If you know about matrices, you can easily construct a inversible matrix that transforms your basis into your second set of vectors, and so conclude that it's also a basis.

But you can also show that each v_i , i=1,...,n , is a linear combination of {cv_1,...,v_n}, and then you've won because as {v_1,...v_n} is a basis, then your space has a dimension of n, so a subset with n vectors that spans your space is a basis.