Editing Bilinear map (section)

== Examples ==
* [[Matrix (mathematics)|Matrix multiplication]] is a bilinear map {{nowrap|M(''m'', ''n'') × M(''n'', ''p'') → M(''m'', ''p'')}}.
* If a [[vector space]] ''V'' over the [[real number]]s <math>\R</math> carries an [[Inner product space|inner product]], then the inner product is a bilinear map <math>V \times V \to \R.</math>
* In general, for a vector space ''V'' over a field ''F'', a [[bilinear form]] on ''V'' is the same as a bilinear map {{nowrap|''V'' × ''V'' → ''F''}}.
* If ''V'' is a vector space with [[dual space]] ''V''<sup>∗</sup>, then the [[Dual_system#Canonical_duality_on_a_vector_space|canonical evaluation map]], {{nowrap|1=''b''(''f'', ''v'') = ''f''(''v'')}} is a bilinear map from {{nowrap|''V''<sup>∗</sup> × ''V''}} to the base field.
* Let ''V'' and ''W'' be vector spaces over the same base field ''F''. If ''f'' is a member of ''V''<sup>∗</sup> and ''g'' a member of ''W''<sup>∗</sup>, then {{nowrap|1=''b''(''v'', ''w'') = ''f''(''v'')''g''(''w'')}} defines a bilinear map {{nowrap|''V'' × ''W'' → ''F''}}.
* The [[cross product]] in <math>\R^3</math> is a bilinear map <math>\R^3 \times \R^3 \to \R^3.</math>
* Let <math>B : V \times W \to X</math> be a bilinear map, and <math>L : U \to W</math> be a [[linear map]], then {{nowrap|(''v'', ''u'') ↦ ''B''(''v'', ''Lu'')}} is a bilinear map on {{nowrap|''V'' × ''U''}}.