<sub>[Cross-posted at Math SE][1]</sub>

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Let $S\subseteq M_{n,m}(\mathbb{C})$ be a $d$-dimensional subspace of the space of $n\times m$ complex matrices (with $n\leq m$, say). I am interested in figuring out conditions on $S$ which guarantee the existence of a full rank matrix in $S$. Since the maximum dimension of a subspace in which every matrix has rank $\leq (n-1)$ is known to be equal to $m(n-1)$ (see the answer to this [question](https://math.stackexchange.com/questions/66877/max-dimension-of-a-subspace-of-singular-n-times-n-matrices/66936#66936) and other linked questions), it is clear that if $d>m(n-1)$, then $S$ must contain a full rank matrix. Are other such simple conditions known which ensure the existence of a full rank matrix in $S$?

According to this [paper](https://arxiv.org/abs/quant-ph/0303055), the $n=m$ case of the stated problem is quite well-known and is known as the Edmond's problem, which has led to exciting developments in the field of matroid theory. Since I'm not familiar with this field, I would love to gather some references for interesting results in this field which are directly related to the mentioned problem. Thanks!

[1]: https://math.stackexchange.com/q/4085327