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  "body": "\n## Summary\n\nA projective subspace of a [[projective space|ZT9A6JK8QF]] $\\mathbb{P}(V)$ is the set of points that come from a [[linear subspace|00YC1D652K]] of $V$.\nPoints, lines and planes of a projective space are its projective subspaces of dimension $0$, $1$ and $2$.\nEquivalently, it is a set of points that contains the whole line through any two of its points.\nThe projective subspaces form a lattice that is isomorphic to the lattice of linear subspaces of $V$.\n\n## Explanations\n\n### L1 human\n\nA projective space is made of lines through the origin of a vector space.\nA plane through the origin contains many of these lines, and they form a projective line.\nIn the same way, every subspace through the origin gives a smaller projective space sitting inside the big one.\nIn the projective plane, the projective subspaces are the empty set, the points, the lines and the whole plane.\n",
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