The ReflectionMatrix(v) command returns the $mxm$ reflection Matrix, R, determined by the Vector v, where $m=\mathrm{Dimension}\left(v\right)$. Thus $R·v=-v$ and $R·w=w$ for all w such that $v·w=0$.

In two-dimensional space, the reflection subspace is a line through the origin.  If the equation of this line is $y=mx$, or $mx-y=0$, then $v=⟨m\,\mathrm{-1}⟩$.

In three-dimensional space, the reflection subspace is a plane through the origin.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{LinearAlgebra}\right]\right)\:$

$\mathrm{ReflectionMatrix}\left(⟨1\,1⟩\right)$

$\left[\begin{array}{cc}0& \mathrm{-1}\\ \mathrm{-1}& 0\end{array}\right]$

$\mathrm{ReflectionMatrix}\left(⟨a\,b\,c⟩\right)$

$\left[\begin{array}{ccc}\frac{-a^2+b^2+c^2}{a^2+b^2+c^2}& -\frac{2ab}{a^2+b^2+c^2}& -\frac{2ac}{a^2+b^2+c^2}\\ -\frac{2ab}{a^2+b^2+c^2}& \frac{a^2-b^2+c^2}{a^2+b^2+c^2}& -\frac{2bc}{a^2+b^2+c^2}\\ -\frac{2ac}{a^2+b^2+c^2}& -\frac{2bc}{a^2+b^2+c^2}& \frac{a^2+b^2-c^2}{a^2+b^2+c^2}\end{array}\right]$

$\mathrm{ReflectionMatrix}\left(⟨1\,2\,3\,4⟩\right)$

$\left[\begin{array}{cccc}\frac{14}{15}& -\frac{2}{15}& -\frac{1}{5}& -\frac{4}{15}\\ -\frac{2}{15}& \frac{11}{15}& -\frac{2}{5}& -\frac{8}{15}\\ -\frac{1}{5}& -\frac{2}{5}& \frac{2}{5}& -\frac{4}{5}\\ -\frac{4}{15}& -\frac{8}{15}& -\frac{4}{5}& -\frac{1}{15}\end{array}\right]$