(a) Define a Givens rotation $\Omega^{[p, q]} \in \mathbb{R}^{m \times m}$ and show that it is an orthogonal matrix.

(b) Define a QR factorization of a matrix $A \in \mathbb{R}^{m \times n}$ with $m \geqslant n$. Explain how Givens rotations can be used to find $Q \in \mathbb{R}^{m \times m}$ and $R \in \mathbb{R}^{m \times n}$.

(c) Let

$$\mathbf{A}=\left[\begin{array}{ccc} 3 & 1 & 1 \\ 0 & 4 & 1 \\ 0 & 3 & 2 \\ 0 & 0 & 3 / 4 \end{array}\right], \quad \mathbf{b}=\left[\begin{array}{c} 98 / 25 \\ 25 \\ 25 \\ 0 \end{array}\right]$$

(i) Find a QR factorization of $A$ using Givens rotations.

(ii) Hence find the vector $\mathbf{x}^{*} \in \mathbb{R}^{3}$ which minimises $\|A \mathbf{x}-\mathbf{b}\|$, where $\|\cdot\|$ is the Euclidean norm. What is $\left\|\mathrm{A} \mathbf{x}^{*}-\mathbf{b}\right\|$ ?