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Endomorphisms of Free Vector Spaces

Suppose $\phi:V\rightarrow V$ and $\psi:W\rightarrow W$ are endomorphisms of free vector spaces $V$ and $W$ over some field $k$. It is easy to see that $\phi\circ \psi$ is another endomorphism of $W$ and that $\phi \circ \psi$ is a $k$-linear map. Also, it is not hard to see that $End_k(W)$ is a ring.
My question is: Is it true that $\phi$ and $\psi$ are determined up to a $k$-linear map in their actions on $V$ and $W$?
More specifically, let $V$ and $W$ be as in the above, and let $T\in End_k(W)$. Does there exist a $k$-linear map $F:V\rightarrow V$ and a $k$-linear map $G:W\rightarrow W$ such that $T=\phi\circ \psi+\psi\circ G+F\circ \psi$?

A:

If $V = W$, then the endomorphisms of $V$ determine all the endomorphisms of $V$ (in the sense that we are given $\phi \in \mathrm{End}(V)$ and must find $F,G$), and you are asking if we can choose $F$ and $G$ to be $k$-linear. I’m afraid the answer is no. For example, if $k = \mathbb{Q}$, consider the homomorphism $f : \mathbb{Q} \rightarrow \mathbb{Q}$ given by $f(n) = n$ for all $n \in \mathbb{Q}$. The only endomorphisms of $\mathbb{Q}$ are the identity map and the trivial map (i.e., the map with $f(n) = 0$ for all \$n

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