Let W ⊂ V with dim V = n. Suppose w1........wn, is an orthogonal basis for W and wm+1,... wn is an orthogonal basis for W⊥.
(a) Prove that the combination w1,...,wn forms an orthogonal basis of V.
(b) Show that if v = c1w1 + ... cn wn is any vector in V, then its orthogonal decomposition v = w + z is given by w = c1 w1 + ... + cm wm ∊ W and z = cm+1 wm+1 + ... + cn wn ∊ W⊥.