# 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.

Consider the linear projection map for the homogeneous coordinates of the central projection described in the first lecture. What are the images of the points on the line $l_\infty$ if we assign homogeneous coordinates  $\svector{b_1\\b_2\\0}$ with $b_1 \neq 0$ to the points on the line?