$E \times_H \mathbb{R}^n$ is isomorphic to the total space of the tautological bundle $\gamma^n$ over $G_n(\mathbb{R}^{n+k})$?

Question

Let $H$ be the subgroup of $\text{GL}(n + k, \mathbb{R})$ consisting of matrices whose lower $n \times k$ block is empty; i.e. consisting of matrices of the form$$\begin{pmatrix} A & * \\ 0 & B\end{pmatrix},$$where $A \in \text{GL}(n, \mathbb{R}$), $B \in \text{GL}(k, \mathbb{R})$, and $*$ is arbitrary. I know that $\text{GL}(n + k, \mathbb{R})$ is a principal $H$ bundle over the Grassmann manifold $G_n(\mathbb{R}^{n+k})$. Let $E$ denote the total space of this principal bundle. Let $\rho: H \to \text{GL}(n, \mathbb{R})$ be the surjective homomorphism obtained by letting $H$ act on the first $n$ coordinates of $\mathbb{R}^{n+k}$, and let $E \times_H \mathbb{R}^n$ be the total space of the $\mathbb{R}^n$ bundle over $G_n(\mathbb{R}^{n+k})$ obtained from this action. How do I see that $E \times_H \mathbb{R}^n$ is isomorphic to the total space of the tautological bundle $\gamma^n$ over $G_n(\mathbb{R}^{n+k})$?

Answer 1

Write an element of $GL(n+k,\mathbb R)$ as $\begin{pmatrix}C & D\\ E& F\end{pmatrix}$, then the principal $H$-bundle sends this element to $\rm{Im}\begin{pmatrix}C \\ E\end{pmatrix}$ in the Grassmannian.

Write elements of $E\times_H\mathbb R^n$ as $\left[ \begin{pmatrix}C & D\\ E& F\end{pmatrix}, x\right]$.

Write elements in the total space of the tautological bundle as $(V,w)$, where $V$ is the $n$-dimensional subspace and $w\in V$.

The iso is then given as $\left[ \begin{pmatrix}C & D\\ E& F\end{pmatrix}, x\right]\mapsto \left( \rm{Im}\begin{pmatrix}C \\ E\end{pmatrix} ,\begin{pmatrix}C \\ E\end{pmatrix}\cdot x\right)$.