Hint
For all , can be expressed univocally in the form ).
So can I let T(x)= lambda[1]*w[1] + ... + lambda[n]*w[n], for some x element of V, then say the right hand side is uniquely determined due to lambda[i] being uniquely determined and as you can write x in terms of lambda[i]*v[i] there has to be a unique map between each v[i] to w[i]?