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 vector $x^{*} \in \mathbb{R}^{n}$ which minimises $\left\|A x^{*}-b\right\|$, where $b \in \mathbb{R}^{m}, m>n$, and the norm is the Euclidean one.

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

Using a Householder transformation, solve the least squares problem for

$$A=\left[\begin{array}{rrr} 1 & -1 & 5 \\ 0 & 1 & 5 \\ 0 & 0 & 3 \\ 0 & 0 & 4 \end{array}\right], \quad b=\left[\begin{array}{r} 1 \\ 2 \\ -1 \\ 2 \end{array}\right]$$

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