Bilinear form

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(λu, v) = λB(u, v)
• B(u, v + w) = B(u, v) + B(u, w)     and     B(u, λv) = λB(u, v)

The dot product on ${\displaystyle \mathbb {R} ^{n}}$ 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.