the problem statement is, Prove that there is a unique bilinear form f, on Um X Vn such that, f(ui,vj)=aij where aij, i=1,...,m j=1,..n are given scalars, {u1,u2,...um} and {v1,v2,...vn} are ordered bases for Um and Vn respectively.

i've got how to prove it is bilinear but how do i prove uniqueness?