Orthogonal reflection/Hyperplane/Eigenspaces/Example

For an orthogonal reflection of ${}\mathbb {R} ^{n}$, there exists an ${}(n-1)$-dimensional linear subspace ${}U\subseteq \mathbb {R} ^{n}$, which is fixed by the mapping and every vector orthogonal to ${}U$ is sent to its negative. If ${}v_{1},\ldots ,v_{n-1}$ is a basis of ${}U$ and ${}v_{n}$ is a vector orthogonal to ${}U$, then the reflection is described by the matrix

${\begin{pmatrix}1&0&\cdots &\cdots &0\\0&1&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&1&0\\0&\cdots &\cdots &0&-1\end{pmatrix}}$

with respect to this basis.