# A linear algebra exercise from Griffiths "Introduction to quantum mechanics" [closed]

(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.) 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 $$S$$.