The joint design of optimal linear Minimum MSE (MMSE) filters at the transmitter and receiver of a Multiple-Input Multiple-Output (MIMO) system arises as a natural problem when transmitting discrete-time continuous-amplitude sources using analog Joint Source Channel Coding (JSCC). In this work, we derive an explicit closed form solution for the MMSE linear precoder and equalizer of a MIMO system under a transmit power constraint. Such a solution is specially suitable for analog JSCC because it preserves the low complexity and negligible delay distinctive of this type of systems. We also show that the obtained solution for the MMSE linear precoder resembles the waterfilling algorithm and provides the optimal MMSE distribution of the transmit power among the different data streams. The performance of the explicit MMSE transceiver is evaluated in the context of analog JSCC transmissions for MIMO-OFDM channels and the obtained results show that the system closely approaches the theoretical limits and outperforms the zero forcing (ZF) precoding. Finally, we derive some interesting properties of MIMO systems with MMSE linear precoder and equalizer, and quantify their superior performance with respect to those that use ZF linear precoding and/or ZF equalization.