We will follow Trefethen and Bau's text pretty closely, chapter by chapter. Here's a slightly different conceptual organization of the course than provided by Trefethen and Bau's table of contents, with rough dates. By the end of the semester, you should have a pretty good grasp of each of the topics mentioned herein.
Fundamentals, Aug 31 – Sept 23
linear algebra review: mat-vec mult, inverses, rank, span, etc.
orthogonality, norms, SVD
Trefethen lectures 1 – 5
Direct methods, Sept 28 – Oct 21
QR factorization: Gram-Schmidt and modified Gram-Schmidt algorithms, Householder algorithm, least squares
conditioning, stability, and accuracy
LU decomposition (a.k.a. Gaussian elimination), pivoting
Trefethen chapters 6-23
Iterative methods, Oct 26 – Nov 25
eigenvalue algorithms: Rayleigh quotient, inverse iteration, QR algorithm, shifts
SVD algorithms
Krylov subspace methods: Arnoldi iteration, GMRES, and conjugate gradients
Trefethen chapters 24-40
Optimization, (time permitting) Nov 30 – Dec 14