A question regarding the vector bundle associated to the frame bundle Let $E\to M$ be a smooth real vector bundle of rank $n$ and $\mathcal{F}E\to M$ be the associated frame bundle. The frame bundle is a principal $GL(n,\mathbb{R})$-bundle and $GL(n,\mathbb{R})$ has a natural action on $\mathbb{R}^n$. Thus we can construct the associated vector bundle $\mathcal{F}E \times_{GL(n,\mathbb{R})} \mathbb{R}^n \to M$. This is again a vector bundle of rank $n$. A natural question is that :

Is there a relation between the two bundles $\mathcal{F}E \times_{GL(n,\mathbb{R})} \mathbb{R}^n \to M$ and $E \to M$ ?

I guess that they would be isomorphic, but I am unable to see if there is a natural isomorphism between them.

You are right that they are isomorphic. The natural isomorphism to take is $\mathcal{F}E \times_{\text{GL}(n, \mathbb{R})} \mathbb{R}^n \to E$ given by
$$[\xi, v] \mapsto \xi(v)$$
(Here I am thinking of $\xi$ as an isomorphism from $\mathbb{R}^n$ to a fiber of $E$). Once you see that this is well-defined (exercise), it is clearly an isomorphism, since it covers the identity and on each fiber it's given by an isomorphism $\xi$.
This observation tells you that the construction which associates to a vector bundle its frame bundle gives an equivalence between vector bundles of rank $n$ and principal $\text{GL}(n, \mathbb{R})$-bundles, which is a neat and useful fact.
