How do I prove that if $S$ is onto then $T$ cannot be one-to-one?

Question

Let $T: \Bbb{R}^3 \to \Bbb{R}^4$ and $S: \Bbb{R}^4 \to \Bbb{R}^2$ be two linear transformations such that the composition $S\circ T=0$.

How can I show that if $S$ is onto, then $T$ cannot be one-to-one?

Answer 1

If $S$ is onto the $\dim\ker S=4-2$. If $T$ is one-to-one then $\dim\operatorname{im}T=3$. If $S\circ T=0$ then $\operatorname{im}T\subseteq \ker S$.