Assume two sets $A$, $B\in R^n$, and define $\bar{A}=\{x\in R^m:Tx\in A\}$, $\bar{B}=\{x\in R^m:Tx\in B\}$, where $T$ is some $(n\times m)$ matrix. Under which conditions does it hold that $A\neq B \Rightarrow \bar{A}\neq\bar{B}$ .
My guess is that $n\leq m$ and full rank is required in order to have a injection when the inverse transformation is applied, however I was not able to show this in a more rigorous way.