Exercise 1.5.1 Let Y = (y1, y2, y3)

 be a random vector. Suppose that E(Y ) ∈

M, whereM is defined by M = {

(a, a −

b, 2b)

|a, b ∈ R}.

(a) Show thatM is a vector space.

(b) Find a basis forM.

(c) Write a linear model for this problem (i.e., find X such that Y = Xβ + e, E

(e) = 0).

(d) If β = (β1, β2)

 in part (c), find two vectors r = (r1, r2, r3)

 and s =

(s1, s2, s3)

 such that E(r Y ) = r Xβ = β1 and E(sY ) = β2. Find another vector t =

(t1, t2, t3)

 with r = t but E(tY ) = β1.