This is Exercise 4.5.Q. of Ravi Vakil's foundations of algebraic geometry
Let $k$ be an algebraically closed field and $V$ be a vector space over $k$ and $V^*$ is its dual. I want to find a natural bijection (base-free) between the closed points (maximal ideals) in $\mathbb PV:=\operatorname {Proj}(\operatorname {Sym}^{\cdot}V^*)$ and one-dimensional linear subspaces of $V$.
I already know a proof using basis: let $V=\operatorname{Span}_k\{x_0,x_1,\dots,x_n\}^*$, then $\operatorname {Sym}^{\cdot}V^*=k[x_0,\dots,x_n]$. On the other hand, every one-dimensional subspace of $V$ corresponds to an array $[a_0,\dots, a_n]$ in "classical" $\mathbb P^n_k$ under the basis. But I can't find a basis-free proof from here