Strange vector spaces


by johnqwertyful
Tags: spaces, strange, vector
johnqwertyful
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Feb20-13, 12:42 AM
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What are some of the strangest vector spaces you know? I don't know many, but I like defining V over R as 1 tuples. Defining vector addition as field multiplication and scalar multiplication as field exponentiation. That one's always cool. Have any cool vector spaces? Maybe ones not over R but over maybe more exotic fields?
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WannabeNewton
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Feb20-13, 01:04 AM
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R over Q is infinite dimensional. That's kinda cool I guess :p.
johnqwertyful
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Feb20-13, 01:11 AM
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Quote Quote by WannabeNewton View Post
R over Q is infinite dimensional. That's kinda cool I guess :p.
Never thought of that, but that's true. That's awesome.

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Feb20-13, 03:52 AM
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Strange vector spaces


I also liked it when I realised that some sets functions are a vector space and you can basically think of them as n-tuples (for [itex]n = 2^{\aleph_0}[/itex]). Was the first time I saw that vector spaces don't need to consist of actual points in [itex]\mathbb{R}^k[/itex].
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Feb20-13, 07:03 PM
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Quote Quote by WannabeNewton View Post
R over Q is infinite dimensional. That's kinda cool I guess :p.
Even cooler is trying to visualize the basis (i.e., the Hamel basis, which exists if and only if you allow the axiom of choice).

In general, I thought it was really cool when starting to learn about field extensions, the epiphany that we can view the larger field as a vector space over the smaller one, and now we can bring in all the machinery of linear algebra to develop the theory. That was a great "ah HA!!" moment.


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