Space-time codes leverage the availability of multiple antennas to enhance the reliability of communication over wireless channels. While space-time codes have initially been designed with a focus on open-loop systems, recent technological advances have enabled the possibility of low-rate feedback from the receiver to the transmitter. The focus of this paper is on the implications of this feedback in a single-user multi-antenna system with a general model for spatial correlation. We assume a limited feedback model, that is, a coherent receiver and statistics along with B bits of quantized channel information at the transmitter. We study space-time coding with a family of linear dispersion (LD) codes that meet an additional orthogonality constraint so as to ensure low-complexity decoding. Our results show that, when the number of bits of feedback (B) is small, a space-time coding scheme that is equivalent to beamforming and does not code across time is optimal in a weak sense in that it maximizes the average received SNR. As B increases, this weak optimality transitions to optimality in a strong sense which is characterized by the maximization of the average mutual information. Thus, from a system designer's perspective, our work suggests that beamforming may not only be attractive from a low-complexity viewpoint, but also from an information-theoretic viewpoint.