"What are recent developments in linear algebra?"
Perhaps, we can broaden the scope and include numerical/randomized linear algebra.
I come from a machine learning background, and the most recent directions I am aware of are in applications of Krylov subspace methods to very large number of data points and low-precision training (both in the context of Gaussian processes).
Still, you could throw out our course and spend a semester teaching people how to really think about singular value decomposition. It's everywhere, critical for example in machine learning, yet pure math types think it's "applied".
Riemannian geometry begins with an inner product at each point (though what's amazing about O'Neill's text is that he slips by you a categorical definition first at every step). As another route up the mountain, one could now understand parallel transport as a local version of the singular value decomposition, and work out the whole theory that way. This requires a fluid, intuitive grasp of singular values, rather than the textbook definition. We should teach that understanding.
Have you heard of geometric algebra? It takes a while to wrap your head around, but unifies a lot of mathematics in a very elegant way. For instance, you can view Euclidean geometry as a special case, and concepts like quaternions and rotations can be explained in a very natural way. There are a few good resources below if you're interested.
[0]: http://geometry.mrao.cam.ac.uk/2020/06/euclidean-geometry-an...
[1]: https://www.youtube.com/watch?v=ipqKvNHnABs
[2]: https://alexkritchevsky.com/2020/10/15/ea-operations.html
[3]: https://www.youtube.com/playlist?list=PLpzmRsG7u_gqaTo_vEseQ...
Gurobi had a white paper mentioning how much their solvers' performance was improved by which techniques (I'll link it here if I can find it).
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.