Semilinear Map

In linear algebra, and particularly in projective geometry, a semilinear map between two vector spaces over a field is a function that behaves like a linear map except for a twist by a field automorphism: it is additive with respect to vector addition, and there exists a field automorphism such that scaling a vector before applying the map equals applying the map and then scaling the result by the automorphism's image of the scalar. When such an automorphism exists and the map is nonzero it is unique, and the invertible semilinear transformations of a vector space, taken over every choice of field automorphism, form a group called the general semilinear group, extending the ordinary general linear group; when the field is the complex numbers and the automorphism is complex conjugation, the resulting semilinear map is called an antilinear map.