On 1/9/06, Gareth McCaughan <gareth.mccaughan@pobox.com> wrote:
[about this theorem:]
Any isometry of R^n that fixes a k-dimensional subspace is the product of at most n-k reflections.
I think the isometries of S^n rather than R^n are intended --- the presence of translations would complicate the argument. Also Lemma 3 carefully says "can be written as", since the conclusion is false unless the reflection hyperplanes are chosen with linearly independent coefficient vectors.
[Dan's proof:] Looks OK to me, but Fred's question wasn't "can someone give me a proof of this?" but "can someone explain the proof of this found in Conway&Smith's book". :-)
Quite so. However, if you're going to use induction [Gareth's proof] --- which C&S doesn't mention --- why not just say that the intersection of n-k independent hyperplanes has dimension exactly k, and a point is fixed just when it lies in this set? [I'm assuming that when Lemma 3 says "fixes", it means pointwise, and implies only that the axis includes the k-flat.] In fairness, this glitch is unimportant and seems to be isolated --- but rather unfortunate that it occurs so early in the book! Fred Lunnon