Projection Theorem Derivation of Property 6.2. Throughout this problem, we use the notation of Property 6.2 and of the Projection Theorem given in Appendix B, where H is L2. If Lk+1 = sp{y1, . . . , yk+1}, and Vk+1 =

sp{yk+1 − yk k+1}, for k = 0, 1, . . . , n − 1, where yk k+1 is the projection of yk+1 on Lk, then, Lk+1 = Lk ⊕ Vk+1. We assume P0 0 > 0 and R > 0.

(a) Show the projection of xk on Lk+1, that is, xk+1 k , is given by xk+1 k = xk k + Hk+1(yk+1 − yk k+1), where Hk+1 can be determined by the orthogonality property E

n xk − Hk+1(yk+1 − yk k+1)

yk+1 − yk k+1
0 o

= 0.

Show Hk+1 = Pk k Φ0 A0 k+1 Ak+1Pk k+1A0 k+1 + R

−1