|
In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every vector in ''W''. Informally, it is called the perp, short for perpendicular complement. It is a subspace of ''V''. ==General bilinear forms== Let be a vector space over a field equipped with a bilinear form . We define to be left-orthogonal to , and to be right-orthogonal to , when . For a subset of we define the left orthogonal complement to be : There is a corresponding definition of right orthogonal complement. For a reflexive bilinear form, where implies for all and in , the left and right complements coincide. This will be the case if is a symmetric or an alternating form. The definition extends to a bilinear form on a free module over a commutative ring, and to a sesquilinear form extended to include any free module over a commutative ring with conjugation.〔Adkins & Weintraub (1992) p.359〕 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「orthogonal complement」の詳細全文を読む スポンサード リンク
|