# Topology - Quotient Spaces

## 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.

## 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!

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

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