TABLE OF CONTENTS
1 Vectors and Matrices
1.1 Vectors
1.2 Lengths and Dot Products
1.3 Planes
1.4 Matrices and Linear Equations
2 Solving Linear Equations
2.1 The Idea of Elimination
2.2 Elimination Using Matrices
2.3 Rules for Matrix Operations
2.4 Inverse Matrices
2.5 Elimination = Factorization: A = LU
2.6 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 of A: Solving Ax = b
3.4 Independence, Basis, and Dimension
3.5 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
7 Linear Transformations
7.1 The Idea of a Linear Transformation
7.2 The Matrix of a Linear Transformation
7.3 Choice of Basis: Similarity and Diagonalization
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
List of MATLAB Teaching Codes