I was reading Vinberg's "A Course in Algebra" and in the opening line of chapter 2 he says, "Fix a field $\mathbb K$. We are going to abuse the language slightly and call elements of $\mathbb K$ numbers."
Anyway, linear algebra and solving systems of equations is present (it seems) in tons of abstract books for various reasons (analysis, algebra, geometry, ...) but the most "abstract" I've seen the discussion become is the use of the field $\mathbb Q$ or $\mathbb C$. I'm really wanting to make solving systems of equations a more abstract thing, where we don't even deal with equations of "numbers".
For example, if $X$ is a set and $2^X$ is the set of all its subsets, it can be shown that $2^X$ is a ring with respect to the operations of symmetric difference: $M \Delta N = (M - N) \cup (N- M)$ and intersection, taken for addition and multiplication, respectively. (The ring is commutative and associative).
Well, what does solving a system of equations of this sort look like? Are there other "abstract" systems of equations that you can think of, which are interesting.
Further, are there any connections of this idea with other things we are familiar with? Positive definiteness? Eigenvalue problem? Uniqueness and Existence of solution, etc...