Originally Posted by
Opalg Assuming you know how to carry out the Gram–Schmidt process, start with a set containing (-5, 2, 2, -1) plus the vectors from some basis. The obvious choice would be the set S = {(-5, 2, 2, -1), (1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)}. Apply the G–S process to this set and it will give you the desired orthogonal basis.
(The set S has five vectors in it, so it isn't a basis. But it does span the whole of R^4, which is what matters. When you apply the G–S process to S, the end product will be another set of five vectors, the first being (-5, 2, 2, -1), and the last one will be the zero vector, which of course you will discard. The first four vectors will constitute the desired basis.)