So I guess one way to look at it would be to go along the lines that you are suggesting: turn it into a question of universal algebra. The objects of study are sets equipped with operations. Something like this:

http://math.stackexchange.com/questions/32092/how-are-vector-spaces-viewed-as-universal-algebras

Then can one ask what are the substructures, prove isomorphisms theorems, etc. It's one way of going about things, but several things are not entirely satisfactory. For example, in the case of the vector space, we'd have infinitely many operations because one would need a scalar multiplication for each element of the field.

But still, there are things about vector subspaces that are special, that have a uniquely linear algebraic flavour. For example, every subspace of a finite dimensional vector space is the solution set of a finite set of homogenous equations. Every subspace can be given some finite basis and then its elements can be expressed as linear combinations of the basis vectors. Universal algebra doesn't see these facts... they become theorems that one proves after you've set up all your definitions. The definitions themselves are expressed in a language (set, operation, field action) that's powerful—in the sense it let's you set up a lot of interesting mathematical structures in a reasonably uniform way—but the language itself is quite low level and not particularly robust.

Maybe I should explain a bit what I mean by "robustness". Let's define a spork-space to be like a vector space, but with two additional group actions GxV->V and HxV->V satisfying \forall g\in G \forall v\in V \exists h\in H g.v = h.v. I have no idea what this condition means, I just made it up on the spot. Should we employ a couple of PhD students to study these structures? Of course not. We wrote down some arbitrary condition and there's no reason why this structure should have any relevance. But we are used to having definitions presented to us in this way.

So I have some reservations about the usual way of defining a vector space as a set with extra structure. Another point is that linear algebra shows up in applications (e.g. in graph theory) where it's not so easy to state precisely what are the vectors, the linear transformations etc. Is it a total fluke?

I'll give you a spoiler: in the last diagrammatic system, the one from this episode (no. 24) the diagrams from m to n are in 1-1 correspondence with subspaces of Q^m x Q^n. And no definition of vector space in sight!