Let $E$ be a complex vector bundle over some space $X$.
Definition 1: Let $E'$ be the complex vector bundle with the same total space as $E$, but with conjugate complex multiplication. That is, if $\cdot$ denotes complex multiplication on $E$ and $*$ denotes complex multiplication on $E'$, then $\lambda * v := \overline{\lambda} \cdot v$ for $\lambda \in \mathbb{C}$ and $v \in E$. (cf. Milnor, Stascheff: Characteristic Classes, p.167)
Definition 2: Let $(U_i)$ be an open cover of $X$ such that $E |_{U_i}$ is trivial for all $i$. Denote the transition functions of $E$ by $g_{ij} \in C^\infty(U_i \cap U_j, GL(n, \mathbb{C}))$. Let $E''$ be the vector bundle with transition functions $\overline{g_{ij}}$. (cf. exercise 3 on this exercise sheet: https://web.archive.org/web/20220525132947/https://www.dpmms.cam.ac.uk/~agk22/complex2.pdf)
Question: Are $E'$ and $E''$ isomorphic as complex vector bundles?