# Any two Lorentz space orthonormal basis are related by a Lorentz transform

**Theorem :** For any two orthonormal basis, $\{ e_\mu \}$ and $\{ e_\mu' \}$, there exists a Lorentz transform $\Lambda \in \mathrm{O}(n-1, 1)$ such that for any two elements of the basis, $\Lambda^\mu_\nu e^i_\mu = e_\nu'$

**Proof :** For an orthonormal basis, we have $\langle e^i, e^j \rangle = \eta^{ij}$. From the signature of our metric, this means we have $n-1$ spacelike vectors

From this proof, we know that between any two such vectors, there exists a Lorentz transform. We can therefore write

\begin{equation} \Lambda^\mu_{\nu, ij} e^i_\mu = (e^j_\nu)' \end{equation}We have $n$ such Lorentz transforms. We therefore need to show that, if two such transformations are different, the new basis would fail to be orthonormal.