In wiki page https://en.wikipedia.org/wiki/Grassmannian of Grassmannian, in the endowment of smooth structure to Grassmannian, I encounter this statement,
"For each ordered set of integers $1 \leq i_1<\cdots<i_k \leq n$, let $U_{i_1, \ldots, i_k}$ be the set of elements $w \in \mathbf{G r}_k(V)$ for which, for any choice of homogeneous coordinate matrix $W$, the $k \times k$ submatrix $W_{i_1, \ldots, i_k}$ whose $j$-th row is the $i_j$-th row of $W$ is nonsingular. "
I can't understand what is the meaning of "for any choice of homogeneous coordinate matrix", I don't know much of homogeneous coordinate, it seems that homogeneous coordinate is an act of adding another coordinate so that we can conduct translation and rotation by using single matrix, and I don't know what is "homogeneous coordinate matrix", so I can't understand this statement.Could you please help me? thank you
It’s easier to do this more abstractly. Given a $k$-plane $K$, choose a basis $e_1,\dots,e_n$ of $V$ such that $e_1,\dots,e_k$ is a basis of $K$. There is an injective map from the vector space $M(n-k,k)$ of all $n-k$ by $k$ matrices to $G(k,V)$, where each $A\in M(n-k,k)$ is mapped to the $k$-plane spanned by $e_i+A_i^\mu e_\mu$, where $1\le i\le k$ and $k+1\le\mu\le n$. This is a coordinate chart. Given any other $k$-plane $K’$, you get another chart. If $K’$ lies in the coordinate chart of $K$, it is easy to calculate the transition map between the two charts and check it’s smooth. Since you can define such a chart for every $k$-plane $K$, the charts cover $G(k,V)$.