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Linear Algebra Done Right

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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.

From the reviews of previous editions:

Altogether, the text is a didactic masterpiece. ― zbMATH

The determinant-free proofs are elegant and intuitive. ― American Mathematical Monthly

The most original linear algebra book to appear in years, it certainly belongs in every undergraduate library ― CHOICE



Author(s): Sheldon Axler
Series: Undergraduate Texts in Mathematics
Edition: 4
Publisher: Springer
Year: 2023

Language: English
Commentary: Mathematics Subject Classification (2020): 15-01, 15A03, 15A04, 15A15, 15A18, 15A21 | Publisher PDF | Publishing Due: 07 December 2023
Pages: xvii, 390
City: Cham
Tags: Dual Spaces; Finite-Dimensional Spectral Theorem; Linear Algebra; Multilinear Algebras; Matrix Theory; Product Spaces; Quotient Spaces; Vector Spaces

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