Euclidean vector space/Isometry/Orthogonal/Fact/Proof

Proof

Suppose first that ${}\varphi$ is an isometry. Then, ${}v_{i}=\varphi (u_{i})$ is an orthonormal basis due to fact. The coordinates of ${}v_{i}$ with respect to ${}u_{i}$ constitute the columns of the describing matrix ${}M$. Therefore, using exercise, we have

${}{v_{i}^{\text{tr}}}v_{j}=\left\langle v_{i},v_{j}\right\rangle =\delta _{ij}\,.$

Read as a matrix equation, this means

${}{M^{\text{tr}}}M={\begin{pmatrix}1&0&\cdots &\cdots &0\\0&1&0&\cdots &0\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&1&0\\0&\cdots &\cdots &0&1\end{pmatrix}}\,.$

This argument can be read backwards to get the reverse implication.

To fact