(a) Define the QR factorization of a rectangular matrix and explain how it can be used to solve the least squares problem of finding an $x^{*} \in \mathbb{R}^{n}$ such that

$$\left\|A x^{*}-b\right\|=\min _{x \in \mathbb{R}^{n}}\|A x-b\|, \quad \text { where } \quad A \in \mathbb{R}^{m \times n}, \quad b \in \mathbb{R}^{m}, \quad m \geqslant n,$$

and the norm is the Euclidean distance $\|y\|=\sqrt{\sum_{i=1}^{m}\left|y_{i}\right|^{2}}$.

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

(c) Using Householder reflection, solve the least squares problem for the case

$$A=\left[\begin{array}{rr} 2 & 4 \\ 1 & -1 \\ 2 & 1 \end{array}\right], \quad b=\left[\begin{array}{l} 1 \\ 5 \\ 1 \end{array}\right]$$

and find the value of $\left\|A x^{*}-b\right\|=\min _{x \in \mathbb{R}^{2}}\|A x-b\|$.