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  "body": "\n## Statement\n\nLet $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $W$ be a vector space over $F$.\nLet $T \\colon V \\to W$ be a [[linear map|0NRYHJPQG2]].\nThen\n$$\\dim \\im T + \\dim \\ker T = \\dim V,$$\nthat is, $\\rank T + \\operatorname{nullity} T = \\dim V$, with the [[rank|0ZZXSBTR4J]] and the [[nullity|0APCFW7639]] of $T$.\n\n## Remarks\n\nFor an $m \\times n$ matrix $A$, this says $\\rank A + \\dim \\{x : Ax = 0\\} = n$.\nA proof extends a basis of $\\ker T$ to a basis of $V$; the images of the new vectors form a basis of $\\im T$.\nThe theorem also follows from the [[first isomorphism theorem|0MTBGVZ4XJ]] $V / \\ker T \\cong \\im T$.\n\n## Explanations\n\n### L1 human\n\nEach dimension of the domain either survives in the image or is crushed into the kernel.\nSo the dimensions of image and kernel add up to the dimension of the domain.\nA map from $\\R^5$ to $\\R^3$ has a kernel of dimension at least $2$.\n",
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