Define the QR factorization of an $m \times n$ matrix $A$ and explain how it can be used to solve the least squares problem of finding the $x^{*} \in \mathbf{R}^{n}$ which minimises $\|A x-b\|$ where $b \in \mathbf{R}^{m}, m>n$, and the norm is the Euclidean one.

Define a Householder (reflection) transformation $H$ and show that it is an orthogonal matrix.

Using a Householder reflection, solve the least squares problem for

$$A=\left(\begin{array}{rrr} 2 & 4 & 7 \\ 0 & 3 & -1 \\ 0 & 0 & 2 \\ 0 & 0 & 1 \\ 0 & 0 & -2 \end{array}\right), \quad b=\left(\begin{array}{r} 9 \\ -7 \\ 3 \\ 1 \\ -1 \end{array}\right)$$

giving both $x^{*}$ and $\left\|A x^{*}-b\right\|$.