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

Question

I have an $n$-dimensional vector bundle $E \to X$ and sections $s_1, \dots, s_n : X \to E$ such that for any $x \in X$, the elements $s_1(x), \dots, s_n(x)$ are linearly independent in the vector space $E_x$.

By using the theorem of fiberwise isomorphism, and I want show that $E \to X$ is isomorphic to the trivial bundle $X \times \mathbb{C}^n \to X$.

Answer 1

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.