Depends on how you define "linear algebra." Representation theory, lie groups, and grassmannians can all be considered "linear algebra" and are active areas of math research, but it probably isn't what you are looking for.
I don't quite care about obscure and beautiful theoretical problems. I am looking for new developments that have shown promise to make big practical impact.
For what it's worth, I am no linear algebra definition police so feel free to list what you think is reasonably relevant.
