Linear Map

Let $K$ be a field and let $V$ and $W$ be vector spaces over $K$.

A map $T:V\to W$ is linear if

1. $T(u+v)=T(u)+T(v)$ for all $u,v\in V$;
2. $T(\lambda v)=\lambda\,T(v)$ for all $\lambda\in K$ and $v\in V$,

where on the left-hand sides the operations are those of $V$ and on the right-hand sides those of $W$.