Suppose $V$ is a vector space over a field $k$. Fixing a linear transformation $T$, it is common to make $V$ a $k[x]$-module by defining $f(x)\cdot v=f(T)(v)$.
Is every possible $k[x]$-module structure over $V$ necessarily induced by some $T\in L(V)$?
If $V$ is some $k[x]$-module, we can define a map $T$ on $V$ by $T(v)=x\cdot v$. Then $T$ is additive. But for scalars, $$T(cv)=x\cdot(cv)=(xc)\cdot v=(cx)\cdot v=c\cdot(x\cdot v)=c\cdot T(v)$$ but I don't think we can assume that multiplication by scalars in $k$ over $V$ as a $k$-vector space needs to be the same as multiplication by scalars in $k$ when viewing $V$ as a $k[x]$-module.