Let $k$ be a field. Show that the $k$-rational points of $\mathbb{P}^n_k=\operatorname{Proj}[x_0,x_1,...,x_n]$ are in bijective correspondence with the points $[a_0,a_1,...,a_n], a_i\in k$ of the projective space.
According to the Wikipedia page on k-rational points I have a morphism $f:\mathbb{P}^n_k\to \operatorname{Spec}(k)$ and a $k$-rational point is a section of it $a:\operatorname{Spec}(k)\to \mathbb{P}^n_k$ such that $f\circ a=id_{\operatorname{Spec}(k)}$. Since $\operatorname{Spec}(k)=0$ I get $\mathbb{P}^n_k\overset{f}{\to}0\overset{a}{\to}\mathbb{P}^n_k$ such that $f(a)=0$. How can I show that correspondence and what is that function $f$ specifically?
Question: "How can I show that correspondence and what is that function f specifically?"
Answer: If $V:=k\{e_0,..,e_n\}$ and $V^*:=k\{x_0,..,x_n\}$ it follows there is a 1-1 correspondence between maps $x: Spec(k) \rightarrow \mathbb{P}(V^*):=Proj(Sym_k^*(V^*))$ commuting with the structure map, and rank one quotients $\phi_x: V^* \rightarrow l^* \rightarrow 0$. A rank one quotient $\phi_x$ corresponds 1-1 to a line
$$\phi_x^*: l_x \rightarrow V$$
hence there is a 1-1 correspndence between $k$-rational points $x \in \mathbb{P}(V^*)$ and lines $l_x \subseteq V$.
This is a usage of the tautological sequence. Let $\pi: \mathbb{P}(V^*) \rightarrow Spec(k)$ be the structure map. There is a surjection
$$ T1.\text{ }\pi^*(V^*) \cong V^* \otimes \mathcal{O}_{\mathbb{P}(V^*)} \rightarrow \mathcal{O}_{\mathbb{P}(V^*)}(1) \rightarrow 1.$$
By Proposition II.7.12 in Hartshorne, there is a 1-1 correspondence between maps
$$ x: Spec(k) \rightarrow \mathbb{P}(V^*)$$
with $x \circ \phi = Id$ and the set of rank one quotients
$$\phi_x: id^*(V^*) \cong V^* \rightarrow l^* \rightarrow 0.$$
A rank one quotient $l^*$ dualize to give a canonical line
$$\phi_x^*: l \rightarrow V.$$
Hence we get a 1-1 correspondence
$$\mathbb{P}(V^*)(k) \cong \{ l_x \subseteq V:\text{$l_x$ is a line in $V$} \}.$$
If you dualize the tautological sequence $T1$ and take the fiber at $x$ you get the corresponding line $l_x$:
$$ \mathcal{O}_{\mathbb{P}(V^*)}(-1)(x) \cong l_x \subseteq V.$$