Linear Algebra Done Right
by Sheldon Axler

Chapter 1. Vector Spaces

1. Complex Numbers
2. Definition of Vector Space
3. Properties of Vector Spaces
4. Subspaces
5. Sums and Direct Sums

Chapter 2. Finite-Dimensional Vector Spaces

1. Span and Linear Independence
2. Bases
3. Dimension

Chapter 3. Linear Maps

1. Definitions and Examples
2. Null Spaces and Ranges
3. The Matrix of a Linear Map
4. Invertibility

Chapter 4. Polynomials

1. Degree
2. Complex Coefficients
3. Real Coefficients

Chapter 5. Eigenvalues and Eigenvectors

1. Invariant Subspaces
2. Polynomials Applied to Operators
3. Upper-Triangular Matrices
4. Diagonal Matrices
5. Invariant Subspaces on Real Vector Spaces

Chapter 6. Inner-Product Spaces

1. Inner Products
2. Norms
3. Orthonormal Bases
4. Orthogonal Projections and Minimization Problems
5. Linear Functionals and Adjoints

Chapter 7. Operators on Inner-Product Spaces

1. Self-Adjoint and Normal Operators
2. The Spectral Theorem
3. Normal Operators on Real Inner-Product Spaces
4. Positive Operators
5. Isometries
6. Polar and Singular-Value Decompositions

Chapter 8. Operators on Complex Vector Spaces

1. Generalized Eigenvectors
2. The Characteristic Polynomial
3. Decomposition of an Operator
4. Square Roots
5. The Minimal Polynomial
6. Jordan Form

Chapter 9. Operators on Real Vector Spaces

1. Eigenvalues of Square Matrices
2. Block Upper-Triangular Matrices
3. The Characteristic Polynomial

Chapter 10. Trace and Determinant

1. Change of Basis
2. Trace
3. Determinant of an Operator
4. Determinant of a Matrix
5. Volume

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More information at the author's own page: http://www.axler.net/LADR.html

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	Linear Algebra Done Right
	by Sheldon Axler