I want to show that $E=X/{\sim}$ (where $X=[0,1]\times\mathbb{R}^n$) is the total space of a smooth vector bundle over $S^1$. Here $\sim$ is defined as follows: fix a linear isomorphism $L$ of $\mathbb{R}^n$ and set $(0,v)\sim(1,Lv)$ for all $v\in\mathbb{R}^n$.
Here's what I've thought so far and my idea for solving this.
There is a smooth map $\pi:E\longrightarrow S^1$ that maps points $[(t,v)]$, $0<t<1$, to points $s(t)\in S^1$, $s(t)\neq \mathbf{1}$, where $\mathbf{1}\in S^1$ is the point where the gluing of $[0,1]$ took place ($s(t)$ is $p(t)$ where $p:[0,1]\longrightarrow S^1$ is just the defining quotient map of $S^1$), and $\pi$ maps $[(0,v)]=[(1,v)]$ to $\mathbf{1}\in S^1$.
One local trivialization is the easier one, which is where there is no gluing so basically it is the passage to the quotient of the identity map of $(0,1)\times \mathbb{R}^n$. I'm stuck with the other trivialization. My idea is to construct a map $[0,\epsilon)\cup (1-\epsilon,1]$ into itself such that $(t,v)$ is sent to $(t,L_t v)$ where $L_t$ is a family of isomorphisms that "approaches $L$" smoothly as $t\to 0$.
Is this a good idea? If so, could you show me how to fill in the details?
Recall how do you construct the Möbius strip.
Localization is easy, the point is how they (two copies of $I\times \mathbb{R^n}$) glued together on the overlap. (I do not think you need to use $L_t$).
Details: Glue $(0,0.6)\times \mathbb{R}^n$ and $(0.5, 1.1)\times \mathbb{R}^n$ via $(x, v)\mapsto (x, v)$ if $x\in (0.5,0.6)$, and $(x, v)\mapsto (x+1, Lv)$ if $x\in (0, 0.1)$.