I am reading Lee's Introduction to Smooth Manifolds and got stuck on the problem 5.6. The question is written here, question 1. (There is a typo in the question. The last sentence should be "Show that F is smoothly isomorphic...")
I do not know how to use the transition function in order to show that these bundles are isomorphic. I tried to find smooth trivializations of F, but I could not.
Edit: This is how Lee defined the Möbius bundle in his book. (Lee, page 105, example 5.2)