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This work examines a novel heterogeneous dynamic spectrum access network where the primary users (PUs) coexist with both underlay and interweave cognitive transmitters (UCTs and ICTs); all terminals being potentially equipped with multiple antennas. UCTs are allowed to transmit concurrently with PUs subject to interference constraints, while the ICTs employ spectrum sensing and are permitted to access the shared spectrum only when both PUs and UCTs are absent. We investigate the design of MIMO precoding algorithms for the UCT that increase the detection probability at the ICTs, while simultaneously meeting a desired Quality-of-Service target to the underlay cognitive receivers (UCRs) and constraining interference leaked to PUs. The objective of such a proactive approach, referred to as prescient precoding, is to minimize the probability of interference from ICTs to the UCRs and primary receivers due to imperfect spectrum sensing. We begin with downlink prescient precoding algorithms for multiple single-antenna UCRs and multi-antenna PUs/ICTs. We then present prescient block-diagonalization algorithms for the MIMO underlay downlink where spatial multiplexing is performed for a plurality of multi-antenna UCRs. Numerical experiments demonstrate that prescient precoding by UCTs provides a pronounced performance gain compared to conventional underlay precoding strategies.