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