= Solution
In position representation,
$$
L_z=-i\hbar\left(x\partial_y-y\partial_x\right),
$$
which differentiates only with respect to $x$ and $y$. Multiplication by $z$ therefore commutes with it:
$$
\boxed{[L_z,z]=0}.
$$
Taking the matrix element of this commutator between angular-momentum eigenstates gives
$$
0=\hbar(m'-m)
\langle n',\ell',m'|z|n,\ell,m\rangle.
$$
Thus the matrix element can be nonzero only when
$$
\boxed{m'=m}.
$$
This is the magnetic part of the <electric-dipole selection rules for hydrogen>.
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