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| Introduction to Vectors | |
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| Vectors and linear combinations | |
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| Lengths and dot products | |
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| Matrices | |
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| Solving Linear Equations | |
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| Vectors and linear equations | |
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| The idea of elimination | |
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| Elimination using matrices | |
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| Rules for matrix operations | |
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| Inverse matrices | |
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| Elimination = factorization: A = LU | |
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| Transposes and permutations | |
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| Vector Spaces and Subspaces | |
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| Spaces of vectors | |
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| The nullspace of A: solving Ax = 0 | |
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| The rank and the row reduced form | |
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| The complete solution to Ax = b | |
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| Independence, basis and dimension | |
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| Dimensions of the four subspaces | |
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| Orthogonality | |
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| Orthogonality of the four subspaces | |
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| Projections | |
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| Least squares approximations | |
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| Orthogonal bases and Gram-Schmidt | |
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| Determinants | |
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| The properties of determinants | |
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| Permutations and cofactors | |
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| Cramer's rule, inverses, and Volumes | |
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| Eigenvalues and Eigenvectors | |
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| Introduction to eigenvalues | |
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| Diagonalizing a matrix | |
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| Applications to differential equations | |
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| Symmetric matrices | |
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| Positive definite matrices | |
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| Similar matrices | |
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| Singular value decomposition (SVD) | |
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| Linear Transformations | |
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| The idea of a linear transformation | |
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| The matrix of a linear transformation | |
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| Diagonalization and the pseudoinverse | |
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| Applications | |
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| Matrices in engineering | |
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| Graphs and networks | |
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| Markov matrices, population, and economics | |
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| Linear programming | |
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| Fourier series: linear algebra for functions | |
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| Linear algebra for statistics and probability | |
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| Computer graphics | |
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| Numerical Linear Algebra | |
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| Gaussian elimination in practice | |
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| Norms and condition numbers | |
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| Iterative methods for linear algebra | |
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| Complex Vectors and Matrices | |
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| Complex numbers | |
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| Hermitian and unitary matrices | |
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| The fast Fourier transform | |
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| Solutions to selected exercises | |
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| Matrix factorizations | |
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| Conceptual questions for review | |
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| Glossary: a dictionary for linear algebra | |
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| Index | |
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| Teaching codes | |