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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