Projective n-space, denoted as P^n, is a mathematical concept that extends the idea of n-dimensional space by adding "points at infinity." In P^n, every line through the origin in R^{n+1} corresponds to a unique point, allowing for a more comprehensive understanding of geometric properties. This space is crucial in projective geometry, where parallel lines meet at a point at infinity. In P^n, points are represented by equivalence classes of non-zero vectors in R^{n+1}. Two vectors are considered equivalent if they are scalar multiples of each other. This structure helps in studying properties of geometric figures, such as lines, planes, and higher-dimensional shapes, in a unified way.