The Problem:
Suppose $V,W$ are vector spaces over $F$ and $T:V \to W, S:W \to V$ are linear transformations. Suppose also that $S \circ T$ is an isomorphism. Prove that $W$ is the direct sum $\text{im}(T) \oplus \ker (S)$.
My Approach:
Put $A = \text{im}(T)$ and $B = \ker (S)$. We need to show that $A \cap B = \{0\}$ and $W = A + B$.
For $A \cap B = \{0\}$ (I think I've got this part): Clearly, $\{0\} \subset A \cap B$ (as $A \cap B$ is a subspace of $W$). Now, if $x \in A \cap B \subset W$, then $x = T(v)$, for some $v \in V$; and so $S(x) = S(T(v)) = 0$, since $x$ is in the kernel of $S$. But since $ST$ is an isomorphism, it must map $0$ to $0$ injectively (right?); and so $T(v)$ must be $0$. Thus, since $T(v) = x$, it follows that $x = 0$; i.e., $A \cap B = \{0\}$.
For $W = A + B$: Let $w \in W$, and write $w = w_1 + w_2$ (which is obviously permitted). Then, since $S$ and $T$ are linear, $S(w) = S(w_1 + w_2) = S(w_1) + S(w_2) \in V$; and so $T(S(w)) = T(S(w_1)) + T(S(w_2)) \in W$. Thus, $S(w_1)$ and $S(w_2)$ are in $A$… I'm not really sure where I'm going with this…
Best Answer
You have not specified what $w_1$ and $w_2$ are, so it is not clear to me what are you claiming to be obviously permitted. As for the first part, you reasoning is correct.
Here's a hint: take $w \in W$. Now, $S(w)$ is an element of $V$, and so by surjectivity of $ST$ we have that $S(w) = ST(y)$ for some $y$ in $V$. Note that if we were indeed successful in writing $w = T(a) + b$ for $a$ in $V$ and $b$ in $\ker S$, we should necessarily have that $S(w) = ST(a)$ and $b = w - T(a)$. Can you take it from here? A solution follows: