linear algebra - Endomorphisms of vector bundles

I can't help but repeat the earlier answers but in my own way:

This is really just a question about linear algebra. It is worth remembering that a vector bundle is just a parameterized family of vector spaces (I like to call differential geometry "parameterized linear algebra"). So anything you can do naturally to a vector space, you can do to a vector bundle.

So to extend the definition of a determinant of an endomorphism of vector spaces to one for an endomorphism of vector bundles, you just need a good natural definition of determinant:

An endomorphism of a vector space $V$ naturally induces an endomorphism of $Lambda^nV$, where $n$ is the dimension of $V$. Since $Lambda^nV$ is 1-dimensional, an endomorphism of $Lambda^nV$ must consist of multiplication by a fixed scalar. That scalar is the determinant of the endomorphism.

The extension to vector bundles then becomes obvious, because you just do the same thing to each fiber.