**Projective space** $\mathbb{P}^n(K)$ of dimension $n$ over a field $K$ is the set $(K^{n+1}\setminus\{0\})/{}\sim{}$, where
$$
(x_0,x_1,\dots,x_n) \sim (y_0,y_1,\dots,y_n) \iff x_0=\lambda y_0, \dots, x_n=\lambda y_n\quad\text{for some}\ \lambda\in K^*.
$$
The equivalence class of $(x_0,x_1,\dots,x_n)$ in $\mathbb{P}^n(K)$ is denoted by $(x_0:x_1:\dots:x_n)$, and the $x_i$ are called **homogeneous coordinates**. Thus
$$
\mathbb{P}^n(K) = \{(x_0:\dots:x_n)\mid x_0,\dots,x_n\in K,\ \text{not all zero}\}.
$$

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Edgar Costa on 2020-10-12 09:22:29

**Referred to by:**

**History:**(expand/hide all)

- 2020-10-12 09:22:29 by Edgar Costa (Reviewed)
- 2020-10-12 04:00:32 by John Cremona
- 2020-10-11 11:21:44 by John Cremona
- 2020-10-11 11:19:11 by John Cremona

**Differences**(show/hide)