morphism between pullback bundles

Question

Let $\alpha_1=(E_1,\rho_1,M)$, $\alpha_2=(E_2,\rho_2,M)$ two smooth vector bundles over M and $f:N\longrightarrow M$ a smooth surjective map. If $f^*\alpha_1$ and $f^*\alpha_2$ (the pullback bundles) are isomorphic as vector bundles, it is true that $\alpha_1$ and $\alpha_2$ are isomorphic as vector bundles?.

Answer 1

No, take two non isomorphic vector bundles $\alpha_1,\alpha_2$ defined over the $n$-torus $T^n$ and $p:\mathbb{R}^n\rightarrow T^n$ the covering map, $p^*\alpha_1$ is isomorphic to $p^*\alpha_2$, since $\mathbb{R}^n$ is contractible, every bundle over $\mathbb{R}^n$ is trivial.