Exercise 10.15 Let C(V0) = C(X) ∩ C(V), C(X) = C(V0, X1), C(V) =

C(V0, V1) with the columns of V0, V1, and X1 being orthonormal. Show that the columns of [V0, V1, X1] are linearly independent.

Hint: Write V0b0 + V1b1 + X1b2 = 0 and show that bi = 0, i = 0, 1, 2. In particular, write 0.5V0b0 + V1b1 = −(0.5V0b0 + X1b2), 0.5V0b0 + V1b1 ∈ C(V) and−(0.5V0b0 + X1b2) ∈ C(X) so the vector is inC(V0) =

C(X) ∩ C(V).