I am studying a proof that requires the use of a projective space, and my textbook describes it as the following:
$\mathbb{PF}^n$ is the set of equivalnce classes of $\mathbb{F}^{n+1}\setminus\{{0\}}$ where two elements are equivalent if on is a rescaling of the other by a factor of $\lambda\in\mathbb{F}^*$. The projective space $\mathbb{PF}^n$can be written as a disjoint union $\mathbb{F}^n\bigcup\mathbb{PF}^{n-1}$. The subset $\mathbb{PF}^{n-1}\subset\mathbb{PF}^n$is called the set of points at infinity.
I need help understanding this entire definition, especially on how I can get a geometric grasp of this type of space.
The equivalence relation that is given is that $$(a_0, \dots , a_n) \sim (b_0, \dots , b_n) \iff (a_0 , \dots, a_n) = (\lambda b_0, \dots , \lambda b_n)$$ for $\lambda \in \Bbb{F}^*$, with the equivalence classes denoted as, say, $[a_0 ,\dots , a_n]$.
$\Bbb{PF}^n$ is then the quotient set $\Bbb{F}^{n+1} \setminus \{0\}/ \sim$.
Define for $i=0, \dots, n$ the set $U_i=\{[a_0, \dots, a_n] \in \Bbb{PF}^n | a_i \not= 0\}$, then define $\phi_i:U_i \to \Bbb{F}^{n}$ by $[a_0 ,\dots, a_n] \mapsto (\frac{a_0}{a_i}, \dots, \hat{\frac{a_i}{a_i}}, \dots, \frac{a_n}{a_i}) $ with the term in the ^ omitted. This is a bijection with inverse $\mu_i : (b_0 \dots , \hat{b_i}, \dots b_n) \mapsto [b_0 , \dots, 1, \dots b_n] $. Usually we fix $i = 0$ and want to view $\Bbb{F}^{n}$ as a subset of $\Bbb{PF}^n$ by identifying $(a_1 ,\dots, a_n)\in \Bbb{F}^{n}$ with $[1, a_1, \dots , a_n] \in \Bbb{PF}^n$.
With this identification, $\Bbb{FP}^n = \mu_0(\Bbb{F}^n) \cup H_{\infty}$, where $\Bbb{FP}^{n-1}$ can be viewed as the hyperplane at infinity $H_{\infty}=\{[a_0 ,\dots , a_n] \in \Bbb{FP}^n| a_0 = 0\}$ and $\Bbb{F}^n$ as $\mu_0(\Bbb{F}^n)$.
Gometrically, the projective $n$-space can be though of as the space all the lines in $\Bbb{F}^{n+1}$ passing through the origin, here line means a linear subspace. An element in the projective space is just an $n+1$-tuple in $\Bbb{F}^{n+1}\setminus \{0\}$ upto multiplication by a non-zero constant, which is equivalent to saying it is a $1$-dimensional $\Bbb{F}$-subvectorspace (a line) of $\Bbb{F}^{n+1}$. In a projective space, two lines will always intersect, and when they're parallel, they intersect at infinity.