How to prove that the first Stiefel-Whitney class $w_1 (E)$ of a real rank $n$ vector bundle over a manifold M is equal to $w_1(\det E)$, where $\det E$ is the $n$-th wedge power of $E$?
(I want to assume the "axiomatic" definition of Stiefel-Whitney classes, as given e.g. in the book by Milnor and Stasheff).
I have just been asked an analogous question by a younger guy, but I think I could only find a proof starting from a different definition of the $w_i$'s. Perhaps I'm just missing something? Of course, feel free to close it if you find it's to homework-ish for MO standards.
Best Answer
$E \oplus det E$ is orientable (its structure group $O(n)$ is represented in $SO(n+1)$), so its $w_1$ vanishes; and $w_1(E \oplus det E) = w_1(E) + w_1(det E)$.