Trefethen & Bau's "Numerical Linear Algebra" is a pretty good primer, though the last few chapters I remember being a bit lacking (it's been about 6 years though).

This book helped me to visualize Krylov space methods better than any other resource I have seen. It got me through my numerical analysis qualifying exam. I do agree that the last chapters are lacking in depth. I also recommend his book "Pseudo-spectral Methods in Matlab" for a thorough and very visual look at things like Gibb's phenomenon and spectral convergence.