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?

An interesting one I saw in Anton Elementary Linear Algebra (10th ed. ch.4 ex. 8: "An Unusual Vector Space").
The set of positive reals as a vector space over the reals.  (I will place a vector over vectors, e.g. $\vec{u}$ and use plain letters for scalars and the values of vectors as reals (so $\vec{u}$ is the vector whose "real number value" is $u$), hopefully this reduces confusion.)
For vectors $\vec{u}$ and $\vec{v}$ and a real number $k$ define:
$\vec{u}+\vec{v} := uv$
$k\cdot\vec{u} := u^k$
We then have:
$\vec{0} = 1$
$-\vec{u} = 1/u$
