= Solution
An $m$ by $n$ matrix has <matrix rank> $m$ exactly when at least one of its $m$ by $m$ minors has nonzero <determinant>. For each multi-index $I$, the set
$$
U_I=\{M:\det M_I\ne0\}
$$
is <open set>[open] in the <vector space> of all real $m$ by $n$ matrices, because the determinant is <continuous function>[continuous]. Their union is $X_{m,n}$, so $X_{m,n}$ is itself open in $\mathbb R^{mn}$. It therefore inherits the standard <smooth manifold> structure and has <dimension of a vector space>[dimension] $mn$. This is the <full-row-rank matrix manifold>.
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