Is there a lower bound for $N$ such that the linear span of $N$ linearly independent matrices $\in C^{n\times n}$ contains at least one rank 1 matrix? Given $N$ linearly independent matrices in $C^{n\times n}$, is there a lower bound known for $N$ as a function of $n$ such that the linear span of those $N$ matrices is guaranteed to contain a rank 1 matrix?
 A: The set of rank $1$ matrices is a projective variety of dimension $2n-2$ inside the projective space over $C^{n \times n}$, which has dimension $n^2-1$. By the projective dimension theorem, it must intersect every projective variety of dimension $k$ whenever $2n-2+k > n^2-1$. The linear span of $N$ linearly independent matrices is a projective variety of dimension $k=N-1$. Therefore, if $N > (n-1)^2$, your span contains a rank $1$ matrix.
This is sharp: a generic subspace of dimension $(n-1)^2$ does not contain a rank $1$ matrix.
This question is fundamental in quantum information theory, where we want to find subspaces of $\mathbf{C}^n \otimes \mathbf{C}^n$ where every element is (very) entangled. See for example Chapter 8.1 of our book.
Aubrun, Guillaume; Szarek, Stanisław J., Alice and Bob meet Banach. The interface of asymptotic geometric analysis and quantum information theory, Mathematical Surveys and Monographs 223. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-3468-7/hbk; 978-1-4704-4172-2/ebook). xxi, 414 p. (2017). ZBL1402.46001.
