Dual bundle ($E^{\ast }$)

The dual bundle $E^{\ast }$ has fibre $\mathrm{Hom}(E_x,\mathbb{F})$ over $x$. Its transition matrices are the inverse transposes of those of $E$.

For a real or complex line bundle $L$, the evaluation isomorphism $L^{\ast }\otimes L\cong \mathrm{End}(L)$ identifies the identity endomorphisms with a nowhere-zero global section. Consequently $L^{\ast }\otimes L$ is trivial.