Each automorphism $\mathbb{C}P^n\rightarrow \mathbb{C}P^n$ (where $\mathbb{C}P^n$ is regarded as a complex manifold) is induced by a linear map $\mathbb{C}^{n+1}\rightarrow \mathbb{C}^{n+1}$.
I know that there exist theorems that will show that any automorphism $\mathbb{C}P^n\rightarrow \mathbb{C}P^n$ as complex manifolds will actually be an automorphism of varieties, and then we can probably reduce to automorphisms of $\mathbb{C}[X_0,\ldots,X_n]$ that preserve grading, which are linear.
Is there instead a quick and dirty way to prove this? (No is also an acceptable answer. I tried to google this quickly, but didn't find anything.)