Full-row-rank matrix manifold
= Full-row-rank matrix manifold
{title2=$X_{m,n}$}
The real $m$ by $n$ matrices of <matrix rank> $m\leq n$ form an open <smooth manifold> $X_{m,n}\subseteq\mathbb R^{mn}$. Indeed, it is the union of the open sets on which a chosen $m$ by $m$ minor has nonzero <determinant>, so its dimension is $mn$.