Give an explicit example of a pair of linear transformations $T : V \to W$ and $S : W \to U$ between vector spaces $V$, $W$, and $U$, so that neither $T$ nor $S$ is the zero linear transformation, but the composition $ST$ is the zero linear transformation.
Hint: What’s the relationship between the range of $T$ and the kernel of $S$?
I am struggling with this problem. Using the rank nullity theorem, I found that the range of $T$ as well as the the kernel of $S$ should have the same dimension as the domain of $T$. However, I'm confused as how to proceed from there. Any tips are appreciated!