# Projective Geometry

Let us start with a general definition for an arbitrary projective space. In this lecture we will almost entirely deal with real projective spaces.

**Definition.** Let $V$ be a vector space over an arbitrary field $F$. The *projective space* $P(V)$ is the set of $1$-dimensional* *vector subspaces of $V$. If $\dim(V) = n+1$, then the *dimension* of the projective space is $n$.

- A $1$-dimensional projective space is a
*projective line.* - A $2$-dimensional projective space is a
*projective plane.*