How to show that one linear map is injective and the other surjective when you compose two of them together

Question

If I have $T\colon \mathbb{C}^{3}\to\mathbb{C}^{2}$ and $S\colon \mathbb{C}^{2}\to\mathbb{C}^{3} $ which are both linear maps and we know that $\mathrm{rank}(ST)=2$. How would I show that $T$ is injective and that $S$ is a surjection?

Answer 1

Do you know that $\min\{ \mathrm{rank\ }S, \mathrm{rank\ }T\} \ge \mathrm{rank}(ST)$? Since both $S$ and $T$ have rank at most $2$, both must have rank exactly equal to $2$. As pointed out in the comments this implies $T$ is surjective and $S$ is injective.