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 $det≠0$

Given any unit vectors, there is a nonsingular rotation matrix that takes the first vector to the other. Given any non-zero, non-unit vector, there is some non-singular scaling matrix that takes the vector to a unit vector. So given two arbitrary non-zero vectors $v$ and $w$, there are $S_v$ and $S_w$ such that $S_vv$ and $S_ww$ are unit vectors, and $R$ such that $R(S_vv)=S_ww$. Then $w=(S_w^{-1}RS_v)v$. So taking $A=S_w^{-1}RS_v$, we have that $v=Aw$, so given any non-zero vectors $v$, $w$, $v$~$w$; the set of non-zero vectors is an equivalence class.
