Cover -- Half title -- Title -- Copyright -- Dedication -- Contents -- Preface -- Chapter 1 Vectors and linear systems -- 1.1 Vectors and linear combinations -- 1.2 Length, angle and dot products -- 1.3 Matrices -- Chapter 2 Solving linear systems -- 2.1 Vectors and linear equations -- 2.2 Matrix operations -- 2.3 Inverse matrices -- 2.4 LU decomposition -- 2.5 Transpose and permutation -- Chapter 3 Vector spaces -- 3.1 Spaces of vectors -- 3.2 Nullspace, row space and column space -- 3.3 Solutions of Ax = b -- 3.4 Rank of matrices -- 3.5 Bases and dimensions of general vector spaces -- Chapter 4 Orthogonality -- 4.1 Orthogonality of the four subspaces -- 4.2 Projections -- 4.3 Least squares approximations -- 4.4 Orthonormal bases and Gram-Schmidt -- Chapter 5 Determinants -- 5.1 Introduction to determinants -- 5.2 Properties of determinants -- Chapter 6 Eigenvalues and eigenvectors -- 6.1 Introduction to eigenvectors and eigenvalues -- 6.2 Diagonalizability -- 6.3 Applications to differential equations -- 6.4 Symmetric matrices and quadratic forms -- 6.5 Positive definite matrices -- Chapter 7 Singular value decomposition -- 7.1 Singular value decomposition -- 7.2 Principal component analysis -- Chapter 8 Linear transformations -- 8.1 Linear transformation and matrix representation -- 8.2 Range and null spaces of linear transformation -- 8.3 Invariant subspaces -- 8.4 Decomposition of vector spaces -- 8.5 Jordan normal form -- 8.6 Computation of Jordan normal form -- Chapter 9 Linear programming -- 9.1 Extreme points -- 9.2 Simplex method -- 9.3 Simplex tableau -- Index