I am new to Linear Algebra, and was asked the following question:
Let $S,T:V\to V$ be linear transformations, while V is a linear space.
If $\mbox{im} ST={0}$ then $\mbox{im} T\subseteq \ker S$
I have tried to reason this through, but I am not sure whether my reasoning is correct:
$$\ker S: S\underline x=0$$
$$\mbox{im} T = \text{ column-space of }T$$
So if I multiply $S \times T$, with the result being $\{0\}$, that would seem to imply that indeed $T$ is a subset of $\ker S$, as it obeys $S\underline x=0$, with $T$ taking the place of $\underline x$.
Can anybody help out?
Thank you
I think you have the right idea - let's just be slightly more precise.
Suppose $y \in \textrm{Im}(T)$, so $y = T(x)$ for some $x \in V$. Then $S(y) = ST(x) = 0$ since $ST = 0$ and hence $y \in \textrm{Ker}(S)$. Therefore $\textrm{Im}(T) \subseteq \textrm{Ker}(S)$.