Lec # | TOPICS | Readings |
---|---|---|

1 | Systems of Linear Equations | Section 0.1: Systems of linear equations, row equivalence |

2 | Echelon Form | Section 0.2: Gaussian and Gauss-Jordan elimination, (reduced) row-echelon form, back-substitution |

3 | Matrices | Section 0.3: Matrices, matrix operations, block multiplication |

4 | Matrices (cont.) | Section 0.4: Matrices and linear systems, elementary matrices, (reduced) row-echelon matrices |

5 | Solution Spaces | Section 0.5: The space of solutions to a homogeneous linear system, uniqueness of the reduced row-echelon form, matrix rank, criterion for existence of solutions |

6 | Inverses and Transposes | Section 0.6: Matrix inverses (right, left), invertible matrices, transpose of a matrix, symmetric matrices |

7 | Fields and Spans | Section 1.1: Definition of a field F, examples: Q, R, C, Z/pZ (see also 1.6. pp. 132-133), linear combinations of vectors, and spans in F^{n} |

8 | Vector Spaces | Section 1.2: Vector spaces, definition and examples, sub-spaces, the row space, column space, and nullspace of a matrix |

9 | Linear Independence | Section 1.3: Linear independent vectors |

10 | Basis and Dimension | Section 1.4: Basis of a vector space, dimension, bases for the row space and column space of a matrix, Rank plus nullity theorem for matrices, Basis extension theorem |

11 | Coordinates | Section 1.5: Coordinates with respect to an ordered basis, change of coordinates matrix |

12 | Review for Quiz 1 | |

13 | Quiz 1 (Chapters 0-1) | |

14 | Determinants | Section 2.1 (pp. 137-143): Determinant function (definition, properties, uniqueness), computing determinants using row-reduction Section 2.1 (pp. 144-146): invertible matrices, det(AB) = det(A)det(B), det(A^{t}) = det (A) |

15 | Permutations | Section 2.2: Permutations and the permutation definition of the determinant |

16 | Determinants (cont.) | Section 2.2: Permutations and the permutation definition of the determinant |

17 | Laplace Expansion | Section 2.3: Cofactor (Laplace) expansion of the determinant, the adjoint of a matrix, finding the inverse using the adjoint, Cramer's rule |

18 | Review for Quiz 2 | |

19 | Quiz 2 (Chapter 2) | |

20 | Linear Transformations | Section 3.1: Linear transformations (definition, examples), matrix associated to a linear transformation |

21 | Rank, Kernel, Image | Section 3.2: Properties of linear transformations, rank, kernel, image, Rank plus nullity theorem for linear transformations, one-one, onto, isomorphism |

22 | Matrix Representations | Section 3.3: Matrix representations for linear transformations, similar matrices |

23 | Eigenspaces | Section 3.4: Eigenvalues, eigenvectors (definitions and examples), eigenspaces, characteristic polynomial |

24 | Eigenspaces (cont.) | Section 3.4: Eigenvalues, eigenvectors (definitions and examples), eigenspaces, characteristic polynomial |

25 | Diagonalization | Section 3.5: Diagonalizable linear operators and matrices |

26 | Cayley-Hamilton Theorem | Section 6.1: Cayley-Hamilton theorem, minimal polynomial |

27 | Jordan Canonical Form | Section 6.4 (pp. 373-376): Jordan form, generalized eigenvectors and Primary decomposition theorem from section 6.5 (see also J. Starr's notes from Fall 2004) |

28 | Review for Quiz 3 | |

29 | Quiz 3 (Chapter 3) | |

30 | Computing Generalized Eigenvectors | Section 6.4 (pp. 376-384): More on Jordan form, computing generalized eigenvectors |

31 | Norms and Inner Products | Section 4.1: Norms, real and hermitian inner products (definitions, examples), projections, Schwartz' inequality |

32 | Norms and Inner Products (cont.) | Section 4.1: Norms, real and hermitian inner products (definitions, examples), projections, Schwartz' inequality |

33 | Orthogonal Bases | Section 4.2: Orthogonal and orthonormal bases, Gram-Schmidt algorithm, QR decomposition |

34 | Orthogonal Projections | Section 4.3: Orthogonal projections, orthogonal complement, direct sums |

35 | Isometries, Spectral Theory | Section 4.5 (pp. 282-285): Isometries, orthogonal and unitary matrices |

36 | Singular Value Decomposition | Section 4.5 (pp. 286-291): Self-adjoint operators, symmetric and hermitian matrices, eigenvalues of self-adjoint operators, Principal axis theorem, Spectral resolution |

37 | Polar Decomposition | Section 4.6: Singular value decomposition, positive (semi)definite matrices, Polar decomposition |

38 | Review for the Final |