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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 Complex conjugate vector space ： ウィキペディア英語版 Complex conjugate vector space In mathematics, the complex conjugate of a complex vector space $V\backslash ,$ is a complex vector space $\backslash overline\; V$, which has the same elements and additive group structure as $V$, but whose scalar multiplication involves conjugation of the scalars. In other words, the scalar multiplication of $\backslash overline\; V$ satisfies :$\backslash alpha\backslash ,$ *\, v = where $$ * is the scalar multiplication of $\backslash overline\; V$ and $\backslash cdot$ is the scalar multiplication of $V$. The letter $v\backslash ,$ stands for a vector in $V\backslash ,$, $\backslash alpha\backslash ,$ is a complex number, and $\backslash overline\backslash alpha$ denotes the complex conjugate of $\backslash alpha\backslash ,$. More concretely, the complex conjugate vector space is the same underlying ''real'' vector space (same set of points, same vector addition and real scalar multiplication) with the conjugate linear complex structure ''J'' (different multiplication by ''i''). ==Motivation== If $V\backslash ,$ and $W\backslash ,$ are complex vector spaces, a function $f\backslash colon\; V\; \backslash to\; W\backslash ,$ is antilinear if :$f(v+v\text{'})\; =\; f(v)\; +\; f(v\text{'})\backslash quad\backslash text\backslash quad\; f(\backslash alpha\; v)\; =\; \backslash overline\backslash alpha\; \backslash ,\; f(v)$ With the use of the conjugate vector space $\backslash overline\; V$, an antilinear map $f\; :\; V\; \backslash to\; W$ can be regarded as an ordinary linear map of type $\backslash overline\; V\; \backslash to\; W$. The linearity is checked by noting: :$f(\backslash alpha$ * v) = f(\overline \alpha \cdot v) = \overline \alpha \cdot f(v) Conversely, any linear map defined on $\backslash overline\; V$ gives rise to an antilinear map on $V\backslash ,$. This is the same underlying principle as in defining opposite ring so that a right $R$-module can be regarded as a left $R^$-module, or that of an opposite category so that a contravariant functor $C\; \backslash to\; D$ can be regarded as an ordinary functor of type $C^\; \backslash to\; D$. 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「Complex conjugate vector space」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.