How about random matrix theory? There has been so much cool stuff there in the past few years? Also the resolution of Horn's conjecture is very major --- ok, I'd rather label that as matrix theory and combinatorics, but it directly impacts linear algebra.

Almost linear time solution of linear systems (Spielman, Teng, et al, and others).

Or these do not meet your bar?

Definitely, some items on the top of my list are:

1. Random matrix theory --- both asymptotic and non asymptotic; including things like semi-circular law, circular law, and so on. Check out Terry Tao's blog for very nice summaries.
2. The resolution of Horn's conjecture (see this nice summary article by R. Bhatia, which also mentions several other nice connections)
3. Randomised linear algebra and progress on fast solutions to linear systems (see e.g., the very readable summary in N. Vishnoi's web book)
4. Advances in quantum information theory? Though I don't know how much of that I would push into just linear algebra
5. Not advances in linear algebra itself, but the gigantic success of basic linear algebra in new areas (machine learning, information retrieval, etc., e.g., Google's PageRank method).