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This is problem 181 in Golan's linear algebra book. I have posted a proposed solution in the comments.

Problem: Let $V$, $W$, and $Y$ be vector spaces over a field $F$ and let $a\in Hom(V,W)$ and $b\in Hom(W,Y)$ satisfy the condition that $im(a)$ has a finitely generated complement in $W$ and $im(b)$ has a finitely generated complement in $Y$. Show that $im(b\circ a)$ has a finitely-generated complement in $Y$.

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Let $\{w_i\}$ be a (finite) basis of the complement of $im(a)$ and $\{y_i\}$ be a basis complement of $im(b)$. We want to show that composition $b\circ a$ "misses" a finite number of basis elements of $Y$. We already know it "misses" everything in $\{y_i\}$. But $\dim \text{im} (b(w_i)$, $w_i\in \{w_i\})$ is finite, and the composition will miss these and only these vectors too, so the total complement of the image of $b\circ a$ is finite.

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