Exercise 1.5.8. Show that

(a) for Example 1.0.1 the perpendicular projection operator onto C(X) is M =

1 6

J6 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

⎤

⎥⎥⎥⎥⎥⎦

.