Background: Measurements of polarization observables for the reactions γ p → K + and γ p → K + 0 have been performed. This is part of a program of measurements designed to study the spectrum of baryon resonances in particular, and nonperturbative QCD in general. Purpose: The accurate measurement of several polarization observables provides tight constraints for phenomenological fits, which allow the study of strangeness in nucleon and nuclear systems. Beam-recoil observables for the γ p → K + 0 reaction have not been reported before now. Method: The measurements were carried out using linearly polarized photon beams incident on a liquid hydrogen target, and the CLAS detector at the Thomas Jefferson National Accelerator Facility. The energy range of the results is 1.71 < W < 2.19 GeV, with an angular range −0.75 < cos θ K < +0.85. Results: The observables extracted for both reactions are beam asymmetry , target asymmetry T , and the beam-recoil double polarization observables O x and O z .