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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 measurementsdesigned to study the spectrum of baryon resonances.  Purpose: The accurate measurement of several polarization observablesprovides tight constraints for phenomenological fits. Beam-recoil observablesfor the $\vec{\gamma} p \rightarrow K^+ \Sigma^0$ reaction have not beenreported before now.  Method: The measurements were carried out using linearly polarized photonbeams and the CLAS detector at the Thomas Jefferson National AcceleratorFacility. The energy range of the results is 1.71\,GeV $<W<$ 2.19\,GeV, with anangular 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 polarizationobservables $O_x$ and $O_z$.  Conclusions: Comparison with theoretical fits indicates that in the regionswhere no previous data existed, the new data contain significant newinformation, and strengthen the evidence for the set of resonances used in thelatest Bonn-Gatchina fit.

Photoproduction of $\Lambda$ and $\Sigma^0$ hyperons using linearly polarized photons