Projective space (reviewed)

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 $$ \mathbb{P}^n(K) = \{(x_0:\dots:x_n)\mid x_0,\dots,x_n\in K,\ \text{not all zero}\}. $$