What Are the Strangest Vector Spaces You Can Imagine?

[johnqwertyful]

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?

 

[WannabeNewton]

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

 

[johnqwertyful]

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

Never thought of that, but that's true. That's awesome.

 

[CompuChip]

I also liked it when I realized that some sets functions are a vector space and you can basically think of them as n-tuples (for n = 2^{\aleph_0}). Was the first time I saw that vector spaces don't need to consist of actual points in \mathbb{R}^k.

 

[jbunniii]

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.