Reflections and Rotations in Rn are One to One and Onto
A linear transformation is one to one and onto if and only if it has an inverse.We can represent a rotation in
\[\mathbb{R}^n\]
by a matrix \[R( \theta )\]
where \[R\]
means rotation and \[\theta\]
means a rotation anticlockwise through an angle \[\theta\]
.The inverse to this transformation is the rotation
\[R(- \theta )\]
, the rotation about the same axis through the same angle in the opposite direction. Hence rotations in \[\mathbb{R}^n\]
are one to one and onto.Reflections in
\[\mathbb{R}^n\]
are also one to one and onto since the inverse of any reflection represented by a matrix \[q ( \theta )\]
is \[q (- \theta )\]
Since reflections are self inverse, they are also one to one and onto.