§ Line bundles, a high level view as I understand them today

§ Why are bundles invertible?

Because locally, they're locally modules. This leads us to

§ Why are modules invertible?

All modules are invertible when tensored with their dual. To simplify further, let's move to linear algbera from ring theory; consider the field R\mathbb R. Over this, we have a vector space of dimension 11, R\mathbb R. Now, if we consider RR\mathbb R \otimes \mathbb R^*, this is isomorphic to R\mathbb R since we can replace (r,f)f(r)(r, f) \mapsto f(r). This amounts to the fact that we can contract tensors. So, RRR\mathbb R \otimes \mathbb R^* \simeq \mathbb R. Generalize to bundles.

§ References