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  "body": "\n## Description\n\nLet $F$ be a field and let $0 \\le r \\le n$ be integers.\nThe *Grassmannian* $\\operatorname{Gr}(r, n)$ is the set of $r$-dimensional [[linear subspaces|00YC1D652K]] of $F^n$.\n\nThe Plücker embedding makes it a subset of a [[projective space|ZT9A6JK8QF]].\nLet $\\bigwedge^r F^n$ be the $r$-th exterior power, a part of the [[exterior algebra|0MDMZ965VE]], of dimension $\\binom{n}{r}$.\nA subspace $W$ with basis $w_1, \\dots, w_r$ is sent to\n$$W \\mapsto [w_1 \\wedge \\cdots \\wedge w_r] \\in \\mathbb{P}\\left(\\bigwedge\\nolimits^r F^n\\right) \\cong \\mathbb{P}^{\\binom{n}{r} - 1}.$$\nA change of basis multiplies $w_1 \\wedge \\cdots \\wedge w_r$ by the determinant of the change of basis matrix, so the point is well defined.\nThe map is injective, since $W = \\set{v \\in F^n : v \\wedge w_1 \\wedge \\cdots \\wedge w_r = 0}$.\nThe coordinates of the image are the Plücker coordinates $p_I$, the $r \\times r$ minors of the $r \\times n$ matrix with rows $w_1, \\dots, w_r$.\n\n## Remarks\n\nWhen $F$ is algebraically closed, the image is the zero set of quadratic equations, the Plücker relations, and it is a [[projective variety|A4H4CZ2V35]] of [[dimension|36X2WTHYCB]] $r(n - r)$.\n\nFor $r = 1$, $\\operatorname{Gr}(1, n) = \\mathbb{P}^{n-1}$.\nFor $r = 2$ and $n = 4$, the Grassmannian of lines in $\\mathbb{P}^3$ is the quadric hypersurface\n$$p_{12} p_{34} - p_{13} p_{24} + p_{14} p_{23} = 0$$\nin $\\mathbb{P}^5$, of dimension $4 = 2 \\cdot (4 - 2)$.\n\nOver $\\R$ or $\\C$, Grassmannians are also compact manifolds, and infinite Grassmannians are [[classifying spaces for vector bundles|RF8WD9TSE7]].\n\n## Explanations\n\n### L1 human\n\nThe Grassmannian is a space whose points are planes, or more generally subspaces of a fixed dimension, inside a bigger space.\nThe lines through the origin of $F^n$ form projective space, so the Grassmannian generalises projective space.\nTo give it coordinates, take a basis of the subspace, write it as the rows of a matrix, and record all the maximal minors of that matrix.\nChanging the basis multiplies all minors by the same number, so the minors give a well-defined point of projective space.\n",
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