Semilinearity

linear-algebra

Definition

Semilinearity

Let $V$ and $W$ be vector spaces over a field $F$. A function $T:V\to W$ is semilinear if it is additive and scales vectors up to a field automorphism.

More precisely, there exists a field automorphism $\sigma :F\to F$ such that for all $x,y\in V$ and $\alpha \in F$,

$$T(x+y)=T(x)+T(y)$$

and

$$T(\alpha x)=\sigma (\alpha )T(x).$$

If $\sigma$ is the identity map, then $T$ is linear.

Semilinearity is therefore a twisted form of homogeneity.