Here we begin a discussion of linear algebra. There are three views on what this is all about.
The mid-level view is that we’re studying the properties of linear maps — homomorphisms — between abelian groups, and particularly between modules or vector spaces, which are just modules over a field. In particular we’ll focus on vector spaces over some arbitrary (but fixed!) field .
The high-level view is that what we’re really studying is the category of -modules. Categories are all about how morphisms between their objects interact, so this is what we’re really after here. And it turns out we already know a lot about these sorts of categories! Specifically, they’re abelian categories. In fact, since we’re working over a field (which is a commutative ring) the properties of -functors tell us that is enriched over itself.
So this tells us that our category of vector spaces has a biproduct — the direct sum — and in particular a zero object — the trivial -dimensional vector space . It also has a tensor product, which makes this a monoidal category, using the one-dimensional vector space itself as monoidal identity. We also know that kernels and cokernels exist, which then (along with biproducts) give us all finite limits and colimits.
The third viewpoint is that we’re talking about solving systems of linear equations, and that’s where “linear algebra” comes from. The connection between these abstract formulations and that concrete one is a bit mysterious at first blush, but we’ll start making it tomorrow.