The group \(\mathbb{R}^*=\mathbb{R}\setminus \{0\}\) acts on \(\mathbb{R}^n\setminus \{0\}\) by scalar multiplication.
Remarks
- The action is free .
- The orbits are the lines through the origin with the origin removed.
- The orbit space \(\mathbb{R}^n\setminus \{0\}/\mathbb{R}^*\) is the real projective space \(\mathbb{P}^n\) .