Consider the reflection maps in \(\mathbb { R } ^ { n }\) . The reflection across the origin transforms every vector \({ \vec { x } } \in \mathbb { R } ^ { n } { \mathrm { ~ t o ~ } } - { \vec { x } } .\) The reflection across one of the basis axis \(O i\) inverts all coordinates of \(\vec { x }\) except \(x _ { i }\) Similarly, one can define the reflections across the planes in \(\mathbb { R } ^ { 3 }\) and higher dimensional hyperplanes in \(\mathbb { R } ^ { n } , n > 3\) Some of these reflections are equivalent to rotations around the origin O, others are not.
Observe that in \(\mathbb { R } ^ { 2 }\) the central reflection is equivalent to the rotation by an angle π around O. Show that no reflections across a line crossing O can be achieved by any rotation.
In \(\mathbb { R } ^ { 3 }\) , find if one can achieve by some rotation around \(O \ 1 )\) the reflection across \(O , 2 )\) the reflection across an arbitrary line crossing \(O , 3 )\) the reflection across an arbitrary plane crossing O.
In \(\mathbb { R } ^ { n }\) , formulate and prove a general statement about the existence of rotations that perform the reflection across a given m-dimensional plane, \(m = 0 , 1 , . . . , n - 1\) , crossing the origin (where \(m = 1\) corresponds to the line, and \(m = 0 \textrm { - } \mathrm { t o }\) the point).