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**Orientation-preserving isometry**

I am preparing for an exam, and would like to have a rigorous definition of the following:

**Orientation-preserving isometry of $R^n$**

I know that it is something like the following (feel free to correct my wording):

When the homomorphism $\pi:M_n \rightarrow O_n$ is applied to the unique representation $t_a\phi$ of an isometry $f$, and $\pi(f)=\phi$, define $\sigma:M_n \rightarrow \pm 1$. This map that sends an **isometry of $R^n$** to $1$ is **orientation-preserving**.

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