CONTENTS Preface Chapter 1,   LINEAR EQUATIONS 1.1 Introduction 1.2 Gaussian Elimination And Matrices 1.3 Gauss-Jordan Method 1.4 Two-Point Boundary-Value Problems 1.5 Making Gaussian Elimination Work 1.6 Ill-Conditioned Systems Chapter 2,   RECTANGULAR SYSTEMS AND ECHELON FORMS 2.1 Row Echelon Form And Rank 2.2 The Reduced Row Echelon Form 2.3 Consistency Of Linear Systems 2.4 Homogeneous Systems 2.5 Nonhomogeneous Systems 2.6 Electrical Circuits
Chapter 3,   MATRIX ALGEBRA
3.1 From Ancient China To Arthur Cayley
3.2 Addition, Scalar Multiplication, And Transposition
3.3 Linearity
3.4 Why Do It This Way
3.5 Matrix Multiplication
3.6 Properties Of Matrix Multiplication
3.7 Matrix Inversion
3.8 Inverses Of Sums and Sensitivity
3.9 Elementary Matrices And Equivalence
3.10 The LU Factorization

Chapter 4,   VECTOR SPACES
4.1 Spaces And Subspaces
4.2 Four Fundamental Subspaces
4.3 Linear Independence
4.4 Basis And Dimension
4.6 Classical Least Squares
4.7 Linear Transformations
4.8 Change Of Basis And Similarity
4.9 Invariant Subspaces

Chapter 5,   NORMS, INNER PRODUCTS, AND ORTHOGONALITY
5.1 Vector Norms
5.2 Matrix Norms
5.3 Inner Product Spaces
5.4 Orthogonal Vectors
5.5 Gram-Schmidt Procedure
5.6 Unitary and Orthogonal Matrices
5.7 Orthogonal Reduction
5.8 The Discrete Fourier Transform
5.9 Complementary Subspaces
5.10 Range-Nullspace Decomposition
5.11 Orthogonal Decomposition
5.12 Singular Value Decomposition
5.13 Orthogonal Projection
5.14 Why Least Squares?
5.15 Angles Between Subspaces

Chapter 6,   DETERMINANTS
6.1 Determinants

Chapter 7,   EIGENVALUES AND EIGENVECTORS
7.1 Elementary Properties Of Eigensystems
7.2 Diagonalization by Similarity Transformations
7.3 Functions Of Diagonalizable Matrices
7.4 Systems Of Differential Equations
7.5 Normal Matrices
7.6 Positive Definite Matrices
7.7 Nilpotent Matrices and Jordan Structure
7.8 The Jordan Form
7.9 Functions Of Non-diagonalizable Matrices
7.10 Difference Equations, Limits, And Summability
7.11 Minimum Polynomials and Krylov Methods

Chapter 8,   Perron-Frobenius Theory Of Nonnegative Matrices
8.1 Introduction
8.2 Positive Matrices
8.3 Nonnegative Matrices
8.4 Stochastic Matrices and Markov Chains

Index