For an assignment I have to count the number of lines in $\mathbb{P}_k^n$ (for a finite field $k$, so for example $F_q$) using the following definition:
Let $k$ be a field and $n>0$. A line in $\mathbb{P}_k^n$ is a set of the form $\{{[x_0 : x_1 : . . . : x_n] ∈ \mathbb{P}_k^n | (x_0, x_1, . . . , x_n) ∈ V − {0}}\}$ where $V$ is a plane (i.e. a 2-dimensional linear subspace) in $k^{n+1} = A_k^{n+1}$.
As far as I understand projective space, lines in $\mathbb{P}_k^n$ correspond to points in $\mathbb{P}_k^{n-1}$. Because lines in $\mathbb{P}_k^n$ correspond to planes in $A_k^{n+1}$, which are lines in $A_k^{n}$, which correspond to points in $\mathbb{P}_k^{n-1}$. Can I instead of counting lines in $\mathbb{P}_k^n$ just count points in $\mathbb{P}_k^{n+1}$?
