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I have a question that I do not understand how to solve that.

Let $V$ be inner product space.

Let {$e_{1},...,e_{n}$} an orthonormal basis for $V$

Let {$z_{1},...,z_{n}$} an orthonormal basis for $V$

I have to show that the matrix represents

the transformation matrix between {$e_{1},...,e_{n}$} to

{$z_{1},...,z_{n}$} is unitary.

How do I do it ?

I got no clue!!

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I'd sketch a proof as follows. Regard the vectors in the orthonormal bases $\left(e_{1},...,e_{n}\right)$ and $\left(z_{1},...,z_{n}\right)$ as column vectors. Then let $\mathbf{U}_e$ and $\mathbf{U}_z$ be the matrices where the rows are the transposes of the column vectors $e_{1},...,e_{n}$ and $z_{1},...,z_{n}$ respectively. Then both $\mathbf{U}_e$ and $\mathbf{U}_z$ are unitary, and the matrix which maps between $\left(e_{1},...,e_{n}\right)$ and $\left(z_{1},...,z_{n}\right)$ is going to be given by $\mathbf{U}_e\mathbf{U}_z^{-1}$, which will be unitary because $\mathbf{U}_e$ and $\mathbf{U}_z$ are unitary.

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