If I have a basis of V, say v1,..,vn and w1,...,wn elements of W, how can I show there is a unique transformation T(vi)=wi for all i 1..n?
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]?