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

## Sheldon Axler: Linear Algebra Done Right (2nd Edition)

Published $\text {1997}$, Springer.

### 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
Exercises
CHAPTER 7: Operators on Inner-Product Spaces
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