In mathematics, a **bilinear form** on a vector space *V* is a bilinear mapping *V* × *V* → *F*, where *F* is the field of scalars. That is, a bilinear form is a function *B*: *V* × *V* → *F* 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**) = λ*B*(**u**,**v**)

The definition of a bilinear form can easily be extended to include modules over a commutative ring, with linear maps replaced by module homomorphisms. When *F* 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.

Read more about Bilinear Form: Coordinate Representation, Maps To The Dual Space, Symmetric, Skew-symmetric and Alternating Forms, Reflexivity and Orthogonality, Different Spaces, Relation To Tensor Products, On Normed Vector Spaces

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