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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?

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    Seems like a duplicate of http://math.stackexchange.com/questions/4694/what-are-some-alternative-definitions-of-vector-addition-and-scalar-multiplicatio/ to me.2010-09-22
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    Oops. Sorry about the duplication. Should I delete this, Qiaochu?2010-09-22
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    If you can reword the question into something sufficiently different it should be fine.2010-09-22
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    Another class of non-examples: (non-identity) cosets of subspaces. So, for example, take a line through the origin $L$. Then $v+L$ is a parallel line. This "looks" like a vector space but does not contain the zero vector (i.e. origin) so it's not.2011-12-24

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