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 vector space of all order $n$ magic squares ($n\times n$ matrices with real entries and all row and column and diagonal sums equal). 
The reals as a vector space over the rationals. ${\bf Q}(\sqrt2)$ as a vector space over the rationals. 
