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  "body": "\n## Statement\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.\nThen the map\n$$\\bar T \\colon V / \\ker T \\to \\im T, \\qquad v + \\ker T \\mapsto T(v),$$\nis well defined, and it is a linear isomorphism.\nHere $V / \\ker T$ is the [[quotient space|0BR3FY5Q8M]] by the [[kernel|0S23E6YE83]], and $\\im T$ is the [[image|0WSFKKT6FF]] of $T$.\n\n## Remarks\n\nIn finite dimensions, it gives the [[rank–nullity theorem|0W4FSW1TVB]]: $\\dim V - \\dim \\ker T = \\dim \\im T$.\nThe same theorem holds for groups, rings and modules.\nFor a surjective map $T$, it says $V / \\ker T \\cong W$.\n\n## Explanations\n\n### L1 human\n\nA linear map identifies vectors that differ by an element of the kernel.\nAfter we glue these vectors together, the map becomes one to one.\nSo the quotient by the kernel is a copy of the image.\n",
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