Editing Bilinear form (section)

==Relation to tensor products==
By the [[universal property]] of the [[tensor product]], there is a canonical correspondence between bilinear forms on {{mvar|V}} and linear maps {{math|''V'' ⊗ ''V'' → ''K''}}. If {{math|''B''}} is a bilinear form on {{mvar|V}} the corresponding linear map is given by
{{block indent|left=1.6|text= {{math|'''v''' ⊗ '''w''' ↦ ''B''('''v''', '''w''')}}}}
In the other direction, if {{math|''F'' : ''V'' ⊗ ''V'' → ''K''}} is a linear map the corresponding bilinear form is given by composing ''F'' with the bilinear map {{math|''V'' × ''V'' → ''V'' ⊗ ''V''}} that sends {{math|('''v''', '''w''')}} to {{math|'''v'''⊗'''w'''}}.

The set of all linear maps {{math|''V'' ⊗ ''V'' → ''K''}} is the [[dual space]] of {{math|''V'' ⊗ ''V''}}, so bilinear forms may be thought of as elements of {{math|(''V'' ⊗ ''V'')<sup>∗</sup>}} which (when {{mvar|V}} is finite-dimensional) is canonically isomorphic to {{math|''V''<sup>∗</sup> ⊗ ''V''<sup>∗</sup>}}.

Likewise, symmetric bilinear forms may be thought of as elements of {{math|(Sym<sup>2</sup>''V'')<sup>*</sup>}}  (dual of the second [[symmetric power]] of {{math|''V''}}) and alternating bilinear forms as elements of {{math|(Λ<sup>2</sup>''V'')<sup>∗</sup> ≃ Λ<sup>2</sup>''V''<sup>∗</sup>}} (the second [[exterior power]] of {{math|''V''<sup>∗</sup>}}). If {{math|char(''K'') ≠ 2}}, {{math|(Sym<sup>2</sup>''V'')<sup>*</sup> ≃ Sym<sup>2</sup>(''V''<sup>∗</sup>)}}.