  Background: Measurements of polarization observables for the reactions
$\vec{\gamma} p \rightarrow K^+ \Lambda$ and $\vec{\gamma} p \rightarrow K^+
\Sigma^0$ have been performed. This is part of a programme of measurements
designed to study the spectrum of baryon resonances.
  Purpose: The accurate measurement of several polarization observables
provides tight constraints for phenomenological fits. Beam-recoil observables
for the $\vec{\gamma} p \rightarrow K^+ \Sigma^0$ reaction have not been
reported before now.
  Method: The measurements were carried out using linearly polarized photon
beams and the CLAS detector at the Thomas Jefferson National Accelerator
Facility. The energy range of the results is 1.71\,GeV $<W<$ 2.19\,GeV, with an
angular range $-0.75 < \cos\theta^{\star}_K < +0.85$.
  Results: The observables extracted for both reactions are beam asymmetry
$\Sigma$, target asymmetry $T$, and the beam-recoil double polarization
observables $O_x$ and $O_z$.
  Conclusions: Comparison with theoretical fits indicates that in the regions
where no previous data existed, the new data contain significant new
information, and strengthen the evidence for the set of resonances used in the
latest Bonn-Gatchina fit.
