---
tag: 2RD6T5KG59
kind: concept
name: Projective subspace
msc: [51A05, 51N15]
sources:
  - url: https://en.wikipedia.org/wiki/Projective_space
---

## Summary

A projective subspace of a [[projective space|ZT9A6JK8QF]] $\mathbb{P}(V)$ is the set of points that come from a [[linear subspace|00YC1D652K]] of $V$.
Points, lines and planes of a projective space are its projective subspaces of dimension $0$, $1$ and $2$.
Equivalently, it is a set of points that contains the whole line through any two of its points.
The projective subspaces form a lattice that is isomorphic to the lattice of linear subspaces of $V$.

## Explanations

### L1 human

A projective space is made of lines through the origin of a vector space.
A plane through the origin contains many of these lines, and they form a projective line.
In the same way, every subspace through the origin gives a smaller projective space sitting inside the big one.
In the projective plane, the projective subspaces are the empty set, the points, the lines and the whole plane.