Define the linear least-squares problem for the equation $A \mathbf{x}=\mathbf{b}$, where $A$ is an $m \times n$ matrix with $m>n, \mathbf{b} \in \mathbb{R}^{m}$ is a given vector and $\mathbf{x} \in \mathbb{R}^{n}$ is an unknown vector.

If $A=Q R$, where $Q$ is an orthogonal matrix and $R$ is an upper triangular matrix in standard form, explain why the least-squares problem is solved by minimizing the Euclidean norm $\left\|R \mathbf{x}-Q^{\top} \mathbf{b}\right\|$.

Using the method of Householder reflections, find a QR factorization of the matrix

$$A=\left[\begin{array}{rrr} 1 & 3 & 3 \\ 1 & 3 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & -1 \end{array}\right]$$

Hence find the solution of the least-squares problem in the case

$$\mathbf{b}=\left[\begin{array}{r} 1 \\ 1 \\ 3 \\ -1 \end{array}\right]$$