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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 [[linear subspace|00YC1D652K]] of $V$.\nFor $v \\in V$, the *coset* of $v$ is $v + W = \\{v + w : w \\in W\\}$.\nThe *quotient space* $V / W$ is the set of all cosets $v + W$ with $v \\in V$, with the operations\n$$(u + W) + (v + W) = (u + v) + W, \\qquad a(v + W) = av + W$$\nfor $u, v \\in V$ and $a \\in F$.\n\n## Remarks\n\nThe operations are well defined, because $W$ is a subspace: the result does not depend on the choice of $u$ and $v$ in their cosets.\nTwo cosets $u + W$ and $v + W$ are equal if and only if $u - v \\in W$.\nThe zero vector of $V / W$ is the coset $0 + W = W$.\nIn $\\R^2$ with $W$ the $x$-axis, the cosets are the horizontal lines, and $V / W$ is isomorphic to $\\R$.\n\n## Explanations\n\n### L1 human\n\nIn the quotient space $V / W$, we treat two vectors as the same when they differ by a vector of $W$.\nEach element of $V / W$ is a copy of $W$ shifted by some vector.\nDividing the plane by a line gives the set of all lines parallel to it, which behaves like a single line.\n",
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