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about the origin (which can be written as the composition of two reflections). Any arbitrary composition of such rotations and reflections can be rewritten as a composition of no more than 2 reflections. Similarly, in three-dimensional
Euclidean space, every orthogonal transformation can be described
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which preserve the value of the bilinear form between every pair of vectors; in
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is also a symmetric bilinear space). The orthogonal transformations in the space are those
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plane, every orthogonal transformation is either a reflection across a
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The notion of a symmetric bilinear space is a generalization of
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not equal to 2. Then, every element of the orthogonal group
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as a single reflection, a rotation (2 reflections), or an
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55:symmetric bilinear space can be described as the
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324:Introduction to quadratic forms over fields
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157:symmetric bilinear space over a
73:whose structure is defined by a
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260:; Lafontaine, Jacques (2004).
101:under composition, called the
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332:American Mathematical Society
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191:Indefinite orthogonal group
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291:Garling, D. J. H. (2011).
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45:orthogonal transformation
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201:Householder reflections
75:symmetric bilinear form
434:Abstract algebra stubs
373:-related article is a
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16:For other uses, see
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262:Riemannian Geometry
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212:References
110:Euclidean
221:(2001).
185:See also
118:rotation
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149:be an
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