A bilinear form on V is a bilinear mapping such that

B(u+u', v) = B(u,v) + B(u',v),

B(u, v+v') = B(u,v) + B(u,v'),

B(tu, v) = B(u, tv) = tB(u,v), where t is a scalar in the field F.

Let u= BX, v=BY, where B = (v_1, v_2, ... , v_n) is a basis of V and X,Y are coordinate vectors.

Then, .

Using bilinearity, , where A is .