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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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Applications: Force Vectors | |
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Code Vectors | |
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Vignette: The Codabar System | |
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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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Direct Methods for Solving Linear Systems | |
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Exploration: Lies My Computer Told Me | |
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Exploration: Partial Pivoting | |
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Exploration: Counting Operations: 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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Linear Economic Models | |
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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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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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Vignette: Robotics | |
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Applications: Markov Chains | |
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Linear Economic Models | |
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Population Growth | |
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Graphs and Digraphs | |
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Error-Correcting Codes. | |
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Eigenvalues and Eigenvectors | |
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Introduction: A Dynamical System on Graphs | |
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Introduction to Eigenvalues and Eigenvectors | |
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Determinants | |
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Vignette: Lewis Carroll's Condensation Method | |
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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 and the Perron-Frobenius Theorem: Markov Chains | |
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Population Growth | |
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The Perron-Frobenius Theorem | |
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Linear Recurrence Relations | |
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Systems of Linear Differential Equations | |
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Discrete Linear Dynamical Systems | |
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Vignette: Ranking Sports Teams and Searching the Internet | |
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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 Equations | |
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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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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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Mathematical Notation and Methods of Proof | |
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Mathematical Induction | |
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Complex Numbers | |
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Polynomials | |