This is a follow up question to the following post: How do we show every linear transformation which is not bijective is the difference of bijective linear transforms?
The previous post dealt with showing a linear transformation which was not bijective was the difference of two bijiective homomorphisms.
Let $F$ be a field and $I$ a nonempty index set (Note: $I$ can be uncountable). Consider the direct sum $V = \oplus_{i \in I} F_i $ where each $F_i$ is isomorphic to $F$. In order to rule out a pathological example we must assume furthermore that $F$ is not isomorphic to $\mathbb{Z}_2$.
How do we show if $\phi: V \rightarrow V$ is a bijective $F$-module homomorphism (linear transformation) then $\phi = f -g$ where $f,g:V \rightarrow V$ are two bijective $F$-module homomorphisms ?
I am not quite sure if the proof from the following post applies and if so does it suffice to check that $F$ is free.