Proving that the Grassmanian is a smooth manifold I am trying to find a differentiable structure on the Grassmannian, which is the set of all $k$-planes in $\mathbb{R}^{n}
 $. To do this, I have to show that for any given $\alpha$, $\beta$, the set $$\left\{ \left(AA_{\alpha}^{-1}\right)_{\alpha'}:A\in\mbox{M}_{k}\left(n\times k,\mathbb{R}\right),A_{\alpha},A_{\beta}\in\mbox{GL}\left(k,\mathbb{R}\right)\right\} $$
is open in the set of  $(n-k)\times k$ matrices.
Here is the notation:
-$\alpha$ is just a $k$-tuple $\left(\alpha_{1},\ldots,\alpha_{k}\right)$
  such that $1\leq\alpha_{1}<\cdots<\alpha_{k}\leq n$, and $\beta$ is another  $k$-tuple $\left(\beta_{1},\ldots,\beta_{k}\right)$
  such that $1\leq\beta_{1}<\cdots<\beta_{k}\leq n$.
-$\mbox{M}_{k}\left(n\times k,\mathbb{R}\right)$ is the set of all $n\times k$ matrices with rank $k$, which is an open subset of the set of all $n\times k$ matrices.
-Given $A\in\mbox{M}_{k}\left(n\times k,\mathbb{R}\right)$, $A_{\alpha}$ is defined to be the $k\times k$ submatrix of $A$ whose $i$th row is the $\alpha _{i}$th row of $A$. We define $A_{\alpha '}$ to be the $(n-k)\times k$ submatrix of $A$ which consists of the remaining rows.  
I am unsure how to proceed.
 A: Fix some $k$-tuples $\alpha$, $\beta$ and let $S = \{ A \in M_k(n \times k, \mathbb{R}) \mid A_\alpha, A_\beta \in \mathrm{GL}(k,\mathbb{R}) \}$. We want to show that the image of the map
$$
\begin{align}
\phi : S &\to M((n-k)\times k,\mathbb{R}) \\
A &\mapsto (A{A_\alpha}^{-1})_{\alpha'}
\end{align}
$$
is open. Hence, it is enough to prove that $\phi$ is an open map.
We can write $\phi$ as a composition $$\phi = p_{\alpha'} \circ m \circ (i,\mathrm{inv}\circ \pi_\alpha)$$
where


*

*$i : S \to M(n \times k)$ is the inclusion map;

*$\pi_\alpha : S \to \mathrm{GL}(k)$ is defined by $\pi_\alpha(A) = A_\alpha$;

*$\mathrm{inv} : \mathrm{GL}(k) \to \mathrm{GL}(k)$ is matrix inversion;

*$m : M(n\times k) \times \mathrm{GL}(k) \to M(n \times k)$ is given by matrix multiplication;

*$p_{\alpha'} : M(n\times k) \to M((n-k) \times k)$ is defined by $p_{\alpha'}(A) = A_{\alpha'}$.


Note that these maps are all open. Indeed,


*

*$i$ is open because $S$ is an open subset of $M(n\times k)$;

*$p_{\alpha'} : M(n\times k) \to M((n-k) \times k)$ is a surjective linear map;

*$\pi_\alpha$ is the restriction of a surjective linear map $M(n\times k) \to M(k \times k)$ to the open subset $S \subset M(n \times k)$;

*$\mathrm{inv} : \mathrm{GL}(k) \to \mathrm{GL}(k)$ is a homeomorphism.


Since a composition of open maps is open, this shows that $\phi$ is open, as desired.
