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Topology - Quotient Spaces

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  1. Mar 5, 2017 #1
    1. The problem statement, all variables and given/known data
    (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.

    2. Relevant equations


    3. 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!
     
  2. jcsd
  3. Mar 5, 2017 #2

    fresh_42

    Staff: Mentor

    If they are linearly dependent, it's easy. If not, why not make a basis with both of them?
     
  4. Mar 5, 2017 #3
    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.
     
  5. Mar 5, 2017 #4

    fresh_42

    Staff: Mentor

    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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