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)

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