Detailed experimental studies have been made of the thermoelectric properties of the two-dimensional electron gas in single-interface GaAs-Al0.3Ga0.7As heterostructures. At temperatures below 10 K the zero-applied-magnetic-field thermopower is linear in temperature and is consistent with the diffusion thermopower for a two-dimensional electron gas. In the presence of a quantizing magnetic field the thermopower tensor is not diagonal and measurements of both components of the thermoelectric tensor have been made as functions of temperature and applied magnetic field. In the samples which have been studied, the experimental results are consistent with Gaussian-broadened Landau levels. At higher temperatures the thermopower is found to decrease with increasing temperature. This effect occurs both in zero field and in the presence of a magnetic field. This effect is similar to what is anticipated if phonon drag were present except the sign of this extra contribution is opposite to that observed in bulk n-type GaAs.