Show that if $\mathbf{u} \in \mathbb{R}^{m} \backslash\{\mathbf{0}\}$ then the $m \times m$ matrix transformation

$$H_{\mathbf{u}}=I-2 \frac{\mathbf{u} \mathbf{u}^{\top}}{\|\mathbf{u}\|^{2}}$$

is orthogonal. Show further that, for any two vectors $\mathbf{a}, \mathbf{b} \in \mathbb{R}^{m}$ of equal length,

$$H_{\mathbf{a}-\mathbf{b}} \mathbf{a}=\mathbf{b} .$$

Explain how to use such transformations to convert an $m \times n$ matrix $A$ with $m \geqslant n$ into the form $A=Q R$, where $Q$ is an orthogonal matrix and $R$ is an upper-triangular matrix, and illustrate the method using the matrix

$$A=\left[\begin{array}{rrr} 1 & -1 & 4 \\ 1 & 4 & -2 \\ 1 & 4 & 2 \\ 1 & -1 & 0 \end{array}\right]$$