Proof unitary matrix vector columns (and rows) are orthonormal using index notation

Question

A matrix $A$ is said to be unitary if it satisfies: $AA^\dagger = A^{\dagger}A = I$

being $A^\dagger$ the conjugate transpose of $A$ and $I$ the identity matrix.

I would like to prove that the colums (rows) of the matrix $A$ form an orthonormal basis, by using index notation if possible.

The thing is that I know how to do it with matrix notation but I struggle with index notation, so if someone could help me with it...

Answer 1

$$ \begin{aligned} I_{ij}&=(A^\dagger A)_ij\\ &=\sum_k A^{\dagger}_{ik}A_{kj}\\ &=\sum_k\overline{A_{ki}}A_{kj}\\ &=\langle a_j,a_i\rangle \end{aligned} $$ You should figure out what $a_i,a_j$ denote.