Exercise 1.5.8 Show that

(a) for Example 1.0.1 the perpendicular projection operator onto C(X) is M = 1 6 J 6 6 + 1 70 ⎡
⎢⎢⎢⎢⎢⎢⎣
25 15 5 −5 −15 −25 15 9 3 −3 − 9 −15 5 3 1 −1 − 3 − 5 − 5 −3 −1 1 3 5 −15 −9 −3 3 9 15 −25 −15 −5 5 15 25

(b) for Example 1.0.2 the perpendicular projection operator onto C(X) is M = ⎡
⎢⎢⎢⎢⎢⎢⎣
1/3 1/3 1/3 0 0 0 1/3 1/3 1/3 0 0 0 1/3 1/3 1/3 0 0 0 0 0 0 1 0 0 0 0 0 1/2 1/2 0 0 0 0 1/2 1/2