Now available in Open Access, this best-selling textbook for a second course in linear algebra is aimed at undergraduate math majors and graduate students. The fourth edition gives an expanded treatment of the singular value decomposition and its consequences. It includes a new chapter on multilinear algebra, treating bilinear forms, quadratic forms, tensor products, and an approach to determinants via alternating multilinear forms. This new edition also increases the use of the minimal polynomial to provide cleaner proofs of multiple results. Also, over 250 new exercises have been added.
The novel approach taken here banishes determinants to the end of the book. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on finite-dimensional vector spaces. The author has taken unusual care to motivate concepts and simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. Beautiful formatting creates pages with an unusually student-friendly appearance in both print and electronic versions.
No prerequisites are assumed other than the usual demand for suitable mathematical maturity. The text starts by discussing vector spaces, linear independence, span, basis, and dimension. The book then deals with linear maps, eigenvalues, and eigenvectors. Inner-product spaces are introduced, leading to the finite-dimensional spectral theorem and its consequences. Generalized eigenvectors are then used to provide insight into the structure of a linear operator.
Conditions of Use
This book is licensed under a Creative Commons License (CC BY-NC-SA). You can download the ebook Linear Algebra Done Right, 4th Edition for free.
- Title
- Linear Algebra Done Right, 4th Edition
- Publisher
- Springer
- Author(s)
- Sheldon Axler
- Published
- 2023-11-20
- Edition
- 4
- Format
- eBook (pdf, epub, mobi)
- Pages
- 407
- Language
- English
- ISBN-10
- 3031410254
- ISBN-13
- 9783031410253
- License
- CC BY-NC-SA
- Book Homepage
- Free eBook, Errata, Code, Solutions, etc.
Linear Algebra Done Right About the Author Contents Preface for Students Preface for Instructors Acknowledgments Vector Spaces R^n and C^n Complex Numbers Lists F^n Digression on Fields Exercises 1A Definition of Vector Space Exercises 1B Subspaces Sums of Subspaces Direct Sums Exercises 1C Finite-Dimensional Vector Spaces Span and Linear Independence Linear Combinations and Span Linear Independence Exercises 2A Bases Exercises 2B Dimension Exercises 2C Linear Maps Vector Space of Linear Maps Definition and Examples of Linear Maps Algebraic Operations on L(V, W) Exercises 3A Null Spaces and Ranges Null Space and Injectivity Range and Surjectivity Fundamental Theorem of Linear Maps Exercises 3B 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 Invertibility and Isomorphisms Invertible Linear Maps Isomorphic Vector Spaces Linear Maps Thought of as Matrix Multiplication Change of Basis Exercises 3D Products and Quotients of Vector Spaces Products of Vector Spaces Quotient Spaces Exercises 3E Duality Dual Space and Dual Map Null Space and Range of Dual of Linear Map Matrix of Dual of Linear Map Exercises 3F Polynomials Zeros of Polynomials Division Algorithm for Polynomials Factorization of Polynomials over C Factorization of Polynomials over R Exercises Eigenvalues and Eigenvectors Invariant Subspaces Eigenvalues Polynomials Applied to Operators Exercises 5A The Minimal Polynomial Existence of Eigenvalues on Complex Vector Spaces Eigenvalues and the Minimal Polynomial Eigenvalues on Odd-Dimensional Real Vector Spaces Exercises 5B Upper-Triangular Matrices Exercises 5C Diagonalizable Operators Diagonal Matrices Conditions for Diagonalizability Gershgorin Disk Theorem Exercises 5D Commuting Operators Exercises 5E Inner Product Spaces Inner Products and Norms Inner Products Norms Exercises 6A Orthonormal Bases Orthonormal Lists and the Gram–Schmidt Procedure Linear Functionals on Inner Product Spaces Exercises 6B Orthogonal Complements and Minimization Problems Orthogonal Complements Minimization Problems Pseudoinverse Exercises 6C Operators on Inner Product Spaces Self-Adjoint and Normal Operators Adjoints Self-Adjoint Operators Normal Operators Exercises 7A Spectral Theorem Real Spectral Theorem Complex Spectral Theorem Exercises 7B Positive Operators Exercises 7C Isometries, Unitary Operators, and Matrix Factorization Isometries Unitary Operators QR Factorization Cholesky Factorization Exercises 7D Singular Value Decomposition Singular Values SVD for Linear Maps and for Matrices Exercises 7E 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 Operators on Complex Vector Spaces Generalized Eigenvectors and Nilpotent Operators Null Spaces of Powers of an Operator Generalized Eigenvectors Nilpotent Operators Exercises 8A Generalized Eigenspace Decomposition Generalized Eigenspaces Multiplicity of an Eigenvalue Block Diagonal Matrices Exercises 8B Consequences of Generalized Eigenspace Decomposition Square Roots of Operators Jordan Form Exercises 8C Trace: A Connection Between Matrices and Operators Exercises 8D Multilinear Algebra and Determinants Bilinear Forms and Quadratic Forms Bilinear Forms Symmetric Bilinear Forms Quadratic Forms Exercises 9A Alternating Multilinear Forms Multilinear Forms Alternating Multilinear Forms and Permutations Exercises 9B Determinants Defining the Determinant Properties of Determinants Exercises 9C 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
