Edit: the original poster is Menny, but the question is CW; the first-person pronoun refers to Menny, not to the most recent editor.

I'm doing an introductory talk on linear algebra with the following aim: I want to give the students a concrete example through which they will be able to see how many notions arise "naturally". Notions such as vector spaces, the zero vector, span, linear dependency and independency, basis, dimension, "good" bases, solving linear equations, and even linear maps and eigenvectors. A related MO question is Linear algebra proofs in combinatorics.

The aim of this post is to find some more "concrete","real" and "natural" examples in this spirit that can interest everyone who loves what we do (and give them motivation to learn new definitions and formalisms). So if you have some ideas - please post them! Thanks, Menny