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  "body": "\n## Statement\n\nLet $k$ and $k'$ be [[fields|04C5ZBG1WT]], and let $n \\ge 2$.\nLet $V$ and $W$ be vector spaces of dimension $n + 1$ over $k$ and $k'$.\nLet $f \\colon \\mathbb{P}(V) \\to \\mathbb{P}(W)$ be a [[collineation|V7PRPN24CV]] between their [[projective spaces|ZT9A6JK8QF]].\nThen there are a field isomorphism $\\sigma \\colon k \\to k'$ and a bijection $T \\colon V \\to W$ such that:\n\n1. $T(u + v) = T u + T v$ and $T(c v) = \\sigma(c) T v$ for all $u, v \\in V$ and $c \\in k$;\n2. $f([v]) = [T v]$ for every nonzero $v \\in V$.\n\nIn particular, every collineation of $\\mathbb{P}^n(\\R)$ with $n \\ge 2$ is a [[projective transformation|P1MKTF7MWQ]].\n\n## Remarks\n\nA map $T$ with property 1 is called $\\sigma$-semilinear.\nThe proof fixes a frame of $n + 2$ points in general position and its image, and normalizes $T$ on the corresponding basis vectors.\nComparing the points of a line through two frame points with the scalars defines $\\sigma$.\nThe configuration theorems that hold in $\\mathbb{P}(V)$ show that $\\sigma$ is additive and multiplicative.\nFor $k = \\R$ the identity is the only field automorphism, so $T$ is linear and $f$ is a projective transformation.\n\nThe hypothesis $n \\ge 2$ is needed: $\\mathbb{P}^1(k)$ has only one line, so every bijection of it is a collineation.\nThe automorphism $\\sigma$ cannot be dropped in general: complex conjugation of all coordinates is a collineation of $\\mathbb{P}^2(\\C)$ that is not a projective transformation.\nThe theorem also holds for vector spaces over division rings.\nThe collineation group of $\\mathbb{P}^n(k)$, for $n \\ge 2$, is therefore the group $\\mathrm{P\\Gamma L}(n + 1, k)$ of maps induced by semilinear bijections.\n\n## Explanations\n\n### L1 human\n\nA collineation only promises to keep points on a line together.\nThe theorem says that, in a projective space of dimension at least $2$, this alone forces the map to come from linear algebra, up to a twist of the scalars.\nOver the real numbers there is no twist, so every line-preserving bijection of the real projective plane is given by an invertible $3 \\times 3$ matrix.\n\n### L2 human\n\nThe theorem identifies the symmetry group of the incidence geometry $\\mathbb{P}^n(k)$ with $\\mathrm{P\\Gamma L}(n + 1, k)$, which is $\\mathrm{PGL}(n + 1, k)$ extended by the automorphisms of $k$.\nTogether with the [[Veblen–Young theorem|NPNNXWMVCC]], it shows that the incidence structure of a projective space of dimension at least $3$ determines its coordinate division ring up to isomorphism.\nThe [[fundamental theorem of affine geometry|W3YYX63R9R]] is the affine counterpart.\nIt is the reason why, in real projective geometry, one may freely replace \"collineation\" by \"projective transformation\".\n",
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