5.8. EXERCISES 141

Show that for each k ≤ n,

span (a1, · · · ,ak) = span (q1, · · · ,qk)

Prove that every subspace of Rn has an orthonormal basis. The procedure just de-scribed is similar to the Gram Schmidt procedure which will be presented later.

14. Suppose QnRn converges to an orthogonal matrix Q where Qn is orthogonal and Rn

is upper triangular having all positive entries on the diagonal. Show that then Qn

converges to Q and Rn converges to the identity.