[wmsiler]

Linear algebra is the sort of topic where it seems like you can always go deeper. I also got an A in my linear algebra class in college. When I got to grad school for math, I took the first year graduate course on algebra and saw linear algebra in terms of algebraic things like modules and representations. I decided that I hadn't really known linear algebra before, but that I did then. Then I took differential geometry, which involves studying infinite collections of vector spaces parameterizes by points on a manifold. I realized I still hadn't know linear algebra, but after that I certainly did. Then I took a functional analysis class, where we did infinite dimensional linear algebra, where a lot of the finite dimensional theory has analogues but everything is more complicated (e.g. instead of finite bases and dot products, you consider things like Fourier analysis and Hilbert spaces).

The same realization that I don't know linear algebra came when taking a Lie algebras class, and again when learning homological algebra, and probably a few other times as well. There are certainly lots of areas of math that I've never explored that take linear algebra in some other direction (for example, I don't have any idea what the applied math guys do with....).

It's really an amazingly vast subject, especially considering that it's usually just thought of as a tool used to study more advanced topics.

[bglazer]

I've been learning (trying to learn) linear algebra by studying Gilbert Strang's "Introduction to Linear Algebra" book. There are some really mind bending sections in which he explicitly treats a concept like linear regression in terms of geometry and then algebra and then calculus and then probability and then ties them all back together.

What's really struck me is when I dive into a Wikipedia rabbit hole of linear algebra, following links for terms I don't know. I start on the Topology page, read the introduction section of 10 articles that start with "x is a generalization of y" and somehow end up back on the Topology page. Shallow exposure to the sheer volume and conceptual depth of stuff like topology and algebra has really made me respect modern math.