I can't believe he didn't mention Gilbert Strang. This should be the starting point for everyone, IMO. The oddball Russian books look very interesting, but for

I can't believe he didn't mention Gilbert Strang. This should be the starting point for everyone, IMO. The oddball Russian books look very interesting, but for a computer guy to learn, I'd suggest Strang, then Trefethen & Bau (my favorite). If there is anything else worth knowing about numeric linear algebra it is in Gollub, or Demmel's applied numerical linear algebra (another unforgivable omission; he only helped write LAPACK!). The problem books can also be important; they're all pretty good if you pick the problems right. Not sure the highly theoretical stuff is actually worth anything as far as getting things done, unless you go on to study Algebra and mathematics in general. Ultimately the "real world" stuff where linear algebra is studied as systems of equations and geometric constructs is going to be closest to what you actually do with linear algebra on a computer.