Canonical orientation of a complex vector bundle
= Canonical orientation of a complex vector bundle
An ordered complex basis of a rank-$m$ complex vector space gives the real basis $(v_1,iv_1,\ldots,v_m,iv_m)$. Every complex change-of-basis matrix has positive real determinant $|\det_{\mathbb C}A|^2$, so these bases define a canonical integral orientation and hence an $R$-orientation for every commutative coefficient ring $R$.