Preface 

xi  
To the Instructor 

xix  
To the Student 

xxv  


1  (56) 

Introduction: The Racetrack Game 


1  (2) 

The Geometry and Algebra of Vectors 


3  (13) 

Length and Angle: The Dot Product 


16  (16) 

Exploration: Vectors and Geometry 


30  (2) 


32  (16) 

Exploration: The Cross Product 


46  (2) 

Code Vectors and Modular Arithmetic 


48  (9) 

Systems of Linear Equations 


57  (74) 


57  (1) 

Introduction to Systems of Linear Equations 


58  (9) 

Exploration: Lies My Computer Told Me 


65  (2) 

Direct Methods for Solving Linear Systems 


67  (22) 

Exploration: Partial Pivoting 


85  (1) 

Exploration: Counting OperationsAn Introduction to the Analysis of Algorithms 


86  (3) 

Spanning Sets and Linear Independence 


89  (12) 


101  (19) 

Iterative Methods for Solving Linear Systems 


120  (11) 


131  (116) 

Introduction: Matrices in Action 


131  (2) 


133  (17) 


150  (10) 


160  (19) 

Subspaces, Basis, Dimension, and Rank 


179  (22) 

Introduction to Linear Transformations 


201  (16) 


217  (30) 

Exploration: The LU Factorization 


241  (6) 

Eigenvalues and Eigenvectors 


247  (103) 

Introduction: A Dynamical System on Graphs 


247  (2) 

Introduction to Eigenvalues and Eigenvectors 


249  (7) 


256  (30) 

Exploration: Geometric Applications of Determinants 


280  (6) 

Eigenvalues and Eigenvectors of nXn Matrices 


286  (10) 

Similarity and Diagonalization 


296  (11) 

Iterative Methods for Computing Eigenvalues 


307  (11) 

Applications and the PerronFrobenius Theorem 


318  (32) 


350  (75) 

Introduction: Shadows on a Wall 


350  (3) 


353  (11) 

Orthogonal Complements and Orthogonal Projections 


364  (11) 

The GramSchmidt Process and the QR Factorization 


375  (14) 

Exploration: The Modified QR Factorization 


384  (3) 

Exploration: Approximating Eigenvalues with the QR Algorithm 


387  (2) 

Orthogonal Diagonalization of Symmetric Matrices 


389  (9) 


398  (27) 


425  (112) 

Introduction: Magic Squares 


425  (2) 

Vector Spaces and Subspaces 


427  (16) 

Linear Independence, Basis, and Dimension 


443  (16) 


459  (11) 


470  (10) 

The Kernel and Range of a Linear Transformation 


480  (16) 

The Matrix of a Linear Transformation 


496  (24) 

Exploration: Tilings, Lattices, and the Crystallographic Restriction 


517  (3) 


520  (17) 

Distance and Approximation 


537  (98) 

Introduction: Taxicab Geometry 


537  (2) 


539  (18) 

Exploration: Geometric Inequalities and Optimization Problems 


552  (5) 

Norms and Distance Functions 


557  (18) 

Least Squares Approximation 


575  (23) 

The Singular Value Decomposition 


598  (22) 


620  (15) 
Appendix A Mathematical Notation and Methods of Proof 

635  (10) 
Appendix B Mathematical Induction 

645  (8) 
Appendix C Complex Numbers 

653  (12) 
Appendix D Polynomials 

665  (12) 
Appendix E Technology Bytes 

677  (46) 
Answers to Selected OddNumbered Exercises 

723  (33) 
Index 

756  