(a) Show that for Example 1.0.1 the perpendicular projection operator onto C(X) is 25 15 5 -5 -15 -25 15 9 3 -3 -9 -15 16 1 5 3 1 -1 -3 -5 M= (/6 + 70 -5 -3 -1 1 3 5

-15 -9 -3 3 9 15

-25 -15 -5 5 15 25

(b) Show that for Example 1.0.2 the perpendicular projection operator onto C(X) is 1/3 1/3 1/3 0 0 0 1/3 1/3 1/3 0 0 0 M= 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