Vectors / 1: |
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 |
Applications: Force Vectors; Code Vectors |
Vignette: The Codabar System |
Systems of Linear Equations / 2: |
Introduction: Triviality |
Introduction to Systems of Linear Equations |
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 |
Spanning Sets and Linear Independence |
Applications: Allocation of Resources |
Balancing Chemical Equations |
Network Analysis |
Electrical Networks |
Linear Economic Models |
Finite Linear Games |
Vignette: The Global Positioning System |
Iterative Methods for Solving Linear Systems |
Matrices / 3: |
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 |
Vignette: Robotics |
Applications: Markov Chains |
Population Growth |
Graphs and Digraphs |
Error-Correcting Codes |
Eigenvalues and Eigenvectors / 4: |
Introduction: A Dynamical System on Graphs |
Introduction to Eigenvalues and Eigenvectors |
Determinants |
Vignette: Lewis Carroll's Condensation Method |
Exploration: Geometric Applications of Determinants |
Eigenvalues and Eigenvectors of n x n Matrices |
Similarity and Diagonalization |
Iterative Methods for Computing Eigenvalues |
Applications and the Perron-Frobenius Theorem: Markov Chains |
The Perron-Frobenius Theorem |
Linear Recurrence Relations |
Systems of Linear Differential Equations |
Discrete Linear Dynamical Systems |
Vignette: Ranking Sports Teams and Searching the Internet |
Orthogonality / 5: |
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 Equations |
Vector Spaces / 6: |
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 |
Distance and Approximation / 7: |
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 |
Mathematical Notation and Methods of Proof / Appendix A: |
Mathematical Induction / Appendix B: |
Complex Numbers / Appendix C: |
Polynomials / Appendix D: |
Vectors / 1: |
Introduction: The Racetrack Game |
The Geometry and Algebra of Vectors |