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Matroids are central objects in combinatorial algebraic geometry due in part to a fundamental relationship between them and the Grassmannian. For example, torus orbit closures in the Grassmannian for a natural torus action correspond exactly to matroids. These toric varieties are almost never smooth. One may extend this action further to flag varieties and the resulting toric varieties correspond to flag matroids. For any matroid, Borovik, Gelfand, Vince, and White associated a canonical flag matroid to it called the underlying flag matroid. In this work, we exhibit a surjective torus equivariant rational map from the Grassmannian onto certain flag varieties. Using this map and a connection to the greedy algorithm for linear optimization on matroids, we argue that toric varieties corresponding to underlying flag matroids are smooth and describe the combinatorics of their moment polytopes. Based on joint work with Raman Sanyal.