Unimodular Form

Unimodular Form

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(λu, v) = B(u, λv) = λB(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.