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 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 3 DETERMINANTS 192

        3.1 Cofactor Expansion 192

        3.2 Properties of Determinants 204

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 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 6 ORTHOGONALITY 347

        6.1 The Geometry of Vectors 347

        6.2 Orthogonal Vectors 360

        6.3 Orthogonal Projections 374

        6.4 Least-Squares Approximations and Orthogonal Projections 388

        6.5 Orthogonal Matrices and Operators 398

        6.6 Symmetric Matrices 412

        6.7∗ Singular Value Decomposition 425

        6.8∗ Principal Component Analysis 443

        6.9∗ Rotations of R3 and Computer Graphics 452

CHAPTER 7 VECTOR SPACES 473

        7.1 Vector Spaces and Their Subspaces 473

        7.2 Linear Transformations 485

        7.3 Basis and Dimension 495

        7.4 Matrix Representations of Linear Operators 505

           7.5 Inner Product Spaces 517

APPENDICES 534

A Sets 534

B Functions 536

C Complex Numbers 539

D Matlab 544

E The Uniqueness of the Reduced Row Echelon Form 558

REFERENCES 561

ANSWERS TO SELECTED EXERCISES 562

LIST OF FREQUENTLY USED SYMBOLS 608

INDEX 609

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