# Prove that if $T: V \rightarrow W$ is an application s.t $T(x+y)=T(x)+T(y)$, then T is a linear transformation [duplicate]

Well, I know right that I could find a counterexample to prove that this is false. Anyone knows one? Note that $$dim(V),\hspace{3.25pt}dim(W)<\infty$$
• Even in the case $V=W=\mathbb R$ (as vector spaces over $\mathbb R$) $T$ need not be linear. – Kavi Rama Murthy Apr 8 at 0:44