- #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!