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 V ⊗ V → F. If B is a bilinear form on V the corresponding linear map is given by
- v ⊗ w ↦ B(v, w)
The set of all linear maps V ⊗ V → F is the dual space of V ⊗ V, so bilinear forms may be thought of as elements of
- (V ⊗ V)* ≅ 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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