What is "bilinear form"

n. (context linear algebra English) A function of two arguments from the same vector space which maps onto a field of scalars, which acts like a linear form with respect to either one of its arguments when the other one is held constant.

In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map , where K is the field of scalars. In other words, a bilinear form is a function 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(λu, v) = B(u, λv) = λB(u, v) The definition of a bilinear form can be extended to include modules over a commutative 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.