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

More information at the author's own page: http://linear.axler.net/