Non-numerical vector space examples

I've recently been thinking about why my peers and other people I've helped learn vector spaces had trouble intuitively understanding the concept, and it occurred to me that non-numerical (i.e. nothing like $\langle 3,2,3 \rangle$ or obvious addition/multiplication operations) examples could reinforce intuition. For example, a huge problem was understanding that a vector space is simply a set of vectors with two operations that follow 10 axioms, and that a zero vector isn't necessarily all zeroes, and so on.

Does anyone have any great examples of vector spaces (and the vectors and operations in them, of course) that are non-numerical, and thus can't lead to those trying to prove their validity to being stuck in ruts (like assuming the zero vector is all zeroes, that the inverse vector is the negative scalar multiple, etc.)? Pictures, letters, and any others would certainly be interesting!

Note: I know that there are questions about vector spaces with unusual (and only partially valid) characteristics or out of the ordinary operations, but I'm looking for examples that have minimal numbers involved, to remove all automatic assumptions that are involved in them.

Solution 1:

A simple example is to take $\mathbb{R}^n$ but to fix a vector $w$ and modify scalar multiplication to $a \otimes v = a (v - w) + w$ and addition to $u \oplus v = u + v - w$. This is just the usual vector space structure on $\mathbb{R}^n$, but shifted by $w$, and in my experience many students have a lot of trouble with these kinds of examples; they have never really learned to think in a translation-invariant way. If nothing else, this example should quickly diagnose the problem you mention about the zero vector (which is of course $w$ here).

Perhaps a more "non-numerical" example is to take the space of solutions to a linear homogeneous differential equation or recurrence relation, such as $y'' - 3y' - 2y = 0$ or $a_{n+3} = a_{n+1} + a_n$. While the zero vector is in some sense "all zeroes" in these examples, I like them because it's not immediately obvious how to write down a basis for these spaces (or, having done so, it's not obvious that you've chosen a useful one).

Solution 2:

I like function spaces for this example (pick your favorite kind). It is easy to see that it is a vector space, but a basis might be out of reach. Restricting to polynomials gives you a nice and obvious basis though.