Bilinear Form - Relation To Tensor Products

Relation To Tensor Products

By the universal property of the tensor product, bilinear forms on V are in 1-to-1 correspondence with linear maps VVF. If B is a bilinear form on V the corresponding linear map is given by

vwB(v, w)

The set of all linear maps VVF is the dual space of VV, so bilinear forms may be thought of as elements of

(VV)* ≅ V*V*

Likewise, symmetric bilinear forms may be thought of as elements of Sym2(V*) (the second symmetric power of V*), and alternating bilinear forms as elements of Λ2V* (the second exterior power of V*).

Read more about this topic:  Bilinear Form

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