  The first measurements of the polarized structure function $\sigma_{LT'}$ for
the reaction $p(\vec e,e'K^+)\Lambda$ in the nucleon resonance region are
reported. Measurements are included from threshold up to $W$=2.05 GeV for
central values of $Q^2$ of 0.65 and 1.00 GeV$^2$, and nearly the entire kaon
center-of-mass angular range. $\sigma_{LT'}$ is the imaginary part of the
longitudinal-transverse response and is expected to be sensitive to
interferences between competing intermediate s-channel resonances, as well as
resonant and non-resonant processes. The results for $\sigma_{LT'}$ are
comparable in magnitude to previously reported results from CLAS for
$\sigma_{LT}$, the real part of the same response. An intriguing sign change in
$\sigma_{LT'}$ is observed in the high $Q^2$ data at $W\approx 1.9$ GeV.
Comparisons to several existing model predictions are shown.
