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  "body": "\n## Definition\n\nLet $F$ be a field, let $V$ and $W$ be vector spaces over $F$, and let $T \\colon V \\to W$ be a [[linear map|0NRYHJPQG2]].\nThe *rank* of $T$ is\n$$\\rank T = \\dim \\im T,$$\nthe [[dimension|03GSZ8RTZJ]] of the [[image|0WSFKKT6FF]] of $T$.\n\n## Remarks\n\nFor the map $x \\mapsto Ax$ of a matrix $A$, the rank is the [[rank of the matrix|0ZC9YTXZRY]].\nThe rank satisfies $\\rank T \\le \\dim V$ and $\\rank T \\le \\dim W$.\nThe [[rank–nullity theorem|0W4FSW1TVB]] relates the rank and the [[nullity|0APCFW7639]].\nThe rank of a composition satisfies $\\rank(T \\circ S) \\le \\min(\\rank T, \\rank S)$.\n\n## Explanations\n\n### L1 human\n\nThe rank of a linear map counts the dimensions of its output.\nA map from space onto a plane has rank $2$.\nThe zero map has rank $0$.\n",
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