Studying algebraic geometry I'm in trouble with an assumption that my teacher uses:
Let $n\geq 1$ and $\mathbb{P}^n$ be the projective n-dimensional space; suppose $H$ is the projective hyperplane in $\mathbb{P}^n$ defined by $$\sum_{i=0}^{n}a_i T_i,$$ i.e. $H=\{(u_0:\cdots:u_n) \in \mathbb{P}^n \,:\, \sum_{i=0}^{n}a_i u_i=0\}$. Then $\mathbb{P}^n\setminus H$ (endowed with the induced Zariski topology of $\mathbb{P}^n$) is an affine variety.
I know that if $H=\{(u_0:\cdots:u_n) \in \mathbb{P}^n \,:\, u_0=0\}$, then the open subset of $\mathbb{P}^n$, $\mathbb{A}_{0}^{n}=\mathbb{P}^n\setminus H$ is an affine variety isomorphic to the affine space $\mathbb{A}^n$ trough the map $i_0: \mathbb{A}_{0}^{n} \longmapsto \mathbb{A}^n$ such that $i_0((u_0:\cdots:u_n))=(u_1/u_0,...,u_n/u_0)$ for all $(u_0:\cdots:u_n) \in \mathbb{A}_{0}^{n}$.
So it suffices to show that given any hyperplane $H$ in $\mathbb{P}^n$, then the complementary $\mathbb{P}^n\setminus H$ is isomorphic to $\mathbb{A}_{0}^{n}$.
Any hint to construct this isomorphism? Is it possible otherwise to construct an isomorphism $f: \mathbb{P}^n \longmapsto \mathbb{P}^n$ that sends $H$ to $\mathbb{A}_{0}^{n}$??
Thanks in advance!