# The Unapologetic Mathematician

## Reflections

Before introducing my main question for the next series of posts, I’d like to talk a bit about reflections in a real vector space $V$ equipped with an inner product $\langle\underline{\hphantom{X}},\underline{\hphantom{X}}\rangle$. If you want a specific example you can think of the space $\mathbb{R}^n$ consisting of $n$-tuples of real numbers $v=(v^1,\dots,v^n)$. Remember that we’re writing our indices as superscripts, so we shouldn’t think of these as powers of some number $v$, but as the components of a vector. For the inner product, $\langle u,v\rangle$ you can think of the regular “dot product” $\langle u,v\rangle=u^1v^1+\dots+u^nv^n$.

Everybody with me? Good. Now that we’ve got our playing field down, we need to define a reflection. This will be an orthogonal transformation, which is just a fancy way of saying “preserves lengths and angles”. What makes it a reflection is that there’s some $n-1$-dimensional “hyperplane” $P$ that acts like a mirror. Every vector in $P$ itself is just left where it is, and a vector on the line that points perpendicularly to $P$ will be sent to its negative — “reflecting” through the “mirror” of $P$.

Any nonzero vector $\alpha$ spans a line $\mathbb{R}\alpha$, and the orthogonal complement — all the vectors perpendicular to $\alpha$ — forms an $n-1$-dimensional subspace $P_\alpha$, which we can use to make just such a reflection. We’ll write $\sigma_\alpha$ for the reflection determined in this way by $\alpha$. We can easily write down a formula for this reflection:

$\displaystyle\sigma_\alpha(\beta)=\beta-\frac{2\langle\beta,\alpha\rangle}{\langle\alpha,\alpha\rangle}\alpha$

It’s easy to check that if $\beta=c\alpha$ then $\sigma_\alpha(\beta)=-\beta$, while if $\beta$ is perpendicular to $\alpha$ — if $\langle\beta,\alpha\rangle=0$ — then $\sigma_\alpha(\beta)=\beta$, leaving the vector fixed. Thus this formula does satisfy the definition of a reflection through $P_\alpha$.

The amount that reflection moves $\beta$ in the above formula will come up a lot in the near future; enough so we’ll want to give it the notation $\beta\rtimes\alpha$. That is, we define:

$\displaystyle\beta\rtimes\alpha=\frac{2\langle\beta,\alpha\rangle}{\langle\alpha,\alpha\rangle}$

Notice that this is only linear in $\beta$, not in $\alpha$. You might also notice that this is exactly twice the length of the projection of the vector $\beta$ onto the vector $\alpha$. This notation isn’t standard, but the more common notation conflicts with other notational choices we’ve made on this weblog, so I’ve made an executive decision to try it this way.