I am currently trying to develop a better understanding of the pullback of a vector bundle (and more generally of fibre bundles), although I am struggling to develop nontrivial examples which can be easily visualised.
As an example, it seems to me that the pullback of the mobius strip (considered as a line bundle over $S^1$) by the map $f:\theta\mapsto2\theta$ should be the trivial bundle, although I am at a loss for how to prove this from the definition given,
$f^{-1}E = \{(n,e)\in N\times E \ \ | f(n) = \pi(e)\}$
With $f:N\to M$ the map we are pulling back over and $\pi:E\to M$ the bundle over $M$.
Also: In trying to get a better sense of the things, I turned to wikipedia and found the line
"It is illuminating to consider the pullback of the degree 2 map from the circle to itself over the degree 3 or 4 map from the circle to itself."
I am at a complete loss for what this means, although I suspect it could prove fruitful, since $S^1$ seems the best candidate for gaining some visual intuition as to what's going on.