PREFACE ix   TO THE STUDENT xv   CHAPTER 1 MATRICES, VECTORS, AND SYSTEMS OF LINEAR EQUATIONS 1  1.1 Matrices and Vectors 1  1.2 Linear Combinations, Matrix—Vector Products, and Special Matrices 11  1.3 Systems of Linear Equations 25  1.4 Gaussian Elimination 39  1.5* Applications of Systems of Linear Equations 54  1.6 The Span of a Set of Vectors 64  1.7 Linear Dependence and Linear Independence 73  Chapter 1 Review Exercises  Chapter 1 MATLAB Exercises   CHAPTER 2 MATRICES AND LINEAR TRANSFORMATIONS 90  2.1 Matrix Multiplication 90  2.2* Applications of Matrix Multiplication 101  2.3 Invertibility and Elementary Matrices 117  2.4 The Inverse of a Matrix 130  2.5* Partitioned Matrices and Block Multiplication 141  2.6* The LU Decomposition of a Matrix 147  2.7 Linear Transformations and Matrices 162  2.8 Composition and Invertibility of Linear Transformations 175  Chapter 2 Review Exercises  Chapter 2 MATLAB Exercises   CHAPTER 3 DETERMINANTS 192  3.1 Cofactor Expansion 192  3.2 Properties of Determinants 204  Chapter 3 Review Exercises  Chapter 3 MATLAB Exercises   CHAPTER 4 SUBSPACES AND THEIR PROPERTIES 218  4.1 Subspaces 218  4.2 Basis and Dimension 232  4.3 The Dimension of Subspaces Associated with a Matrix 245  4.4 Coordinate Systems 254  4.5 Matrix Representations of Linear Operators 266  Chapter 4 Review Exercises  Chapter 4 MATLAB Exercises   CHAPTER 5 EIGENVALUES, EIGENVECTORS, AND DIAGONALIZATION 282  5.1 Eigenvalues and Eigenvectors 282  5.2 The Characteristic Polynomial 291  5.3 Diagonalization of Matrices 302  5.4* Diagonalization of Linear Operators 314  5.5* Applications of Eigenvalues 323  Chapter 5 Review Exercises  Chapter 5 MATLAB Exercises     CHAPTER 6 VECTOR SPACES 473  6.1 Vector Spaces and Their Subspaces 473  6.2 Linear Transformations 485  6.3 Basis and Dimension 495  6.4 Matrix Representations of Linear Operators 505  6 .5 Inner Product Spaces 517  Chapter 6 Review Exercises  Chapter 6 MATLAB Exercises   CHAPTER 7 ORTHOGONALITY 347  7.1 The Geometry of Vectors 347  7.2 Orthogonal Vectors 360  7.3 Orthogonal Projections 374  7.4 Least-Squares Approximations and Orthogonal Projections 388  7.5 Orthogonal Matrices and Operators 398  7.6 Symmetric Matrices 412 &

Examples in book are accompanied by similar practice problems that enable students to test their understanding of the material Complete solutions to the practice problems are included with the text All computational exercises are designed so that the calculations involve “nice” numbers. Most sections include approximately twenty true/false exercises designed to test a students understanding of the conceptual ideas in each section. (Answers to every true/false are given in the book. Each chapter ends with a set of review exercises that provide practice with all the main topics of each chapter. For a proof-oriented course, the authors have included a significant number of accessible exercises requiring proofs. They are ordered according to difficulty. Book includes wide variety of applications such as: Economics (the Leontief input-output modelElectrical networks (current flow through an electrical network)Population change (the Leslie matrix)Traffic flowSchedulingGoogle searches The authors have added an appendix providing an introduction to MATLAB.