Elementary Linear Algebra

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