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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 *nullity* of $T$ is\n$$\\operatorname{nullity} T = \\dim \\ker T,$$\nthe [[dimension|03GSZ8RTZJ]] of the [[kernel|0S23E6YE83]] of $T$.\nThe nullity of a matrix $A$ is the nullity of the map $x \\mapsto Ax$, the dimension of the [[null space|0PHDYNR24D]] of $A$.\n\n## Remarks\n\nThe nullity is $0$ if and only if $T$ is injective.\nFor an $m \\times n$ matrix, the nullity is the number of free variables of $Ax = 0$.\nBy the [[rank–nullity theorem|0W4FSW1TVB]], $\\rank T + \\operatorname{nullity} T = \\dim V$ when $V$ is finite-dimensional.\n\n## Explanations\n\n### L1 human\n\nThe nullity counts the dimensions that the map crushes to zero.\nThe projection of space onto a plane crushes one direction, so its nullity is $1$.\nRank counts what survives, and nullity counts what is lost.\n",
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