Exercise 4.4.36

$[Q,R]=qr(A)$ produces from $A$ ( $m$ by $n$ of rank $n$ ) a "full-size" square $Q=\left[\begin{array}{cc}{Q}_1\hfill & Q_2\hfill \end{array}\right]$ and $\left[\begin{array}{c}R\hfill \\ 0\hfill \end{array}\right]$. The columns of $Q_1$ are the orthonormal basis from Gram-Schmidt of the column space of $A$. The $m-n$ columns of $Q_2$ are an orthonormal basis for the left nullspace of $A$. Together the columns of $Q=\left[\begin{array}{cc}{Q}_1\hfill & Q_2\hfill \end{array}\right]$ are an orthonormal basis for $R^m$.