Since every map of vector spaces is a map of abelian groups, I was wondering if the converse also holds:
Given an additive map $\phi: V \to W$ between two vector spaces, does it follow that $\phi$ is also $\mathbf{k}-$linear? I'm interested in the case of $\mathbf{k}$ having characteristic zero, specially if $\mathbf{k}$ is a famous field like rational or real or complex numbers.
I'm guessing it's false, but I tried to come up with counter-examples for $\mathbf{k} = \mathbf{Q}, \mathbf{R}$ and couldn't find any. Finding a counter-example in characteristic $p>0$ may not be that hard, for instance since taking $p-$th power is additive. However that's not the case I care most about. Appreciate any help!