Linearity of Certain Reflections in $\Bbb R^n$ Let $f: \mathbb{R^n} \rightarrow \mathbb{R^n}$ be a reflection about a hyperplane passing through $\vec 0$. Is $f$ always a linear transformation? If so, how can the matrix of the reflection be determined based what is known about the hyperplane?

Let's do the general case directly. 
Suppose the normal direction of this hyperplane is $\mathbf{a} = (a_1, a_2, \ldots, a_n)^T \neq \mathbf{0}$. Since the hyperplane passes through the origin, it has the representation:
$$P: \quad a_1 x_1 + a_2 x_2 + \cdots a_n x_n = 0.$$
Given a point $\mathbf{z} = (z_1, z_2, \ldots, z_n)^T \in \mathbb{R}^n$, it's reflection $\mathbf{z'} = (z_1', z_2', \ldots, z_n')^T$ should satisfy:
\begin{align*}
& \frac{1}{2}(\mathbf{z} + \mathbf{z'}) \in P, \tag{1} \\
& \mathbf{z'} - \mathbf{z} \parallel \mathbf{a}. \tag{2}
\end{align*}
By $(2)$, there exists a constant $k$ such that 
$$z_i' = z_i + ka_i, \; i = 1, 2, \ldots, n. \tag{3}$$
Substitute $(3)$ into $(1)$ gives
$$\sum_{i = 1}^n a_i\left(\frac{2z_i + ka_i}{2}\right) = 0.$$
Solve this for $k$, we have
$$k = -\frac{2\mathbf{a}^T\mathbf{z}}{\mathbf{a}^T\mathbf{a}}.$$
Therefore, use $(3)$ again:
$$\mathbf{z'} = \mathbf{z} - \frac{2\mathbf{a}^T\mathbf{z}}{\mathbf{a}^T\mathbf{a}}\mathbf{a} = \left(I - \frac{2\mathbf{a}\mathbf{a}^T}{\mathbf{a}^T\mathbf{a}}\right)\mathbf{z}.$$
Thus the reflection is a linear transformation, the transformation matrix is given by $I - \frac{2\mathbf{a}\mathbf{a}^T}{\mathbf{a}^T\mathbf{a}}$.
