- #1

mr.tea

- 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!