| 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 | |
| Writing Project: Origins of the Dot Product and the Cross Product | |
| Applications | |
| Systems Of Linear Equations | |
| Introduction: Triviality | |
| Introduction to Systems of Linear Equations | |
| Direct Methods for Solving Linear Systems | |
| Writing Project: A History of Gaussian Elimination | |
| Explorations: Lies My Computer Told Me; Partial Pivoting; Counting Operations: An Introduction to the Analysis of Algorithms | |
| Spanning Sets and Linear Independence | |
| Applications | |
| Vignette: The Global Positioning System | |
| Iterative Methods for Solving Linear Systems | |
| 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 | |
| Vignette: Robotics | |
| Applications | |
| Eigenvalues And Eigenvectors | |
| Introduction: A Dynamical System on Graphs | |
| Introduction to Eigenvalues and Eigenvectors | |
| Determinants | |
| Writing Project: Which Came First-the Matrix or the Determinant? Vignette: Lewis Carroll's Condensation Method | |
| Exploration: Geometric Applications of Determinants | |
| Eigenvalues and Eigenvectors of n x n Matrices | |
| Writing Project: The History of Eigenvalues | |
| Similarity and Diagonalization | |
| Iterative Methods for Computing Eigenvalues | |
| Applications and the Perron-Frobenius Theorem.Vignette: Ranking Sports Teams and Searching the Internet | |
| Orthogonality | |
| Introduction: Shadows on a Wall | |
| Orthogonality in Rn | |
| Orthogonal Complements and Orthogonal Projections | |
| The Gram-Schmidt Process and the QR Factorization | |
| Explorations: The Modified QR Factorization; Approximating Eigenvalues with the QR Algorithm | |
| Orthogonal Diagonalization of Symmetric Matrices | |
| Applications | |
| Vector Spaces | |
| Introduction: Fibonacci in (Vector) Space | |
| Vector Spaces and Subspaces | |
| Linear Independence, Basis, and Dimension | |
| Writing Project: The Rise of Vector Spaces | |
| 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 | |
| Distance And Approximation | |
| Introduction: Taxicab Geometry | |
| Inner Product Spaces | |
| Explorations: Vectors and Matrices with Complex Entries; Geometric Inequalities and Optimization Problems | |
| Norms and Distance Functions | |
| Least Squares Approximation | |
| The Singular Value Decomposition | |
| Vignette: Digital Image Compression | |
| Applications | |
| Codes | |
| (Online)Code Vectors | |
| Vignette: The Codabar System | |
| Error-Correcting Codes | |
| Dual Codes | |
| Linear Codes | |
| The Minimum Distance of a Code | |
| Mathematical Notation and Methods of Proof | |
| Mathematical Induction | |
| Complex Numbers | |
| Polynomials | |
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