# Bilinear form

## Scalar-valued bilinear function / From Wikipedia, the free encyclopedia

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In mathematics, a **bilinear form** is a bilinear map *V* × *V* → *K* on a vector space V (the elements of which are called *vectors*) over a field *K* (the elements of which are called *scalars*). In other words, a bilinear form is a function *B* : *V* × *V* → *K* that is linear in each argument separately:

*B*(**u**+**v**,**w**) =*B*(**u**,**w**) +*B*(**v**,**w**) and*B*(*λ***u**,**v**) =*λB*(**u**,**v**)*B*(**u**,**v**+**w**) =*B*(**u**,**v**) +*B*(**u**,**w**) and*B*(**u**,*λ***v**) =*λB*(**u**,**v**)

The dot product on $\mathbb {R} ^{n}$ is an example of a bilinear form.^{[1]}

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

When K 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.