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  "body": "\n## Definition\n\nLet $V$ be a vector space over a field $k$, and let $\\mathbb{P}(V)$ be the [[projective space|ZT9A6JK8QF]] of one-dimensional subspaces of $V$.\nFor a nonzero vector $v$, write $[v]$ for the point spanned by $v$.\nA *projective subspace* of $\\mathbb{P}(V)$ is a set of the form\n$$\\mathbb{P}(W) = \\set{[w] : w \\in W, \\ w \\ne 0},$$\nwhere $W$ is a [[linear subspace|00YC1D652K]] of $V$.\nIts *projective dimension* is $\\dim W - 1$.\n\n## Remarks\n\nEquivalently, a set $S$ of points is a projective subspace when, for all $[v], [w] \\in S$ and all scalars $a, b$ with $a v + b w \\ne 0$, the point $[a v + b w]$ lies in $S$.\nIn words, $S$ contains the whole line through any two of its distinct points.\n\nExample: in $\\mathbb{P}^2(\\R)$, the set $\\set{[x_0 : x_1 : x_2] : x_0 = 0}$ is a projective subspace of dimension $1$, a line.\nCheck: it is $\\mathbb{P}(W)$ for the plane $W = \\set{x_0 = 0}$ of $\\R^3$, which has [[dimension|03GSZ8RTZJ]] $2$.\n\nNon-example: the set $\\set{[1:0:0], [0:1:0]}$ is not a projective subspace.\nThe line through its two points also contains $[1:1:0]$, which is not in the set.\n\nThe empty set is $\\mathbb{P}(0)$, of dimension $-1$, and $\\mathbb{P}(V)$ itself is a projective subspace.\nFor finite-dimensional $V$, a projective subspace of dimension $\\dim V - 2$ is a hyperplane.\n",
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