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 &
