= Finite reflection decomposition of a Euclidean isometry
Every <Euclidean isometry> of $\mathbb R^n$ is a product of at most $n+1$ operations of <reflection in a hyperplane>. An isometry fixing zero is orthogonal: the <polarization identity> preserves inner products, and its values on an orthonormal basis determine its linear action. An <orthogonal transformation> is a product of at most $n$ linear reflections, by reflecting the image of the first basis vector back to that vector and inducting on its orthogonal complement. For an arbitrary isometry, first reflect its image of zero back to zero, if necessary, and apply the orthogonal result. Affine reflection hyperplanes, rather than only hyperplanes through zero, are essential here.
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