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  "body": "\n## Statement\n\nLet $F$ be a field, let $V$ and $U$ be vector spaces over $F$, and let $W$ be a subspace of $V$.\nLet $\\pi \\colon V \\to V / W$ be the [[quotient map|0N4F5FMQZW]].\nLet $T \\colon V \\to U$ be a [[linear map|0NRYHJPQG2]] with $W \\subseteq \\ker T$.\nThen there is exactly one linear map $\\bar T \\colon V / W \\to U$ with $\\bar T \\circ \\pi = T$.\nIt is given by $\\bar T(v + W) = T(v)$.\n\n## Remarks\n\nThe condition $W \\subseteq \\ker T$ makes $\\bar T$ well defined.\nThe kernel of $\\bar T$ is $\\ker T / W$, and its image is $\\im T$.\nFor $W = \\ker T$, the map $\\bar T$ is injective, which gives the [[first isomorphism theorem|0MTBGVZ4XJ]].\n\n## Explanations\n\n### L1 human\n\nA linear map that kills every vector of $W$ cannot tell apart two vectors that differ by an element of $W$.\nSo it makes sense on the cosets, and it becomes a map from the quotient space.\nThis property describes the quotient space completely.\n",
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