## 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.