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 equipped with an inner product . If you want a specific example you can think of the space consisting of -tuples of real numbers . Remember that we’re writing our indices as superscripts, so we shouldn’t think of these as powers of some number , but as the components of a vector. For the inner product, you can think of the regular “dot product” .
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 -dimensional “hyperplane” that acts like a mirror. Every vector in itself is just left where it is, and a vector on the line that points perpendicularly to will be sent to its negative — “reflecting” through the “mirror” of .
Any nonzero vector spans a line , and the orthogonal complement — all the vectors perpendicular to — forms an -dimensional subspace , which we can use to make just such a reflection. We’ll write for the reflection determined in this way by . We can easily write down a formula for this reflection:
It’s easy to check that if then , while if is perpendicular to — if — then , leaving the vector fixed. Thus this formula does satisfy the definition of a reflection through .
The amount that reflection moves in the above formula will come up a lot in the near future; enough so we’ll want to give it the notation . That is, we define:
Notice that this is only linear in , not in . You might also notice that this is exactly twice the length of the projection of the vector onto the vector . 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.