1 Introduction

1.1 Vectors and Matrices
1.2 Lengths and Dot Products

2 Solving Linear Equations

2.1 Linear Equations
2.2 The Idea of Elimination
2.3 Elimination Using Matrices
2.4 Rules for Matrix Operations
2.5 Inverse Matrices
2.6 Elimination = Factorization: A = LU
2.7 Transposes and Permutations

3 Vector Spaces and Subspaces

3.1 Spaces of Vectors
3.2 The Nullspace of A: Solving Ax = 0
3.3 The Rank and the Row Reduced Form
3.4 The Complete Solution to Ax=b
3.5 Independence, Basis, and Dimension
3.6 Dimensions of the Four Subspaces

4 Orthogonality

4.1 Orthogonality of the Four Subspaces
4.2 Projections
4.3 Least Squares Approximations
4.4 Orthogonal Bases and Gram-Schmidt

5 Determinants

5.1 The Properties of Determinants
5.2 Permutations and Cofactors
5.3 Cramer's Rule, Inverses, and Volumes

6 Eigenvalues and Eigenvectors

6.1 Introduction to Eigenvalues
6.2 Diagonalizing a Matrix
6.3 Applications to Differential Equations
6.4 Symmetric Matrices
6.5 Positive Definite Matrices
6.6 Similar Matrices
6.7 The Singular Value Decomposition

7 Linear Transformations

7.1 The Idea of a Linear Transformation
7.2 The Matrix of a Linear Transformation
7.3 Change of Basis
7.4 Diagonalization and the Pseudoinverse

8 Applications

8.1 Graphs and Networks
8.2 Markov Matrices and Economic Models
8.3 Linear Programming
8.4 Fourier Series: Linear Algebra for Functions
8.5 Computer Graphics

9 Numerical Linear Algebra

9.1 Gaussian Elimination in Practice
9.2 Norms and Condition Numbers
9.3 Iterative Methods for Linear Algebra

10 Complex Vectors and Complex Matrices

10.1 Complex Numbers
10.2 Hermitian and Unitary Matrices
10.3 The Fast Fourier Transform

Solutions to Selected Exercises
Index