Let $K$ be a \reftext{def:field-c54-2026b}{field} and let $V$ and $W$ be \reftext{def:vector-space-2026a}{vector spaces over $K$}.

A map $T:V\to W$ is \textbf{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 $W$.