In the solution a a problem in quantum computation I saw this line: $$U_{ij}=\langle\psi_i|\left(\sum_k|\phi_k\rangle\!\langle\psi_k|\right) |\psi_j\rangle.$$
Where $U_{ij}$ are the entries of a unitary matrix $U$ and $\phi_k$ and $\psi_k$ form two different orthonormal basis vectors.
I already succeed to prove that $U$ can be written as the sum of $\phi_k$ and $\psi_k$ vectors but I don't understand why the expression above is the entry $U_{ij}$.
Can someone help? Thanks!
That's how you define the entries of the matrix of $U$ with respect to the basis $\{\psi_j\}$. The entries of $U$ with respect to that basis are defined as the numbers that satisfy
$$ U|\psi_j\rangle=\sum_k U_{kj}|\,\psi_k\rangle. $$ As the basis is orthonormal, you get $$ \langle \psi_i|U|\psi_j\rangle=U_{ij}. $$