For some reason my book distinguishes the two names.
If a set is an orthogonal set, doesn't that make it immediately a basis for some subspace $W$ since all the vectors in the orthogonal set are linearly independent anyways? So why do we have two different words for the same thing?
Best Answer
When you say orthogonal basis you mean that the set is a basis for the whole given space. Every orthogonal set is a basis for some subset of the space, but not necessarily for the whole space.