We consider the coexistence of a multiple-input multiple-output secondary system with a multiple-input single-output primary link with different degrees of coordination between the systems. First, for the uncoordinated underlay cognitive radio scenario, we fully characterize the optimal parameters that maximize the secondary rate subject to a primary rate constraint for a transmission strategy that combines rate splitting and interference cancellation. Second, we establish a model for the coordinated overlay cognitive radio scenario that consists of a message-learning phase followed by a communication phase. We then propose a transmission strategy that combines techniques for cooperative communication and for the classical cognitive radio channel. We optimize our system to maximize the rate of communication for the secondary users under a primary-user rate constraint and find efficient algorithms to compute the optimal system parameters. Finally, we compare both cognitive radio strategies to assess their relative merits and to evaluate the effect of the message-learning phase. We observe that for closely located transmitters, the overlay strategy outperforms the underlay strategy. In this situation, learning the primary message is very beneficial for the secondary systems, especially if they are interference-limited rather than power-limited. The situation is reversed when the distance between the transmitters is large. In either case, we observe that there is room for significant improvement if the transmitter implements both strategies and decides adaptively which one to use according to the channel conditions. We conclude our work with a discussion on the extension to the coexistence with multiple-input multiple-output primaries.