Solution
= Solution
{c}
A map $T:V\to W$ between <vector spaces> over the same <field> $\mathbb F$ is a <linear map> precisely when it preserves <vector addition> and <scalar multiplication>:
$$
T(u+v)=T(u)+T(v),\qquad T(av)=aT(v)
\quad(u,v\in V,\ a\in\mathbb F).
$$
Equivalently, $T(au+bv)=aT(u)+bT(v)$ for all $u,v\in V$ and $a,b\in\mathbb F$.