Let $V$ and $W$ be vector spaces. The mapping $T:V\to W$ is called a linear transformation if and only if

$T(c\vec{u}+\vec{v}) = cT(\vec{u}) + T(\vec{v}),$

for every choice of $\vec{u},\vec{v}\in V$ and scalar $c\in\mathbb{R}$.

When $V=W$, $T$ is called a linear operator [DG09, p. 202].