We study the achievable rate of an uplink MIMO cognitive radio system where the primary user (PU) and the secondary user (SU) aim to communicate to the closest primary base station (BS) via a multi-access channel through the same unmanned aerial vehicle (UAV) relay. The SU message is then forwarded from the primary BS to the secondary network with a certain incentive reward as a part of the cooperation protocol between both networks. A special linear precoding scheme is proposed to enable the SU to exploit the PU free eigenmodes. We analyze two scenarios in which the UAV relay gain matrix is either fixed or optimized. We derive the optimal power allocation that maximizes the achievable rate of the SU respecting power budget, interference, and relay power constraints. Numerical results highlight the cognitive rate gain of our proposed scheme with respect to various problem parameters. We also highlight the effect of UAV altitude on the SU and PU rates. Finally, when the relay matrix is optimized, we show that the PU rate is remarkably enhanced and that the SU rate is only improved at high power regime.