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Linear transceiver design for multiple access channels (MACs) with spatial correlation at both transmitter and receiver is investigated in the presence of inaccurate channel state information (CSI). We consider a training-based channel estimation at the receiver while a limited-rate feedback channel conveys the transmitter information. Imperfect knowledge comes from the channel estimation errors and the quantization noise. Restricting the decoder to be linear yields to minimize of the sum-mean square error (sum-MSE) subject to individual power constraints. Although no closed-form solution is possible in a multi-user setting, an efficient iterative algorithm relying on the KKT conditions is derived. Numerical results show sum-MSE and BER performance to measure the sensitivity of a mismatched design as well as the effect of quantization noise. Furthermore, the study of channel uncertainty enables to assess the relative impact of imperfect CSI at both ends.