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This paper focuses on linear transceiver design for rate optimization in multiuser Gaussian multple-input multiple-output (MIMO) systems. We focus on two design criteria: 1) maximizing the weighted sum rate subject to a total power constraint; 2) maximizing the minimum user rate subject to a total power constraint. For these problems, new power allocation strategies are derived, which can be formulated as geometric programs (GPs) involving mean square errors (MSEs). Based on these solutions, we propose iterative algorithms, where each iteration contains the optimization of the uplink power, uplink receive filters, and downlink receive filters. Monotonic convergence of the algorithms is proved. Simulations show that the algorithms outperform existing linear schemes. Additionally, we extend the results to other variations of the problems, such as, the problem of sum-rate constrained or user-rate constrained power minimization, and the problem of sum-rate maximization subject to user-rate constraints and a total power constraint.