I'm dealing with Exercise 7.8 of Gathmann's 2021 notes of Algebraic Geometry. The exercise is:
Show that every morphism $f\colon \mathbb{P}^n\to \mathbb{P}^m$ must be of the form $$ f\colon \mathbb{P}^n\to \mathbb{P}^m, x\mapsto (f_0(x):\cdots :f_m(x)) $$ as in Lemma 7.4, i.e. with $f_0,\cdots,f_m\in K[x_0,\cdots, x_n]$ homogeneous polynomials of the same degree such that $V_p(f_0,\cdots, f_m)=\varnothing$.
Some friends told me that this can be shown using ''line bundle'', but I never learned that and there is nothing about that on Grathmann's notes near that exercise. There should be some elementary argument that would work.
I once thought that this can be shown by restricting to the affine coordinates as in my previous post. This would work for $n=1$, thanks to that the open subsets in $\mathbb{A}^1$ are all principal, so that we have good global description for regular functions on them. However, for $n>1$, the open subsets in $\mathbb{A}^n$ would be wild and the regular functions only have local descriptions, so I doubt if it is the right way to approach this problem.
Thanks in advance for any help.
EDIT: It seems that every open subset of $\mathbb{A}^n$ is a finite union of principal ones would make the previous argument work for here. The argument seems tedious so I haven't thought about it carefully, but am I in the right way?