Vivid examples of vector spaces? When teaching abstract vector spaces for the first time, it is handy to have some really weird examples at hand, or even some really weird non-examples that may illustrate the concept. For example, a physicist friend of mine uses "color space" as a (non) example, with two different bases given essentially {red, green, blue} and {hue, saturation and brightness} (see http://en.wikipedia.org/wiki/Color_space). I say this is a non-example for a number of reasons, the most obvious being the absence of "negative color". 
Anyhow, what are some bizarre and vivid examples of vector spaces you've come across that would be suitable for a first introduction?

The solutions of the differential equation $y''+p y' +q y=0$ on some interval $I\subset{\mathbb R}$ form a vector space $V$ of functions $f:I\to{\mathbb R}$. What is the dimension of this space? Physical intuition or the fundamental existence and uniqueness theorem for differential equations tell you that this dimension is 2: Consider the two initial problems $y(0)=1, y'(0)=0$ and $y(0)=0, y'(0)=1$. The two corresponding solutions $y_1(\cdot)$, $y_2(\cdot)$ form a basis of $V$. Now comes the upshot: You can "guess" explicit solutions of the form $y(t):=e^{\lambda t}$ for suitable $\lambda$'s (apart from special cases), and in this way you obtain a completely different basis of $V$.
Of course this is not "weird", but it is an instance of a finite-dimensional vector space which does not have a "natural" basis to begin with.
