Let $x$ denote a $k$-dimensional subspace of $\mathbb{C}^n$ and let $A_x$ be a $k\times n$ matrix whose rows are a basis for $x$. The matroid $M_x$ on the columns of $A_x$ is invariant under a change of basis for $x$. What can we say about the set $\Gamma_x$ of all $k$-dimensional subspaces $y$ such that $M_y = M_x?$. We will explore this question algebraically, showing that for some matroids that arise geometrically many non-trivial equations vanishing on $\Gamma_x$ can be derived geometrically. This is joint work with Will Traves and Ashley Wheeler.