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In this paper, we propose two solutions to the problem of joint transmit-receive antenna selection in a multiple-input multiple-output (MIMO) cognitive radio (CR) system. Our objective is to maximize CR data rates and satisfy interference constraints at the primary user (PU) receiver(s). In the first we approximate the original non-convex optimization problem using an iterative approach solving a series of smaller convex problems. Second we present a novel, norm-based transmit receive antenna selection technique that simultaneously improves throughput while maintaining the PU interference constraints. We show that this approach yields near optimal results with massive complexity reductions. We make a performance comparison between the proposed approaches and the optimal exhaustive search approach. We provide an analysis of the exhaustive search and relate selection gains to system parameters such as the shadow fading standard deviation, the path loss exponent and the number of PUs per square kilometer. Our results establish that antenna selection is a promising option for future MIMO CR devices in sparse PU environments.