What Are the Strangest Vector Spaces You Can Imagine?

Click For Summary

Discussion Overview

The discussion explores unusual or interesting vector spaces, particularly those defined over various fields, including real numbers and rational numbers. Participants share their thoughts on the dimensionality of certain vector spaces and the implications of field extensions in linear algebra.

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification

Main Points Raised

  • One participant proposes defining a vector space V over R as 1-tuples, with vector addition as field multiplication and scalar multiplication as field exponentiation.
  • Another participant notes that R over Q is infinite dimensional, expressing excitement about this property.
  • A later reply confirms the infinite dimensionality of R over Q and expresses enthusiasm about this realization.
  • One participant mentions that certain sets of functions can form a vector space and can be conceptualized as n-tuples for n = 2^{\aleph_0}, highlighting that vector spaces need not consist of points in \mathbb{R}^k.
  • Another participant finds it intriguing to visualize the Hamel basis of R over Q, noting that it exists if the axiom of choice is accepted.
  • One participant shares their excitement about learning field extensions and viewing larger fields as vector spaces over smaller ones, connecting this to linear algebra concepts.

Areas of Agreement / Disagreement

Participants express agreement on the interesting properties of vector spaces, particularly regarding R over Q being infinite dimensional. However, there are multiple perspectives on the definitions and implications of vector spaces, indicating that the discussion remains exploratory without a consensus.

Contextual Notes

Some claims depend on the acceptance of the axiom of choice, and there may be limitations in the definitions of vector spaces discussed, particularly regarding the nature of the fields involved.

johnqwertyful
Messages
396
Reaction score
14
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?
 
Physics news on Phys.org
R over Q is infinite dimensional. That's kinda cool I guess :p.
 
WannabeNewton said:
R over Q is infinite dimensional. That's kinda cool I guess :p.

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

Similar threads

Undergrad About the definition of vector space of infinite dimension
  • · Replies 18 ·
Replies
18
Views
4K
Undergrad Two ways to define operations in a vector space
  • · Replies 7 ·
Replies
7
Views
2K
High School Vector Space over Field of Real Numbers
  • · Replies 7 ·
Replies
7
Views
2K
High School Understanding Bases of a Vector Space
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad When is a subset a subspace in a vector space?
  • · Replies 7 ·
Replies
7
Views
3K
MHB Is This a Valid Vector Space with Unusual Operations?
  • · Replies 5 ·
Replies
5
Views
2K
MHB Some basic question about vector spaces
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Proving inequality about dimension of quotient vector spaces
  • · Replies 25 ·
Replies
25
Views
4K
Undergrad Terminology for Substructures in Vector Spaces
  • · Replies 2 ·
Replies
2
Views
2K
MHB Are $V_1$ and $V_2$ Vector Spaces According to Defined Properties?
  • · Replies 14 ·
Replies
14
Views
4K