Register to reply

Strange vector spaces

by johnqwertyful
Tags: spaces, strange, vector
Share this thread:
johnqwertyful
#1
Feb20-13, 12:42 AM
P: 358
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?
Phys.Org News Partner Science news on Phys.org
What lit up the universe?
Sheepdogs use just two simple rules to round up large herds of sheep
Animals first flex their muscles
WannabeNewton
#2
Feb20-13, 01:04 AM
C. Spirit
Sci Advisor
Thanks
WannabeNewton's Avatar
P: 5,616
R over Q is infinite dimensional. That's kinda cool I guess :p.
johnqwertyful
#3
Feb20-13, 01:11 AM
P: 358
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.

CompuChip
#4
Feb20-13, 03:52 AM
Sci Advisor
HW Helper
P: 4,300
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].
jbunniii
#5
Feb20-13, 07:03 PM
Sci Advisor
HW Helper
PF Gold
jbunniii's Avatar
P: 3,288
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.


Register to reply

Related Discussions
Linear spaces vs. vector spaces Linear & Abstract Algebra 19
Vector Spaces: Verify whether a set is a vector space Calculus & Beyond Homework 1
Position vector and tangent vector in Riemannian spaces Differential Geometry 13
Vector spaces: column spaces Calculus & Beyond Homework 1
Vector Spaces / Sub Spaces Linear & Abstract Algebra 4