Prove that {v_1,...,v_n} is linearly independent also and use the following theorem:

Let V and U be vector space over field K. Let {v_1,...,v_n} a basis of V and {u_1,...,u_n} a basis of U. Then exist a unique linear transformation F:V-->U which:

F(v_1)=u_1, ... , F(v_n)=u_n.