Quote by HallsofIvy
If you add nk independent vectors to the given k vectors to make a basis for R^n and then apply GramSchmidt, the vectors {u_(k+1),..., u_n} are not necessarily orthogonal to the original set of k vectors so does NOT solve the problem.

Why not? The given set of k vectors and the set obtained by applying GramSchmidt have the same span.