Linear Map

Definition

For $V,W$ vector spaces over field $F$, $T:V\to W$ is linear if $\forall x,y\in V$ and $\alpha \in F$,

T(x + y) = T(x) +T(y), \quad T(\alpha x) = \alpha T(x), \quad \forall x, y \in V, \forall \alpha \in F