1. VECTORS. Introduction: The Racetrack Game. The Geometry and Algebra of Vectors. Length and Angle: The Dot Product. Exploration: Vectors and Geometry. Lines and Planes. Exploration: The Cross Product. Code Vectors and Modular Arithmetic. Vignette: The Codabar System. Chapter Review. 2. SYSTEMS OF LINEAR EQUATIONS. Introduction: Triviality. Introduction to Systems of Linear Equations. Exploration: Lies My Computer Told Me. Direct Methods for Solving Linear Systems. Exploration: Partial Pivoting. Exploration: Counting Operations - An Introduction to the Analysis of Algorithms. Spanning Sets and Linear Independence. Applications: Allocation of Resources; Balancing Chemical Equations; Network Analysis; Electrical Networks; Finite Linear Games. Vignette: The Global Positioning System. Iterative Methods for Solving Linear Systems. Chapter Review. 3. MATRICES. Introduction: Matrices in Action. Matrix Operations. Matrix Algebra. The Inverse of a Matrix. The LU Factorization. Subspaces, Basis, Dimension, and Rank. Introduction to Linear Transformations. Applications: Markov Chains; Population Growth; Graph Theory; Error-Correcting Codes. Vignette: Robotics. Chapter Review. 4. EIGENVALUES AND EIGENVECTORS. Introduction: A Dynamical System on Graphs. Eigenvalues and Eigenvectors. Determinants. Exploration: Geometric Applications of Determinants. Eigenvalues and Eigenvectors of n x n Matrices. Similarity and Diagonalization. Iterative Methods for Computing Eigenvalues. Applications: Markov Chains; Population Growth; the Perron-Frobenius Theorem; Linear Recurrence Relations; Systems of Linear Differential Equations. Vignette: Ranking Sports Teams and Searching the Internet. Chapter Review. 5. ORTHOGONALITY. Introduction: Shadows on a Wall. Orthogonality in Rn. Orthogonal Complements and Orthogonal Projections. The Gram-Schmidt Process and the QR Factorization. Exploration: The Modified QR Factorization. Exploration: Approximating Eigenvalues with the QR Algorithm. Orthogonal Diagonalization of Symmetric Matrices. Applications: Dual Codes; Quadratic Forms; Graphing Quadratic Forms. Chapter Review. 6. VECTOR SPACES. Introduction: Fibonacci in (Vector) Space. Vector Spaces and Subspaces. Linear Independence, Basis, and Dimension. Exploration: Magic Squares. Change of Basis. Linear Transformations. The Kernel and Range of a Linear Transformation. The Matrix of a Linear Transformation. Exploration: Tilings, Lattices and the Crystallographic Restriction. Applications: Homogeneous Linear Differential Equations; Linear Codes. Chapter Review. 7. DISTANCE AND APPROXIMATION. Introduction: Taxicab Geometry. Inner Product Spaces. Exploration: Vectors and Matrices with Complex Entries. Exploration: Geometric Inequalities and Optimization Problems. Norms and Distance Functions. Least Squares Approximation. The Singular Value Decomposition. Vignette: Digital Image Compression. Applications: Approximation of Functions; Error-Correcting Codes. Chapter Review. Appendix A: Mathematical Notation and Methods of Proof. Appendix B: Mathematical Induction. Appendix C: Complex Numbers. Appendix D: Polynomials.
