Rank $n$ vector bundle with $n$ pointwise linearly independent sections is trivial

Define a map $\varphi : X\times\mathbb{C}^n \to E$ by $\varphi(x, (a_1, \dots, a_n)) = a_1s_1(x) + \dots + a_ns_n(x)$. You can (and should) check that this defines an isomorphism between $E$ and the trivial rank $n$ complex vector bundle.