Let $U,V,W$ be $F$-vector spaces. Let $T$ be a linear mapping from $U$ to $V$. Let $S$ be a linear mapping from $V$ to $W$ where $\dim(\text{range}(T)) > \dim(\text{null}(S))$. Prove that $ST$ does not map every vector in its domain to the $0$-vector.
I tried doing a proof by contradiction. I showed that if $ST$ is the $0$-map, then $\text{range}(T)$ is a subspace $\text{null}(S)$. How do I use the fact that $\dim(\text{range}(T)) > \dim(\text{null}(S))$ to show that I have a contradiction? Thanks in advance!