Suppose $U$ is a subspace of $V$ with $U \ne V$. Suppose $S \in \mathcal L(U,W)$ and $S \ne 0$ (Which means that $Su \ne 0$ for some $u \in U$). Define $T : V \to W$ by

Prove that $T$ is not a linear map on $V$.

Let $u \in U$ such that $Su \ne 0$. Let $v \in V, v \notin U$ and consider how $u+v \notin U$ because if it were in $U$ adding $-u$ would imply $v \in U$ which it isn't. Since $T(u + v) = 0 \ne Tu + Tv$ we conclude $T$ is not linear.

In essence this exercise was to show you can't naively extend a linear map on a subspace to the full space by setting it to zero on the full space.