Preface 

vii  
To the Instructor 

xvii  
To the Student 

xxiii  


1  (57) 

Introduction: The Racetrack Game 


1  (2) 

The Geometry and Algebra of Vectors 


3  (12) 

Length and Angle: The Dot Product 


15  (16) 

Exploration: Vectors and Geometry 


29  (2) 


31  (16) 

Exploration: The Cross Product 


45  (2) 

Code Vectors and Modular Arithmetic 


47  (11) 

Vignette: The Codabar System 


55  (1) 


56  (2) 

Systems of Linear Equations 


58  (76) 


58  (1) 

Introduction to Systems of Linear Equations 


59  (9) 

Exploration: Lies My Computer Told Me 


66  (2) 

Direct Methods for Solving Linear Systems 


68  (22) 

Exploration: Partial Pivoting 


86  (1) 

Exploration: Counting Operations: An Introduction to the Analysis of Algorithms 


87  (3) 

Spanning Sets and Linear Independence 


90  (11) 


101  (21) 


101  (2) 

Balancing Chemical Equations 


103  (1) 


104  (2) 


106  (3) 


109  (10) 

Vignette: The Global Positioning System 


119  (3) 

Iterative Methods for Solving Linear Systems 


122  (12) 


132  (2) 


134  (118) 

Introduction: Matrices in Action 


134  (2) 


136  (16) 


152  (9) 


161  (17) 


178  (11) 

Subspaces, Basis, Dimension, and Rank 


189  (20) 

Introduction to Linear Transformations 


209  (19) 


224  (4) 


228  (24) 


228  (5) 


233  (2) 


235  (5) 


240  (10) 


250  (2) 

Eigenvalues and Eigenvectors 


252  (111) 

Introduction: A Dynamical System on Graphs 


252  (1) 

Introduction to Eigenvalues and Eigenvectors 


253  (9) 


262  (27) 

Exploration: Geometric Applications of Determinants 


283  (6) 

Eigenvalues and Eigenvectors of nxn Matrices 


289  (9) 

Similarity and Diagonalization 


298  (10) 

Iterative Methods for Computing Eigenvalues 


308  (14) 

Applications and the PerronFrobenius Theorem 


322  (41) 


322  (5) 


327  (2) 

The PerronFrobenius Theorem 


329  (3) 

Linear Recurrence Relations 


332  (5) 

Systems of Linear Differential Equations 


337  (8) 

Discrete Linear Dynamical Systems 


345  (8) 

Vignette: Ranking Sports Teams and Searching the Internet 


353  (8) 


361  (2) 


363  (68) 

Introduction: Shadows on a Wall 


363  (2) 


365  (10) 

Orthogonal Complements and Orthogonal Projections 


375  (10) 

The GramSchmidt Process and the QR Factorization 


385  (12) 

Exploration: The Modified QR Factorization 


393  (2) 

Exploration: Approximating Eigenvalues with the QR Algorithm 


395  (2) 

Orthogonal Diagonalization of Symmetric Matrices 


397  (8) 


405  (26) 


405  (6) 


411  (7) 

Graphing Quadratic Equations 


418  (11) 


429  (2) 


431  (107) 

Introduction: Fibonacci in (Vector) Space 


431  (2) 

Vector Spaces and Subspaces 


433  (14) 

Linear Independence, Basis, and Dimension 


447  (20) 

Exploration: Magic Squares 


464  (3) 


467  (9) 


476  (9) 

The Kernel and Range of a Linear Transformation 


485  (16) 

The Matrix of a Linear Transformation 


501  (21) 

Exploration: Tilings, Lattices, and the Crystallographic Restriction 


519  (3) 


522  (16) 

Homogeneous Linear Differential Equations 


522  (7) 


529  (7) 


536  (2) 

Distance and Approximation 


538  (96) 

Introduction: Taxicab Geometry 


538  (2) 


540  (21) 

Exploration: Vectors and Matrices with Complex Entries 


552  (4) 

Exploration: Geometric Inequalities and Optimization Problems 


556  (5) 

Norms and Distance Functions 


561  (16) 

Least Squares Approximation 


577  (22) 

The Singular Value Decomposition 


599  (20) 

Vignette: Digital Image Compression 


616  (3) 


619  (15) 

Approximation of Functions 


619  (7) 


626  (5) 


631  (3) 
Appendix A Mathematical Notation and Methods of Proof 

634  (9) 
Appendix B Mathematical Induction 

643  (7) 
Appendix C Complex Numbers 

650  (11) 
Appendix D Polynomials 

661  (10) 
Answers to Selected OddNumbered Exercises 

671  (35) 
Index 

706  