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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Semilinear transformation ： ウィキペディア英語版 Semilinear transformation 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 $T\backslash colon\; V\; \backslash to\; W$ that is: * linear with respect to vector addition: $T(v+v\text{'})\; =\; T(v)+T(v\text{'})$ * semilinear with respect to scalar multiplication: $T(\backslash lambda\; v)\; =\; \backslash lambda^\backslash theta\; T(v),$ where θ is a field automorphism of ''K,'' and $\backslash lambda^\backslash theta$ means the image of the scalar $\backslash lambda$ 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 $\backslash operatorname(V),$ 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 $q=p^i,$ then ''k'' is $\backslash mathbf/p\backslash mathbf.$ Given a field automorphism $\backslash theta$ of ''K'', a function $f\backslash colon\; V\; \backslash to\; W$ between two ''K'' vector spaces ''V'' and ''W'' is $\backslash theta$-semilinear, or simply semilinear, if for all $x,y$ in ''V'' and $l$ in ''K'' it follows: # $f(x+y)=f(x)+f(y),$ # $f(lx)=l^\backslash theta\; f(x),$ where $l^\backslash theta$ denotes the image of $l$ under $\backslash theta.$ Note that $\backslash theta$ must be a field automorphism for ''f'' to remain additive, for example, $\backslash theta$ must fix the prime subfield as :$n^\backslash theta\; f(x)=f(nx)=f(x+\backslash dots\; +x)=nf(x)$ Also :$(l\_1+l\_2)^\backslash theta\; f(x)=f((l\_1+l\_2)x)=f(l\_1\; x)+f(l\_2\; x)=(l\_1^\backslash theta+l\_2^\backslash theta)f(x)$ so $(l\_1+l\_2)^\backslash theta=l\_1^\backslash theta+l\_2^\backslash theta.$ Finally, :$(l\_1\; l\_2)^\backslash theta\; f(x)=f(l\_1\; l\_2\; x)=l\_1^\backslash theta\; f(l\_2x)=l\_1^\backslash theta\; l\_2^\backslash theta\; f(x)$ 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 $\backslash theta$-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 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.