Abstract
We study projections onto a subspace and reflections with respect to a subspace in an arbitrary vector space with an inner product. We give necessary and sufficient conditions for two such transformations to commute. We then generalize the result to affine subspaces and transformations.
Highlights
Two lines 1 and 2 in 2 are considered
Supposing that U is a vector space equipped with an inner product, V ⊂ U is a linear subspace of U
Given a vector u ∈U, we know from linear algebra [1] [2] that u can be decomposed uniquely = as u where pV (u ) ∈V is the projection of the vector u onto V and u′ ⊥ V, i.e. U= V ⊕ V ⊥
Summary
Two lines 1 and 2 in 2 are considered. When is the reflection over 1 followed by the reflection over 2 the same as the reflection over 2 followed by the reflection over 1 ? It is easy to see that it is the case if and only if 1 ⊥ 2 or 1 = 2. When considering subspaces of 3 , we can ask similar questions for lines, for planes or for the mixed case of one line and one plane. Instead of addressing those cases one by one, we generalize the situation of arbitrary two linear subspaces of a vector space with an inner product
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