As we tell our undergraduates, if K is a field, then there are a great number of different ways to describe a linear subspace of K^n. If the base is an algebraic object with less structure than a field, linear algebra becomes more subtle, and some of these descriptions cease to agree. One such setting is tropical geometry. Tropical geometers have reached consensus as to what the "correct" notion of tropical linear subspace is (one way to get it is by a vector of determinants). My subject will be one of the "wrong" descriptions, namely row spaces of matrices, which only produces a subset of the tropical linear spaces. Applications include generalisations of Mason's results from the '70s on presentations of transversal matroids, and a construction in the new area of tropical ideal theory.

This work is variously joint with Felipe Rinc\'on, Jorge Alberto Olarte, and Jeffrey and Noah Giansiracusa.