If X is any set of vectors in an inner product space V, define
X⊥ = {v | v in V, (v, x) = 0 for all x in X}.
(a) Show that X⊥ is a subspace of V.
(b) If U = span, (u1,u2..., um}, show that (U⊥ = {u1,...,um.
(c) If X⊂ Y, show that Y⊥ ⊂ X⊥.
(d) Show that X⊥ ∩ Y⊥ = (X ∪ Y) ⊥