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  
  by a matrix  
\[R( \theta )\]
  means rotation and  
  means a rotation anticlockwise through an angle  
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  
  are one to one and onto.
Reflections in  
  are also one to one and onto since the inverse of any reflection represented by a matrix  
\[q ( \theta )\]
\[q (- \theta )\]

Since reflections are self inverse, they are also one to one and onto.

Add comment

Security code