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.