Guaranteed for all vector spaces $V,W$

Zero Transformation

$Z:V\to W$ given by $Z(v)=0w$

Proof

Check that $Z$ is linear

So, we check that $Z$ satisfies $Z((a⊡u)⊞(b⊡v))=(a⊡Z(u))\oplus (b\odot Z(\overrightarrow{v}))$ By definition of $Z$, we have: $Z((a⊡u)⊞(b⊡v))=\hat{0}w$

For the right side, we have: $(a⊡Z(\overrightarrow{u}))\oplus (b\odot Z(v))$ $=(a\odot 0w)\oplus (b\odot 0w)$ by defn of $Z$ $=(0w)\oplus (0w)$ for all $c\in F$, $C\odot 0w=0w$ $=0w$ by defn of $0w$ We get that both sides are equal, thus we an conclude that is is linear