I am assuming the definition of Grassmannian is known. Reference is Vector bundles and K-theory page no 28.
I am trying to prove that the map $p:E_n(\mathbb{R}^k)\rightarrow G(\mathbb{R}^k)$ is a vector bundle.
In particular I want to prove it is locally trivial.
Given $l\in G(\mathbb{R}^k)$ I want to produce an open set $U$ containing $l$ and some local trivialisation. That notes has already mentioned that theory choice of $U$ is $$\{l’\in G(\mathbb{R}^k):\pi_l(l’)\text{ has dimension } n\}$$ where $\pi_l:\mathbb{R}^k\rightarrow l$ is orthogonal projection onto $l$.
I fail to see what is the motivation for choosing such $U$.
Any comments are welcome.
It might be easier to understand the case of the tautological vector bundle over $\mathbb{R}P^k$ first, rather than the general case of $G_n(\mathbb{R}^k)$. (So, let $n=1$.) We think of $\mathbb{R}P^k$ to be the space of $1$-dimensional vector subspaces of $\mathbb{R}^k$.
We have $E_1(\mathbb{R}^k)=\{(\ell,v):\ell\in \mathbb{R}P^k\text{ and }v\in\ell\}$ and a map $p:E_1(\mathbb{R}^k)\to\mathbb{R}P^k$ defined by $p(\ell,v)=\ell$. The hope is that $p$ forms a vector bundle, whose fibers over a point $\ell\in\mathbb{R}P^k$ consist of the vectors comprising $\ell$.
What we wish to do is, for a given $\ell$, "measure" the vectors in nearby fibers relative to the vector space $p^{-1}(\ell)$. One might think of all the nearby fibers as being somewhat parallel, so orthogonal projection of a vector from some $p^{-1}(\ell')$ onto $p^{-1}(\ell)$ ought to be invertible. And it's a linear map.
Let $\pi_\ell:\mathbb{R}^k\to \ell$ be orthogonal projection onto $\ell$. The map $p^{-1}(\ell')\to p^{-1}(\ell)$ defined by $(\ell',v)\mapsto(\ell,\pi_\ell(v))$ is invertible exactly when $\pi_\ell(\ell')$ has dimension $1$.
The largest trivialization containing $\ell$ one can choose using this idea is $U=\{\ell'\in\mathbb{R}P^k:\pi_\ell(\ell')\text{ has dimension }1\}$. Along this set, you can effectively measure the vectors in $p^{-1}(U)$ using vectors of $p^{-1}(\ell)$.
(There are more details in Milnor and Stasheff's book.)
Exercise. Think about how a basis for $p^{-1}(\ell)$ looks when carried over to the other fibers of $U$.