamela Gorkin, Mathematics Department, Bucknell University, Lewisburg, PA, USA Jessica Sidman, Mathematics and Statistics, Amherst College, Amherst, MA, USA Advisory Board Colin Adams, Williams College, Williamstown, MA, USA Jayadev S. Athreya, University of Washington, Seattle, WA, USA Nathan Kaplan, University of California, Irvine, CA, USA Jill Pipher, Brown University, Providence, RI, USA Jeremy Tyson, University of Illinois at Urbana-Champaign, Urbana, IL, USA

Undergraduate Texts in Mathematics are generally aimed at third- and fourthyear undergraduate mathematics students at North American universities. These texts strive to provide students and teachers with new perspectives and novel approaches. The books include motivation that guides the reader to an appreciation of interrelations among different aspects of the subject. They feature examples that illustrate key concepts as well as exercises that strengthen understanding.

Sheldon Axler

Linear Algebra Done Right Fourth Edition

Sheldon Axler San Francisco, CA, USA

ISSN 0172-6056 ISSN 2197-5604 (electronic) Undergraduate Texts in Mathematics ISBN 978-3-031-41025-3 ISBN 978-3-031-41026-0 (eBook) https://doi.org/10.1007/978-3-031-41026-0 Mathematics Subject Classification (2020): 15-01, 15A03, 15A04, 15A15, 15A18, 15A21 © Sheldon Axler 1996, 1997, 2015, 2024. This book is an open access publication. Open Access This book is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/), which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made. The images or other third party material in this book are included in the book’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the book’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. This work is subject to copyright. All commercial rights are reserved by the author(s), whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Regarding these commercial rights a non-exclusive license has been granted to the publisher. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland. Paper in this product is recyclable.

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