If you've got some

vector space V and a

field **F**, then a

*bilinear form* is map M(u,v) from V×V to

**F** which is

linear in each argument.

That is, for all

vectors u,v,w and all

scalars a,b:

- M(u+v,w) = M(u,w)+M(v,w)
- M(u,v+w) = M(u,v)+M(u,w)
- M(av,bw) = abM(v,w)