Exclusive Formula omitted-meson electroproduction is studied by the HERMES experiment, using the 27.6 GeV longitudinally polarized electron/positron beam of HERA and a transversely polarized hydrogen target, in the kinematic region 1.0 GeV Formula omitted 7.0 GeV Formula omitted, 3.0 GeV Formula omitted 6.3 GeV, and Formula omitted 0.4 GeV Formula omitted. Using an unbinned maximum-likelihood method, 25 parameters are extracted. These determine the real and imaginary parts of the ratios of several helicity amplitudes describing Formula omitted-meson production by a virtual photon. The denominator of those ratios is the dominant amplitude, the nucleon-helicity-non-flip amplitude Formula omitted, which describes the production of a longitudinal Formula omitted-meson by a longitudinal virtual photon. The ratios of nucleon-helicity-non-flip amplitudes are found to be in good agreement with those from the previous HERMES analysis. The transverse target polarization allows for the first time the extraction of ratios of a number of nucleon-helicity-flip amplitudes to Formula omitted. Results obtained in a handbag approach based on generalized parton distributions taking into account the contribution from pion exchange are found to be in good agreement with these ratios. Within the model, the data favor a positive sign for the Formula omitted transition form factor. By also exploiting the longitudinal beam polarization, a total of 71 Formula omitted spin-density matrix elements is determined from the extracted 25 parameters, in contrast to only 53 elements as directly determined in earlier analyses.