[[Projective space]]
# Projectivization

A projectivization is one way of creating a [[projective space]].
Let $V$ be a vector space over $\mathbb{K}$.
The **projectivization** $\mathrm{P}(V)$ is the [[orbit space]] $V \setminus \{ 0 \} / \mathbb{K}^\times$ of scalar multiplication by $\mathbb{K}^\times$. #m/def/geo 
The following notations are used
$$
\begin{align*}
\mathrm{P}(\mathbb{K}^{n+1}) = \mathrm{P}^n \mathbb{K} = \mathrm{PG}(n,\mathbb{K})
\end{align*}
$$
Projective points are thus equivalence classes of nonzero vectors related by scaling,
and may be denoted by [[Homogenous coördinates]].
If $V$ is a [[topological vector space]], then $\mathrm{P}(V)$ is itself a [[topological space]].
If $\mathbb{K}$ is a [[Galois field]], the projective space has a [[Galois geometry]].

#
---
#state/develop | #lang/en | #SemBr