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  "body": "\n## Definition\n\nLet $F$ be a field, let $V$ be a vector space over $F$, and let $W$ be a subspace of $V$.\nThe *quotient map* is the map\n$$\\pi \\colon V \\to V / W, \\qquad \\pi(v) = v + W,$$\nfrom $V$ to the [[quotient space|0BR3FY5Q8M]] $V / W$.\n\n## Remarks\n\nThe map $\\pi$ is linear and surjective, and its kernel is $W$.\nEach linear map $T \\colon V \\to U$ with $W \\subseteq \\ker T$ factors through $\\pi$, see [[universal property of the quotient space|0ZERJX6PH6]].\nIn Mathlib, it is `Submodule.mkQ`.\n\n## Explanations\n\n### L1 human\n\nThe quotient map sends each vector to the class of all vectors that differ from it by an element of $W$.\nIt forgets exactly the information in $W$.\nThe vectors that it sends to zero are exactly the vectors of $W$.\n",
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