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In linear algebra, particularly projective geometry, a semilinear transformation between vector spaces ''V'' and ''W'' over a field ''K'' is a function that is a linear transformation "up to a twist", hence ''semi''-linear, where "twist" means "field automorphism of ''K''". Explicitly, it is a function that is: * linear with respect to vector addition: * semilinear with respect to scalar multiplication: where θ is a field automorphism of ''K,'' and means the image of the scalar under the automorphism. There must be a single automorphism θ for ''T,'' in which case ''T'' is called θ-semilinear. The invertible semilinear transforms of a given vector space ''V'' (for all choices of field automorphism) form a group, called the general semilinear group and denoted by analogy with and extending the general linear group. Similar notation (replacing Latin characters with Greek) are used for semilinear analogs of more restricted linear transform; formally, the semidirect product of a linear group with the Galois group of field automorphism. For example, PΣU is used for the semilinear analogs of the projective special unitary group PSU. Note however, that it is only recently noticed that these generalized semilinear groups are not well-defined, as pointed out in – isomorphic classical groups ''G'' and ''H'' (subgroups of SL) may have non-isomorphic semilinear extensions. At the level of semidirect products, this corresponds to different actions of the Galois group on a given abstract group, a semidirect product depending on two groups and an action. If the extension is non-unique, there are exactly two semilinear extensions; for example, symplectic groups have a unique semilinear extension, while SU(''n'',''q'') has two extension if ''n'' is even and ''q'' is odd, and likewise for PSU. == Definition == Let ''K'' be a field and ''k'' its prime subfield. For example, if ''K'' is C then ''k'' is Q, and if ''K'' is the finite field of order then ''k'' is Given a field automorphism of ''K'', a function between two ''K'' vector spaces ''V'' and ''W'' is -semilinear, or simply semilinear, if for all in ''V'' and in ''K'' it follows: # # where denotes the image of under Note that must be a field automorphism for ''f'' to remain additive, for example, must fix the prime subfield as : Also : so Finally, : Every linear transformation is semilinear, but the converse is generally not true. If we treat ''V'' and ''W'' as vector spaces over ''k,'' (by considering ''K'' as vector space over ''k'' first) then every -semilinear map is a ''k''-linear map, where ''k'' is the prime subfield of ''K.'' 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Semilinear transformation」の詳細全文を読む スポンサード リンク
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