Linear Algebra with Applications

Table of Contents Foreward Preface 1 Systems of Linear Equations 1.1 Solutions and Elementary Operations 1.2 Gaussian Elimination 1.3 Homogeneous Equations 1.4 An Application to Network Flow 1.5 An Application to Electrical Networks 1.6 An Application to Chemical Reactions Supplementary Exercises for Chapter 1 2 Matrix Algebra 2.1 Matrix Addition, Scalar Multiplication, and Transposition 2.2 Matrix-Vector Multiplication 2.3 Matrix Multiplication 2.4 Matrix Inverses 2.5 Elementary Matrices 2.6 Linear Transformations 2.7 LU-Factorization 2.8 An Application to Input-Output Economic Models 2.9 An Application to Markov Chains Supplementary Exercises for Chapter 2 3 Determinants and Diagonalization 3.1 The Cofactor Expansion 3.2 Determinants and Matrix Inverses 3.3 Eigenvalues and Eigenvectors 3.4 Diagonalization 3.5 Linear Dynamical Systems 3.6 An Application to Linear Recurrences 3.7 An Application to Systems of Differential Equations 3.8 Proof of the Cofactor Expansion Theorem Supplementary Exercises for Chapter 3 4 Vector Geometry 4.1 Vectors and Lines 4.2 Projections and Planes 4.3 More on the Cross Product 4.4 Linear Operators on R3 4.5 An Application to Computer Graphics Supplementary Exercises for Chapter 4 5 Vector Space Rn 5.1 Subspaces and Spanning 5.2 Independence and Dimension 5.3 Orthogonality 5.4 Rank of a Matrix 5.5 Similarity and Diagonalization 5.6 Best Approximation and Least Squares 5.7 An Application to Correlation and Variance Supplementary Exercises for Chapter 5 6 Vector Spaces 6.1 Examples and Basic Properties 6.2 Subspaces and Spanning Sets 6.3 Linear Independence and Dimension 6.4 Finite Dimensional Spaces 6.5 An Application to Polynomials 6.6 An Application to Differential Equations Supplementary Exercises for Chapter 6 7 Linear Transformations 7.1 Examples and Elementary Properties 7.2 Kernel and Image of a Linear Transformation 7.3 Isomorphisms and Composition 7.4 A Theorem about Differential Equations 7.5 More on Linear Recurrences 8 Orthogonality 8.1 Orthogonal Complements and Projections 8.2 Orthogonal Diagonalization 8.3 Positive Definite Matrices 8.4 QR-Factorization 8.5 Computing Eigenvalues 8.6 The Singular Value Decomposition 8.6.1 Singular Value Decompositions 8.6.2 Fundamental Subspaces 8.6.3 The Polar Decomposition of a Real Square Matrix 8.6.4 The Pseudoinverse of a Matrix 8.7 Complex Matrices 8.8 An Application to Linear Codes over Finite Fields 8.9 An Application to Quadratic Forms 8.10 An Application to Constrained Optimization 8.11 An Application to Statistical Principal Component Analysis 9 Change of Basis 9.1 The Matrix of a Linear Transformation 9.2 Operators and Similarity 9.3 Invariant Subspaces and Direct Sums 10 Inner Product Spaces 10.1 Inner Products and Norms 10.2 Orthogonal Sets of Vectors 10.3 Orthogonal Diagonalization 10.4 Isometries 10.5 An Application to Fourier Approximation 11 Canonical Forms 11.1 Block Triangular Form 11.2 The Jordan Canonical Form A Complex Numbers B Proofs C Mathematical Induction D Polynomials Selected Exercise Answers Index