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If $V=C^n$ (Complex numbers) unitary, inner product vector space such that $dim_CV=n$ and $\beta=${$u_1,u_2,...,u_n$} basis. Prove that $\beta$ is orthonormal if and only if the matrix

so $$A=\begin{pmatrix}\leftrightarrow&A_1 &\leftrightarrow\\\ \cdots & \cdots& \cdots\\\leftrightarrow&A_n& \leftrightarrow\end{pmatrix}=\begin{pmatrix}\leftrightarrow&u_1 &\leftrightarrow\\\ \cdots & \cdots& \cdots\\\leftrightarrow&u_n& \leftrightarrow\end{pmatrix} \in M^{nxn}(C)$$

is unitary, i.e, A is inversible and $A^{-1}=A^*$

I would be glad if I can get some help, I don't even know where to start, I know that a basis is orthonormal if $<u_i*u_j>=0$ $ \forall i\neq j$ and if $||u_k||=1$ $ \forall k$

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  • $\begingroup$ Yes I do, I had no idea how to represent that in a better way in Latex $\endgroup$ – Juju9704 Jun 12 at 12:03
  • $\begingroup$ Just note that $(AA^{*})_{ij}=\langle u_i,u_j\rangle$ $\endgroup$ – Peter Melech Jun 12 at 12:06
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Hint: Consider what each entry of $AA^*$ is, and while you do, look hard at your definition of orthonormal basis.

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  • $\begingroup$ I think that for $AA^*$, the entries are $\delta_{ij}$, right? $\endgroup$ – Juju9704 Jun 12 at 12:07
  • $\begingroup$ @Juju9704 All the time? $\endgroup$ – Arthur Jun 12 at 12:23

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