Table of Contents
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Systems of Linear Equations
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Introduction to Systems of Linear Equations
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Augmented Matrix Notation and Elementary Row Operations
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Gaussian Elimination and Rank
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Iterative Methods for Solving Linear Systems
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Big Ideas about Vectors
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Linear Combinations of Vectors
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Span
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Linear Independence
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Matrices
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Addition and Scalar Multiplication of Matrices
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Matrix Multiplication
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Block Matrix Multiplication
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Transpose of a Matrix
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Linear Systems as Matrix and Linear Combination Equations
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Homogeneous Linear Systems
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The Inverse of a Matrix
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Elementary Matrices
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LU Factorization
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Subspaces of ℝn
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ℝn and Subspaces of ℝn
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Introduction to Bases
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Bases and Dimension
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Subspaces of ℝn Associated with Matrices
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Linear Transformations
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Matrix Transformations
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Geometric Transformations of the Plane
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Introduction to Linear Transformations
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Standard Matrix of a Linear Transformation from ℝn to ℝm
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Composition and Inverses of Linear Transformations
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Image and Kernel of a Linear Transformation
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The Determinant
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Finding the Determinant
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Determinants, Areas, and Volumes
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Elementary Row Operations and the Determinant
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Properties of the Determinant
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Tedious Proofs Concerning Determinants
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Determinants and Inverses of Nonsingular Matrices
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Eigenvalues and Eigenvectors
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Describing Eigenvalues and Eigenvectors
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The Characteristic Equation
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Similar Matrices and Their Properties
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Diagonalizable Matrices and Multiplicity
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Gershgorin’s Theorem
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The Power Method and the Dominant Eigenvalue
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Orthogonality
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Orthogonality and Projections
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Least-Squares Approximation
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Gram-Schmidt Orthogonalization
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Orthogonal Complements and Decompositions
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Orthogonal Matrices and Symmetric Matrices
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Positive Definite Matrices
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QR Factorization
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SVD Decomposition
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Vector Spaces
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Abstract Vector Spaces
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Bases and Dimension of Abstract Vector Spaces
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Linear Transformations of Abstract Vector Spaces
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Existence of the Inverse of a Linear Transformation
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Isomorphic Vector Spaces
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Matrices of Linear Transformations with Respect to Arbitrary Bases
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Inner Product Spaces
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Applications
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Application to Network Flow
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Application to Electrical Networks
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Application to Chemical Equations
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Application to Input-Output Economic Models
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Application to Markov Chains
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Curve Fitting
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Application to Computer Graphics
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Appendix
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Triangle Inequality
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Complex Numbers
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Complex Matrices
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INDEX
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Index of GeoGebra Interactives
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Overall
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