Sep. 2: Introduction to AMS505. The geometry of linear equations.
Examples of Gaussian elimination. (1.2 - 1.3)
Sep. 4: Matrix notation and matrix multiplication. Triangular factors
and row exchanges. (1.4 - 1.5)
Sep. 9: Inverses and transposes. Special matrices and applications.
(1.6
- 1.7)
Sep. 11: Vector spaces and subspaces. Solving Ax = 0
and Ax = b. (2.1 - 2.2)
Sep. 16: Linear independence, basis, and dimension. (2.3)
Sep. 18: Four fundamental subspaces. (2.4)
Sep. 23: Linear transformations. (2.5 - 2.6)
Sep. 25: Orthogonal vectors and subspaces. Projections onto lines. (3.1
- 3.2)
Oct. 2: Projections and least squares. (3.3)
Oct. 7: Orthogonal bases and the Gramm-Schmidt method.
Oct. 14: Determinants and their properties. (4.1 - 4.2)
Oct. 16: Formulas for the determinant. Applications of determinants.
(4.3 - 4.4)
Oct. 21: Overview of Chapters 1-4. Oct. 23: Midterm exam
Oct. 28: Overview of the Midterm exam
Oct. 30: The eigenvalue problem. Diagonalization of a matrix. (5.1 -
5.2)
Nov. 4: Difference equations and powers of Ak . (5.3)
Nov. 6: Differential equations and exp(At). (5.4)
Nov. 11: Compex matrices. The spectral theorem. (5.5)
Nov. 13: Similarity transformations. Schur's lemma and the Jourdan
form.
(5.6, see also Schaum's Outlines in Linear Algebra)
Nov. 18: Positive definite forms and matrices. (6.1 - 6.2)
Nov. 20: Singular value decomposition. (6.3)
Nov. 25: Least squares via SVD. Minimum principles. (6.3 - 6.4)
Dec. 2: Matrix norm and condition number. (7.1 - 7.2)
Dec. 4: Computation of eigenvalues. (7.3)
Dec. 9: Iterative methods for Ax = b. (7.4, see also W. Briggs, V. E.
Henson, S. McCormick, A Multigrid Tutorial, SIAM, 2nd edition)
Dec. 11: Overview of Chapters 5 - 7
Dec. 23, 5:00 - 7:30 pm: Final Exam