Exercise 4.4.37

This question describes the next $q_{n+1}$ in Gram-Schmidt using the matrix $Q$ with the columns $q_1,\dots ,q_n$ (instead of using those $q$ ’s separately). Start from $a$, subtract its projection $p=Q{Q}^{T}a$ onto the earlier $q$ ’s, divide by the length of $e=a-Q{Q}^{T}a$ to get the next $q_{n+1}=e∕\parallel e\parallel$.