Exercise 6.4 Consider an inner product space X and a subspace X0. Suppose that any vector y ∈X can be written uniquely as y = y0+y1 with y0 ∈X0 and y1 ⊥

X0. Let M(x) be a linear operator on X in the sense that for any x ∈ X , M(x) ∈

X and for any scalars a1,a2 and any vectors x1, x2, M(a1x1 +a2x2) = a1M(x1)+

a2M(x2). M(x) is defined to be a perpendicular projection operator onto X0 if for any x0 ∈X0, M(x0) = x0, and for any x1 ⊥X0, M(x1) = 0. Using Definition A.11

(Alternate), show that for any vector y, M(y) is the perpendicular projection of y into X0.