Suppose we have a vector bundle $E$ on $\mathbb{P}^n$ such that $h^0(E)=\text{rank }E=r$ and $h^i(E)=0$ for $i>0$. Then is $E$ the trivial vector bundle $O^{\oplus r}$. We know that there is an injective morphism $$0\rightarrow O^{\oplus r}\rightarrow E$$
Is the cokernel zero. Or do we have non trivial bundles?
Definitely not. For instance, $E = O(1) \oplus O(-1)$ on $\mathbb{P}^1$ has rank and $h^0$ equal to 2, but is not trivial. By the way, even the map $O^{\oplus r} \to E$ is not injective in this example.