In this paper, a new transmit antenna selection algorithm for spatial multiplexing ultra-wideband multiple-input multiple-output systems is proposed. It utilizes the null space of the channel matrix spatially and temporally combined for available transmit antennas. A selection strategy is used to successively find and remove the transmit antennas having the largest contribution to the null space. Its error performance is shown to be close to that of the optimal antenna selection algorithm with low complexity.