  We present the first calculation in lattice QCD of the process $\gamma K \to
K\pi$ in which the narrow $K^*$ vector resonance appears. Using a lattice on
which the pion has a mass of 284 MeV, we determine the transition amplitude at
128 points in the $(Q^2, E_{K\pi})$ plane, and find suitable resonant
scattering descriptions. We demonstrate the need to account for $S$--wave
$K\pi$ elastic scattering when converting the finite-volume matrix elements
computed in lattice QCD to the physically relevant infinite-volume matrix
elements, even when we are primarily interested in the $P$--wave amplitude.
Analytically continuing parameterizations of the $\gamma K \to K\pi$ amplitude
to the $K^*$ resonance pole, we obtain the $K^{*+} \to K^+ \gamma$ transition
form-factor, and compare the $Q^2=0$ value to the corresponding value extracted
from the experimental partial-decay width.
