• solving linear systems and
• finding eigenvalues

are accomplished every day on large systems by computer.  This course will place these topics in the correct abstract, finite-dimensional vector space framework but also describe how actual matrices can be handled in a stable, fast, and accurate manner.  Many of the ideas relate to famous matrix decompositions, theorems, and algorithms:

• spectral theorem  and Schur decomposition
• singular value decomposition (SVD)
• the QR method for eigenvalues
• Krylov subspace methods.