Let $\mathbb{P}^n = \bigcup_{i=0}^n \mathbb{A}_i^n$, where each $\mathbb{A}_i^n$ is an affine space $\mathbb{A}_i^n = \mathbb{A}^n$, and where the $\mathbb{A}_i^n$ is dense in $\mathbb{P}^n$.
Suppose $(a_1,...,a_n)\in \mathbb{A}_i^n$ and $(b_1,...,b_n)\in \mathbb{A}_j^n$ for some $i < j$. Write down necessary and sufficient conditions for these points to be equal in $\mathbb{P}^n$.
attempt: Here $\mathbb{P}^n$ is viewed as $\mathbb{A}^n$ along with an extra point for every possible slope of a line. To see that $\mathbb{A}_i^n$ is dense in $\mathbb{P}^n$, notice that we can pick a point $a \in \mathbb{P}^n/\mathbb{A}^n$ such that we can find an open set containing $a$, but not intersecting $\mathbb{A}^n$.
Recall that $\mathbb{A}^n$ is the affine space and is defined as $\mathbb{A}^n = {\{a = (a_1,..,a_n): a_i \in K}\}$ . Then consider if $i=2$ and $j = 3$. Then using $\mathbb{A}_i^n = \mathbb{A}^n$, we have $\mathbb{P}^n = \bigcup_{i=0}^n \mathbb{A}_i^n = \mathbb{A}_1^n \cup \mathbb{A}_2^n = \mathbb{A}^n \cup \mathbb{A}^n$, and a point in $\mathbb{A}_1^n$ looks as $(a_1)$, and in $\mathbb{A}_2^n$ as $(a_1, a_2)$, thus it's intersection is $(a_1,a_2)$. And if $j = 3$ , we have $\mathbb{P}^n = \bigcup_{i=0}^n \mathbb{A}_i^n = \mathbb{A}_1^n \cup \mathbb{A}_2^n \cup \mathbb{A}_3^n = \mathbb{A}^n \cup \mathbb{A}^n \cup \mathbb{A}^n$, where a point in $\mathbb{A}_1^n$ looks as $(b_1)$ and in $\mathbb{A}_2^n$ looks as $(b_1,b_2)$ and in in $\mathbb{A}_3^n$ looks as $(b_1,b_2,b_3)$, so their intersection is $(b_1,b_2,b_3)$. Thus, for $(a_1,a_2)$ and $(b_1,b_2,b_3)$ to be equal, we would need an extra coordinate, $\mathbb{P}^n$ . I am not sure what would the necessary and sufficient conditions for these two points to be equal. Could someone please help? Than you in advance.
For simplicity, let us assume $i<j$ (if $i=j$ the the problem is trivial). The point in $\Bbb P^n$ associated to $(a_1, \dots, a_n)$ is $[a_1, \dots, a_{i-1}, 1, a_i, \dots, a_n]$ with $1$ on the $i$-th position and $[,]$ denoting projective coordinates. The point in $\Bbb P^n$ associated to $(b_1, \dots, b_n)$ is $[b_1, \dots, b_{j-1}, 1, b_j, \dots, b_n]$ with $1$ on the $j$-th position. Now remember that two $(n+1)$-uples represent the same point if and only if one is a non-zero multiple of the other. Therefore, it is necessary and sufficient to require that there exist $\lambda \ne 0$ such that
$$[a_1, \dots, a_{i-1}, 1, a_i, \dots, a_n] = [\lambda b_1, \dots, \lambda b_{j-1}, \lambda , \lambda b_j, \dots, \lambda b_n] .$$
Equating the $i$-th coordinates gives us $1 = \lambda b_i$. This imposes that $b_i \ne 0$ and $\lambda = \dfrac 1 {b_i}$, whence the above condition may be rewritten as
$$[a_1, \dots, a_{i-1}, 1, a_i, \dots, a_n] = \left[ \frac {b_1} {b_i}, \dots, \frac {b_{j-1}} {b_i}, \frac 1 {b_i}, \frac {b_j} {b_i}, \dots, \frac {b_n} {b_i} \right] ,$$
whence by a componentwise comparison it follows that the necessary and sufficient conditions are
$$\begin{cases} a_k = \dfrac {b_k} {b_i}, & k \le i-1, \\ a_k = \dfrac {b_{k+1}} {b_i}, & i \le k \le j-2, \\ a_{j-1} = \dfrac 1 {b_i}, \\ a_k = \dfrac {b_k} {b_i}, & k \ge j .\end{cases}$$