How to find the equivalence classes on a relation? Suppose we define the relation $∼$ by $v∼w$ (where $v$ and $w$ are arbitrary elements in $R^n$) if there exists a matrix $$A∈ GL_n(R)$$ such that $v=Aw$.  What are the equivalence classes for $∼$ in this case? NOTE:$GL_n(R)$ is a set that contains all the $n×n$ matrices with $det≠0$

Hint: There are two equivalence classes: $\{0\}$ and $\Bbb R^n\setminus \{0\}$.
