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  &