## 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.