The question is how to prove the title, this is actually exercise 13.2.C. from Vakil's notes.
The hint is to use the structure theorem for f.g. module over PID. Since to be quasi-coherent is a local property and locally free sheaf is quasi-coherent, we know the sheaf on $A^1_k$ is given by some $k[x]$-module M, and note that $k[x]$ is a PID.
So I guess we first need to prove M is finitely generated. Is it true that being a finitely generated module is a local property? Also, how should I prove M is actually free? Is being free module also a local property? At least for module over PID? It seems to me that this is what the hint about. Could someone help? Thanks!