Exercise 3.2.59

$R=\left[\begin{array}{ccc}1\hfill & 2\hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill \end{array}\right]$has $R^{T}R=\left[\begin{array}{ccc}1\hfill & 2\hfill & 0\hfill \\ 2\hfill & 4\hfill & 0\hfill \\ 0\hfill & 0\hfill & 1\hfill \end{array}\right]$and this matrix row reduces to

Always $R^{T}R$ has the same nullspace as $R$, so its row reduced form must be $R$ with $n-m$ extra zero rows. $R$ is determined by its nullspace and shape!