Chapter 5.6, Problem 11E

LetAbean $m\times 3$ matrix.LetQRbetheQRfactorization obtained when the classical Gram−Schmidt process is applied to the column vectors of A, and let $\tilde{Q}\tilde{R}$ be the factorization obtained when the modified Gram−Schmidt process is used. Show that if all computations were carried out using exact arithmetic then we would have $\tilde{Q}=Q$ and $\tilde{R}=R$ and show that when the computations are done in finite-precision arithmetic, ${\tilde{r}}_{23}$ will not necessarily be equal to $r_{23}$ and consequently ${\tilde{r}}_{33}$ and ${\tilde{q}}_3$ will not necessarily be the same as $r_{33}$ and $q_3.$