1. VECTORS.1.0 Introduction: The Racetrack Game. 1.1 The Geometry and Algebra of Vectors. 1.2 Length and Angle: The Dot Product. Exploration: Vectors and Geometry. 1.3 Lines and Planes. Exploration: The Cross Product. 1.4 Applications: Force Vectors; Code Vectors. Vignette: The Codabar System. 2. SYSTEMS OF LINEAR EQUATIONS. 2.0 Introduction: Triviality. 2.1 Introduction to Systems of Linear Equations. 2.2 Direct Methods for Solving Linear Systems. Exploration: Lies My Computer Told Me. Exploration: Partial Pivoting. Exploration: Counting Operations: An Introduction to the Analysis of Algorithms. 2.3 Spanning Sets and Linear Independence. 2.4 Applications: Allocation of Resources; Balancing Chemical Equations; Network Analysis; Electrical Networks; Linear Economic Models; Finite Linear Games. Vignette: The Global Positioning System. 2.5 Iterative Methods for Solving Linear Systems. 3. MATRICES. 3.0 Introduction: Matrices in Action. 3.1 Matrix Operations. 3.2 Matrix Algebra. 3.3 The Inverse of a Matrix. 3.4 The LU Factorization. 3.5 Subspaces, Basis, Dimension, and Rank. 3.6 Introduction to Linear Transformations. Vignette: Robotics. 3.7 Applications: Markov Chains; Linear Economic Models; Population Growth; Graphs and Digraphs; Error-Correcting Codes. 4. EIGENVALUES AND EIGENVECTORS. 4.0 Introduction: A Dynamical System on Graphs. 4.1 Introduction to Eigenvalues and Eigenvectors. 4.2 Determinants. Vignette: Lewis Carroll's Condensation Method. Exploration: Geometric Applications of Determinants. 4.3 Eigenvalues and Eigenvectors of n x n Matrices. 4.4 Similarity and Diagonalization. 4.5 Iterative Methods for Computing Eigenvalues. 4.6 Applications and the Perron-Frobenius Theorem: Markov Chains; Population Growth; The Perron-Frobenius Theorem; Linear Recurrence Relations; Systems of Linear Differential Equations; Discrete Linear Dynamical Systems. Vignette: Ranking Sports Teams and Searching the Internet. 5. ORTHOGONALITY. 5.0 Introduction: Shadows on a Wall. 5.1 Orthogonality in Rn. 5.2 Orthogonal Complements and Orthogonal Projections. 5.3 The Gram-Schmidt Process and the QR Factorization. Exploration: The Modified QR Factorization. Exploration: Approximating Eigenvalues with the QR Algorithm. 5.4 Orthogonal Diagonalization of Symmetric Matrices. 5.5 Applications: Dual Codes; Quadratic Forms; Graphing Quadratic Equations. 6. VECTOR SPACES. 6.0 Introduction: Fibonacci in (Vector) Space. 6.1 Vector Spaces and Subspaces. 6.2 Linear Independence, Basis, and Dimension. Exploration: Magic Squares. 6.3 Change of Basis. 6.4 Linear Transformations. 6.5 The Kernel and Range of a Linear Transformation. 6.6 The Matrix of a Linear Transformation. Exploration: Tilings, Lattices and the Crystallographic Restriction. 6.7 Applications: Homogeneous Linear Differential Equations; Linear Codes. 7. DISTANCE AND APPROXIMATION. 7.0 Introduction: Taxicab Geometry. 7.1 Inner Product Spaces. Exploration: Vectors and Matrices with Complex Entries. Exploration: Geometric Inequalities and Optimization Problems. 7.2 Norms and Distance Functions. 7.3 Least Squares Approximation. 7.4 The Singular Value Decomposition. Vignette: Digital Image Compression. 7.5 Applications: Approximation of Functions; Error-Correcting Codes. Appendix A: Mathematical Notation and Methods of Proof. Appendix B: Mathematical Induction. Appendix C: Complex Numbers. Appendix D: Polynomials.Appendix D: PolynomialsAppendix E: Technology Bytes Online OnlyAnswers to Selected Odd-Numbered ExercisesIndex