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