Every orthogonal transformation of a finite-dimensional nondegenerate quadratic space over a field of characteristic different from two is a product of at most its dimension many hyperplane reflections. In the real positive-definite case, a reflection normal to sends to whenever . Composing with that reflection fixes and leaves an orthogonal transformation of , so induction proves the bound. Each reflection has determinant minus one, hence orientation-preserving transformations use an even number of reflections.
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The Cartan–Dieudonné theorem is a result in differential geometry and linear algebra that characterizes elements of a projective space using linear combinations of certain vectors. Specifically, it is often described in the context of the geometry of vector spaces and the projective spaces constructed from them.