I have to find a necessary and sufficient condition on $\ker(T)$ for the existence of a linear transformations $S: \Bbb R^3 \to \Bbb R^2$ such that $ST = I$ where $T: \Bbb R^2 \to \Bbb R^3$. So I found the condition for $\ker(S)$ but still struggling to find $\ker(T)$. If $S(Tv)=v$ for $v$ in $\Bbb R^2$, then $$\dim \Bbb R^2 = \text{rank } S = 2,\ \ker S = \dim \Bbb R^3 - \text{rank } S = 1$$
How could I relate my finding for $\ker S$ to $\ker T$? If they were irrelevant, what would be the right way to solve this question?
Thank you in advance!
I don't see any way to use your finding regarding the rank of $S$. As for how to approach the problem,
Hint: $\operatorname{rank}(ST) \leq \operatorname{rank}(T)$.