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Topology - Quotient Spaces

  • #1
102
12

Homework Statement


(part of a bigger question)
For ##x,y \in \mathbb{R}^n##, write ##x \sim y \iff## there exists ##M \in GL(n,\mathbb{R})## such that ##x=My##.
Show that the quotient space ##\mathbb{R}\small/ \sim## consists of two elements.

Homework Equations




The Attempt at a Solution


Well, it is easy to see that the first equivalence class is ##\{0\}##. This means that the second equivalence class should be ##\mathbb{R}^n - \{ 0 \} ##.
But I don't know how to show that given ##x,y\in \mathbb{R}^n##, I can find a matrix that connects between them.
Any suggestions will be great.

Thank you!
 

Answers and Replies

  • #2
12,357
8,753
If they are linearly dependent, it's easy. If not, why not make a basis with both of them?
 
  • #3
102
12
If they are linearly dependent, it's easy. If not, why not make a basis with both of them?
Thank you for the answer.
However, I still don't get how it will help me with taking them both and complete it to a basis.
 
  • #4
12,357
8,753
The matrix (expressed in the coordinates of the new basis) that simply swaps the two and leaves the rest invariant shouldn't be that hard to figure out.
 

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