Linear Transformations Preserve Addition and Scaling

Proving $⟹$

Suppose $T:V\to W$ is linear $T((a⊡\overrightarrow{u})⊞(b⊡\overrightarrow{v}))$ $=T(a⊡\overrightarrow{u})\oplus T(b⊡\overrightarrow{v})$ As $T$ preserves adding $=(a\odot T(u))\oplus (b\odot T(\overrightarrow{v}))$ as $T$ preseves scaling

Proving $\Leftarrow$

Suppose $T$ satisfies $T((a⊡\overrightarrow{u}))⊞(b⊡\overrightarrow{v})$ $=(a\odot T(\overrightarrow{u}))\oplus (b\odot T(\overrightarrow{v}))$ For all $a,b\in F$ and $\overrightarrow{u},\overrightarrow{v}\in V$ We check $T$ preserves addition Suppose $a=b=1\in F$, this gives: $T((1⊡\overrightarrow{u}))⊞(1⊡\overrightarrow{v})$ $=(1⊡T(\overrightarrow{u}))⊞(1\odot T(\overrightarrow{v}))$ $⟺T(\overrightarrow{u}⊞\overrightarrow{v})=T(\overrightarrow{u})+T(\overrightarrow{v})$ Thus, we have shown both sides