On page 119 of Eisenbud and Harris' "The Geometry of Schemes," they construct the Grassmanian by gluing. We start by identifying $k$-dimensional subspaces of an $n$-dimensional space $K^n$ as the set of $k \times n$ matrices modulo left-multiplication by $k \times k$ invertible matrices. That makes sense. We then proceed via gluing:
Consider the space of all $k \times n$ matrices $W$. For each subset $I$ of cardinality $k$ in $\{1,..,n\}$, denote $W_I$ as the closed subspace of matrices whose $I^{th}$ submatrix $M_{I}$ is the identity. Then, for each $J \neq I$ (I assume also of cardinality $k$), let $W_{I, J}$ be the open subspace of matrices whose $J^{th}$ minor is nonzero. The authors then suggest that
$$W_{I, J} \to W_{J, I}$$
given by multiplication on the left by $M_J \cdot M_{I}^{-1}$ is an isomorphism.
How does this make sense? If I start with a matrix $M$ on the left-hand side, then $M_I$ is the identity, so $M_{I}^{-1}$ is the identity. Evidently I am interpreting some piece of notation incorrectly, and I would like to find out what that is.
Relevant: in the paragraphs above this construction, the authors note that if you take a $k \times n$ matrix $M$ whose $I^{th}$ minor (again, assume $I$ has cardinality $k$) is nonzero and multiply it by the inverse of its $I^{th}$ submatrix $M_I$, then you end up with a matrix $M'$ whose $I^{th}$ submatrix is the identity matrix. That's all well and good, but I don't see how it specifically gives us what we want above in a way that makes sense.