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The Ingleton inequalities are the inequality constraints known to be required of representable matroids. For a matroid of n elements, there are 16n Ingleton inequalities. In this paper, we show that many of these inequalities are redundant. We explicitly determine the unique minimal set of Ingleton inequalities, which number on the order of 6n/4- O(5n). In information theory, these inequalities are required of the entropy functions of certain random variables constructed from linear subspaces. As a result, these inequalities appear as constraints in linear programming outer bounds for the capacity region of multisource network coding where the codes are required to be linear. Ingleton inequalities have also played an instrumental role in demonstrating the insufficiency of linear codes for multisource network coding. The reduction that we obtain for Ingleton inequalities is sufficiently large to meaningfully reduce the complexity of these linear programming bounds.