TITLE: Linear algebra in terms of abstract nonsense?
QUESTION [39 upvotes]: 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.

REPLY [2 votes]: Here is an example (more along the lines of "linear algebra ~ $K$-theory of a point"):
The categorical characterization of dimension: Let $\chi: \operatorname{Obj}(\operatorname{fVct}_{\mathbb{K}})\to G$ be a function from f.d. $\mathbb{K}$-vector spaces to an abelian group $G$ satisfying

$V\cong W \implies \chi(V)=\chi(W)$,
If $0\to  U \to V \to W \to 0$ is exact, then $\chi(U)-\chi(V)+\chi(W)=0$.

Then $\chi=\chi(\mathbb{K}^1)\, \dim_\mathbb{K}$.
This is from Kostrikin & Manin's Linear Algebra and Geometry (p.89); it includes other basic categorical statements as well.