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(Edited so that it obeys the rules of homework questions) I am stuck on this linear algebra problem from Griffiths's "Introduction to quantum mechanics". Can somebody give me some guidance? (I am interested to the sentence that is highlighted with red.)

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Let's assume you start with an orthonormal basis $|e_{i}\rangle \, , \, i = 1, ...,N$ , where $N$ is the dimensionality of your vector space. Then we would like for the new basis elements to be defined as:

\begin{equation} |\tilde{e}_{i}\rangle = S|e_{i}\rangle \end{equation}

EDIT:

Since it's not allowed to provide a complete proof and I have been compelled to only provide a guideline here, what I can say is that you should then consider what kind of mathematical condition orthonormality implies. That of course should apply to both the old and new basis. You should also think how the Hermitian conjugate of such a matrix appears when dealing with a complex vector space such as this. From these it's fairly easy to deduce the unitarity of $S$.

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    $\begingroup$ This is a complete (or nearly complete) answer to a homework-like question. We don't answer that kind of question here. We're not here to do people's homework for them. Instead, we give guidance when a question asks for help on a specific idea or principle. The question does not ask for that kind of help, it asks for an answer, which is contrary to the kind of questions allowed here. The question will be closed eventually. Until then, please rephrase your answer to provide guidance or a direction. Alternatively, you can delete your answer. $\endgroup$
    – garyp
    Jun 21 at 2:29
  • $\begingroup$ Well, I cannot delete it anymore (perhaps this is a privilege I haven't unlocked yet), but I can edit accordingly. $\endgroup$
    – rhomaios
    Jun 21 at 6:50
  • $\begingroup$ @rhomaios I think the problem is that your answer was marked as accepted - and, as far as I know, you cannot delete it yourself in this case. But the OP could remove the check mark such that you could, if you wanted to, delete the answer. $\endgroup$ Jun 21 at 10:10
  • $\begingroup$ @JasonFunderberker Since it's edited now there would be little point in further deleting it. But thanks for pointing it out. $\endgroup$
    – rhomaios
    Jun 21 at 10:14
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    $\begingroup$ @rhomaios Yes, I just wanted to inform you. $\endgroup$ Jun 21 at 10:18

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