let V be finite -dimentional and T:V->V*(V* is the dual space of V with same dimension as V) ,let ei be the bases of V,e^i be the bases of V*,consider the linear bijection :K:V->V* defined K(ei)=e^i,show that this bijection depends on the original choice of basis.

My proof is :

choose ei to be the bases first,then prove K under this bases is bijetion.then choose ei' to be the bases, and then prove K under the new bases is bijection , is my proof right or wrong??? can someone give the ideas?