Book:Sheldon Axler/Linear Algebra Done Right/Second Edition

Subject Matter

 * Linear Algebra

Contents

 * Preface to the Instructor
 * Preface to the Student
 * Acknowledgements


 * CHAPTER 1: Vector Spaces
 * Complex Numbers
 * Definition of Vector Space
 * Properties of Vector Spaces
 * Subspaces
 * Sums and Direct Sums
 * Exercises


 * CHAPTER 2: Finite-Dimensional Vector Spaces
 * Span and Linear Independence
 * Bases
 * Dimension
 * Exercises


 * CHAPTER 3: Linear Maps
 * Definitions and Examples
 * Null Spaces and Ranges
 * The Matrix of a Linear Map
 * Invertibility
 * Exercises


 * CHAPTER 4: Polynomials
 * Degree
 * Complex Coefficients
 * Real Coefficients
 * Exercises


 * CHAPTER 5: Eigenvalues and Eigenvectors
 * Invariant Subspaces
 * Polynomials Applied to Operators
 * Upper-Triangular Matrices
 * Diagonal Matrices
 * Invariant Subspaces on Real Vector Spaces
 * Exercises


 * CHAPTER 6: Inner-Product Spaces
 * Inner Products
 * Norms
 * Orthonormal Bases
 * Orthogonal Projections and Minimization Problems
 * Linear Functionals and Adjoints
 * Exercises


 * CHAPTER 7: Operators on Inner-Product Spaces
 * Self-Adjoint and Normal Operators
 * The Spectral Theorem
 * Normal Operators on Real Inner-Product Spaces
 * Positive Operators
 * Isometries
 * Polar and Singular-Value Decompositions
 * Exercises


 * CHAPTER 8: Operators on Complex Vector Spaces
 * Generalized Eigenvectors
 * The Characteristic Polynomial
 * Decomposition of an Operator
 * Square Roots
 * The Minimal Polynomial
 * Jordan Form
 * Exercises


 * CHAPTER 9: Operators on Real Vector Spaces
 * Eigenvalues of Square Matrices
 * Block Upper-Triangular Matrices
 * The Characteristic Polynomial
 * Exercises


 * CHAPTER 10: Trace and Determinant
 * Change of Basis
 * Trace
 * Determinant of an Operator
 * Determinant of a Matrix
 * Volume
 * Exercises


 * Symbol Index
 * Index