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.