Let $S$ = $\{(x,y)^T : x,y \in \Bbb{R}\}$.
Define a relation $R$ on $S$ by $aRb$ iff there exists a $2\times2$ invertible matrix $A$ such that $Aa$ = $b$.
I have shown that $R$ is an equivalence relation.
I need to show that $R$ partitions $S$ into $4$ distinct classes.
Let $ a \in S$ and $[a]$ denote the equivalence class of $a$.
Then by definition,
$[a]$ = $\{b \in S: \exists\;\; 2\times2 \text{ invertible matrix } A \text{ s.t } Aa = b\}$
I can't understand how can I proceed after this.