The linear map $H: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ represents reflection in the plane through the origin with normal $\mathbf{n}$, where $|\mathbf{n}|=1$, and $\mathbf{n}=\left(n_{1}, n_{2}, n_{3}\right)$ referred to the standard basis. The map is given by $\mathbf{x} \mapsto \mathbf{x}^{\prime}=\mathbf{M} \mathbf{x}$, where $\mathbf{M}$ is a $(3 \times 3)$ matrix.

Show that

$$M_{i j}=\delta_{i j}-2 n_{i} n_{j}$$

Let $\mathbf{u}$ and $\mathbf{v}$ be unit vectors such that $(\mathbf{u}, \mathbf{v}, \mathbf{n})$ is an orthonormal set. Show that

$$\mathbf{M n}=-\mathbf{n}, \quad \mathbf{M u}=\mathbf{u}, \quad \mathbf{M v}=\mathbf{v}$$

and find the matrix $\mathbf{N}$ which gives the mapping relative to the basis $(\mathbf{u}, \mathbf{v}, \mathbf{n})$.