
#1
Feb2013, 12:42 AM

P: 308

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?




#2
Feb2013, 01:04 AM

C. Spirit
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P: 4,926

R over Q is infinite dimensional. That's kinda cool I guess :p.




#3
Feb2013, 01:11 AM

P: 308





#4
Feb2013, 03:52 AM

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P: 4,301

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 ntuples (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].




#5
Feb2013, 07:03 PM

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P: 2,913

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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