|
In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space ''V'' is a bilinear map , where ''K'' is the field of scalars. In other words, a bilinear form is a function which is linear in each argument separately: : * ''B''(u + v, w) = ''B''(u, w) + ''B''(v, w) : * ''B''(u, v + w) = ''B''(u, v) + ''B''(u, w) : * ''B''(''λ''u, v) = ''B''(u, ''λ''v) = ''λB''(u, v) The definition of a bilinear form can be extended to include modules over a commutative ring, with linear maps replaced by module homomorphisms. When ''K'' is the field of complex numbers C, one is often more interested in sesquilinear forms, which are similar to bilinear forms but are conjugate linear in one argument. ==Coordinate representation== Let be an ''n''-dimensional vector space with basis Define the matrix ''A'' by . If the ''n'' × 1 matrix ''x'' represents a vector v with respect to this basis, and analogously, ''y'' represents w, then: : Suppose is another basis for ''V'', such that: : (..., f''n'' ) = (..., e''n'' )''S'' where . Now the new matrix representation for the bilinear form is given by: ''S''T''AS''. 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「bilinear form」の詳細全文を読む スポンサード リンク
|