Exercise 10.11 Use Definition B.31 and Proposition 10.4.6 to show that MC(X)∩C(V) = M − MW where C(W) = C[M(I − MV )].

Another way to find V0 is due to Rao and Mitra (1971, pp. 118, 119). V0 =

X ˜U where C( ˜U ) = C[X

(I − MV )]⊥. Note that (I − MV )X ˜U = 0, so C(X ˜U ) ⊂

C(V0). To see that C(V0) ⊂ C(X ˜U ), take v ∈ C(V0). Then v = Xb for some b but also (I − MV )Xb = 0, so b ∈ C[X

(I − MV )]⊥ and b = ˜U γ for some γ , hence v = X ˜U γ . Note that (I − MV ) can be replaced with any matrix B having C(B) =

C(V)

⊥ and also that r (V0) = r (X) + r (V) − r (X, V ).