Question

how to check function R(z)=(e^itheta)z is one-to-one

Answer 1

R(z)=\[e ^{i \theta}z\]

Answer 2

\[let \ z_0,z_1 \in \mathbb{C}
\\assume\\R(z_0)=R(z_1)
\\\text{we need to show that this implies }z_0=z_1\\so\\R(z_0)=R(z_1)\implies e^{i\theta }z_0=e^{i\theta}z_1\iff z_1=z_0\]