Can one know the rank of a matrix product given the rank of one of them?

Idea: combine $$ \text{rank}\,\hat Q=\text{rank}\,QT\le\text{rank}\,Q $$ and $$ \text{rank}\,Q=\text{rank}\,\hat QT^{-1}\le\text{rank}\,\hat Q. $$

The rank is the dimension of the image. Since $T$ is injective and injective maps preserve dimension (as they map linearly independent sets to linearly independent sets), it follows that $T$ is also surjective. So the image of $T$ is $\mathbb R^n$. It follows that the image of $QT$ is equal to the image of $Q$, so in particular $$ \text{rank}\, Q=\text{rank}\,QT. $$