Linearity

algebra

Definition

Linearity

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

A map $T:V\to W$ is linear if it is semilinear with $\sigma =\mathrm{id}$, i.e.

$$T(x+y)=T(x)+T(y),T(\alpha x)=\alpha T(x)$$

for all $x,y\in V$, $\alpha \in F$.

Equivalently, $T$ is homogeneous and additive, respecting superposition.