The transverse polarization in a plane perpendicular to the nuclear spin axis has been determined for electrons emitted in the \ensuremath{\beta} decay of polarized $^{8}\mathrm{Li}$. Such a spin component signals a violation of time reversal symmetry. Using a sample of $^{8}\mathrm{Li}$ nuclei with vector polarization of \ensuremath{\sim}0.11 and a polarimeter with average analyzing power S=-0.10, the asymmetry in the Mott scattering of decay electrons was measured with an accuracy of \ifmmode\pm\else\textpm\fi{}4\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}5}$. From the asymmetry, the transverse spin polarization of the electrons has been determined with an accuracy of \ifmmode\pm\else\textpm\fi{}4\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}4}$, and the amplitude R=(-0.2\ifmmode\pm\else\textpm\fi{}4.0) \ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}3}$ of the triple correlation between nuclear spin, momentum, and spin of the electron has been obtained. These results represent the most precise measurement of the transverse polarization of leptons emitted in weak decays. Time reversal violating part of the correlation amplitude, ${\mathit{R}}_{\mathrm{TRV}}$=(-0.9\ifmmode\pm\else\textpm\fi{}4.0) \ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}3}$, is deduced. It provides the first direct determination of the imaginary, charged weak tensor interaction -0.022(${\mathit{C}}_{\mathit{T}}$+${\mathit{C}}_{\mathit{T}}^{\ensuremath{'}}$)/${\mathit{C}}_{\mathit{A}}$0.017 or alternatively -0.004(${\mathit{a}}_{\mathit{RL}}^{\mathit{T}}$)0.005 (90% C.L.). These are the tightest limits for exotic (nonvector or axial vector) time reversal violating couplings in semileptonic weak decays. A report is presented with emphasis on experimental details and data analysis. Relations of this study to other tests of time reversal violation are discussed.