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

Explain how to construct $Q$ and $R$ by the Gram-Schmidt procedure. Why is this procedure not useful for numerical factorization of large matrices?

Let

$$A=\left[\begin{array}{rrr} 5 & 6 & -14 \\ 5 & 4 & 4 \\ -5 & 2 & -8 \\ 5 & 12 & -18 \end{array}\right], \quad b=\left[\begin{array}{l} 1 \\ 1 \\ 1 \\ 0 \end{array}\right]$$

Using the Gram-Schmidt procedure find a QR decomposition of A. Hence solve the least squares problem giving both $x^{*}$ and $\left\|\mathrm{A} x^{*}-b\right\|$.