Proof of uniqueness of a linear transformation:
We can assume that {e1,e2,...} is the basis of the vector space V.
Suppose T and T' two linear transformations such that
T(e1) = u1 and T'(e1) = u1
T(e2) = u2 and T'(e2) = u2
T(e3) = u3 and T'(e3) = u3, etc.
Then, for each v in V
T(v) = T(k.e1+l.e2+m.e3+...)
= k.u1 + l.u2 + m.u3 + ... for some real k,l,m, ...
and
T'(v)= T'(k.e1 + l.e2 + m.e3+ ...)
= T'(k.e1) + T'(l.e2) + T'(m.e3) + ...
= k.T'(e1) + l.T'(e2) + m.T'(e3) + ...
= k.u1 + l.u2 + m.u3 + ...
Hence, T = T'
Hope this answers your problem.