Let $V$ be an $n$-dimensional $\mathbb{R}$-vector space with an inner product. Let $W$ be an $m$-dimensional subspace of $V$ and $W^{\perp}$ its orthogonal complement, so that every element $v \in V$ can be uniquely written as $v=w+w^{\prime}$ for $w \in W$ and $w^{\prime} \in W^{\perp}$.

The reflection map with respect to $W$ is defined as the linear map

$$f_{W}: V \ni w+w^{\prime} \longmapsto w-w^{\prime} \in V$$

Show that $f_{W}$ is an orthogonal transformation with respect to the inner product, and find its determinant.