Gareth wrote: << [I wrote:]

Er, guys, would one of you be so kind as to state what theorem is being discussed here?

The answer: Any isometry of R^n that fixes a k-dimensional subspace is the product of at most n-k reflections.

...

OK, I'll give it a shot: The proposition is equivalent to saying that any isometry of R^p with no nontrivial fixed subspace is the product of at most p reflections. Any isometry of R^n preserves mutually orthogonal subspaces: of dim = 2, and another one of dim = 1 (if n is odd). (This follows readily from complexification.) The cases of 1 dimension, and one reflection in 2D, are trivial. And a rotation of R^2 of angle theta is the product of a reflection about the x-axis, and one about the line at angle theta/2. QED. --Dan