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

Define a Givens rotation $\Omega^{[p, q]}$ and show that it is an orthogonal matrix.

Using a Givens rotation, solve the least squares problem for

$$A=\left[\begin{array}{lll} 2 & 1 & 1 \\ 0 & 4 & 1 \\ 0 & 3 & 2 \\ 0 & 0 & 0 \end{array}\right], \quad b=\left[\begin{array}{l} 2 \\ 3 \\ 1 \\ 2 \end{array}\right]$$

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