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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 Symmetric bilinear form ： ウィキペディア英語版 Symmetric bilinear form A symmetric bilinear form on a vector space is a linear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinear function $B$ that maps every pair $(u,v)$ of elements of the vector space $V$ to the underlying field such that $B(u,v)=B(v,u)$ for every $u$ and $v$ in $V$. They are also referred to more briefly as just symmetric forms when "bilinear" is understood. Symmetric bilinear forms on finite-dimensional vector spaces precisely correspond to symmetric matrices given a basis for ''V''. Among bilinear forms, the symmetric ones are important because they are the ones for which the vector space admits a particularly simple kind of basis known as an orthogonal basis (at least when the characteristic of the field is not 2). Given a symmetric bilinear form ''B'', the function is the associated quadratic form on the vector space. Moreover, if the characteristic of the field is not 2, ''B'' is the unique symmetric bilinear form associated with ''q''. == Formal definition == Let '' V'' be a vector space of dimension ''n'' over a field ''K''. A map $B\; :\; V\backslash times\; V\backslash rightarrow\; K$ is a symmetric bilinear form on the space if: * $B(u,v)=B(v,u)\; \backslash \; \backslash quad\; \backslash forall\; u,v\; \backslash in\; V$ * $B(u+v,w)=B(u,w)+B(v,w)\backslash \; \backslash quad\; \backslash forall\; u,v,w\; \backslash in\; V$ * $B(\backslash lambda\; v,w)=\backslash lambda\; B(v,w)\backslash \; \backslash quad\; \backslash forall\; \backslash lambda\; \backslash in\; K,\backslash forall\; v,w\; \backslash in\; V$ The last two axioms only imply linearity in the first argument, but the first axiom then immediately implies linearity in the second argument as well. 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「Symmetric bilinear form」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.