Linear Algebra Done Right
Sheldon Axler
Contents
Preface to the Instructor
Preface to the Student
Acknowledgments
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