The cool thing about numerical linear algebra is, as far as numerical analysis is concerned, it's probably one of the most accessible fields to the would-be citizen scientist.
It's been a long time since I've taken linear algebra but if you can find an OCW course, or a really good book I'd start there. You'll want some rigor under your belt. Once you do that, any good numerical analysis text will usually cover the basics (you'll need them) before going into the fascinating world of algorithms as they relate to linear algebra.
Numerical Linear Algebra by Trefethen and Bau is the resource I've seen recommended most often. I've only read the first few chapters but they are good and useful.
head to ulaff.net, they have textbooks, a set of computer exercises, a code repository, youtube lectures, the whole nine yards really starting with linear algebra 101 and culminating with numerical linear algebra
their treatment of numerical linear algebra is not as comprehensive as say the one you'd find in Watkins' Fundamentals of Matrix Computations but it's free
