In this paper we propose fast and efficient transmit antenna selection algorithms for wireless spatial multiplexing (SM) systems with linear receivers. Motivated by matrix inverse properties, a recursive matrix inverse calculation is developed for fast transmit antenna selection. Two selection criteria are proposed based on the recursive matrix inverse updating: one is the minimum mean square error (MMSE) and another is the maximum signal to interference noise ratio (SINR). Different from the conventional optimal antenna selection algorithms, the proposed algorithms are capable of computing matrix inverse and selection criteria from the previous computational results via a simple recursive procedure, so that the algorithmic complexity is reduced significantly. Simulation results show that our algorithms perform very close to the optimal exhaustive search algorithms in terms of bit error rate and capacity in both uncorrelated and highly correlated MIMO channels, while the computational complexity is much lower.