semilinear map

A semilinear map (older term: semilinear transformation) from a left vector space VV over a division ring kk to a left vector space WW over a division ring kk' is an additive map f:VWf: V\to W of the abelian groups of vectors together with a homomorphism of rings α:kk\alpha : k\to k' such that the following weaker form of homogeneity holds:

f(λv)=α(λ)f(v),ak,vV f (\lambda v) = \alpha(\lambda) f(v), \,\,\,\,\,a\in k, v\in V

This notion is very much used in the classical affine and projective geometry over division rings, for example in the standard definition of a homotethy.

Created on May 4, 2016 at 08:46:58. See the history of this page for a list of all contributions to it.