# Topology - Quotient Spaces

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1. Mar 5, 2017

### mr.tea

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. Mar 5, 2017

### Staff: Mentor

If they are linearly dependent, it's easy. If not, why not make a basis with both of them?

3. Mar 5, 2017

### mr.tea

However, I still don't get how it will help me with taking them both and complete it to a basis.

4. Mar 5, 2017

### 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.