📂Geometry

Vector Space

Definition¹

In a Euclidean space $\mathbb{R}^{n+1}$, the set of all lines passing through the origin $\mathbf{0}$ is denoted by $\mathbb{P}^{n}$ and referred to as the projective space.

$$\mathbb{P}^{n} := \left\{ \text{all straight lines passing through in } \mathbb{R}^{n+1} \right\}$$

Explanation

An easy example of a moduli space.

$$(x_{1}, \dots, x_{n+1}) \sim (\lambda x_{1}, \dots, \lambda x_{n+1}),\quad \lambda \in \mathbb{R}\setminus \left\{ 0 \right\}$$

Since the points on a line passing through the origin are scalar multiples of each other, the equivalence relation $\sim$ can be given as above. Also, a line passing through the origin is determined by any one point on that line. Therefore, it can be understood that the quotient space $(\mathbb{R}^{n+1} - \left\{ 0 \right\}) / \sim$ is equivalent to an $n$-dimensional projective space.

$$\mathbb{P}^{n} = (\mathbb{R}^{n+1} - \left\{ 0 \right\} )/ \sim$$