The categories of vector spaces and finite dimensional vector spaces are pretty much as nice as can be, I think.

I was wondering what portions of basic linear algebra (first couple of courses) fall out by saying "big"(er) words, and also what standard facts admit a clarifying categorical phrasing.

What are some interesting examples of facts about vector spaces and linear maps that admit a nice categorical formulation?

Edit. I'm not looking for (completely) elementary things like definitions of universal constructions by universal properties instead of concrete ad hoc realizations, though I can't think of any "nonelementary" things. If you write an elementary property of the category of vectors spaces, e.g a property of any abelian category, please give some nice examples of where it lends its power.