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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 subspace of $V$.\nThen the [[quotient space|0BR3FY5Q8M]] $V / W$ is finite-dimensional, and\n$$\\dim(V / W) + \\dim W = \\dim V,$$\nwhere $\\dim$ is the [[dimension|03GSZ8RTZJ]].\n\n## Remarks\n\nThe number $\\dim(V / W) = \\dim V - \\dim W$ is the *codimension* of $W$ in $V$.\nThe formula is the [[rank–nullity theorem|0W4FSW1TVB]] for the [[quotient map|0N4F5FMQZW]] $V \\to V / W$.\nFor example, a plane $W$ in $\\R^3$ gives $\\dim(\\R^3 / W) = 1$.\n\n## Explanations\n\n### L1 human\n\nDividing by a subspace removes its dimensions.\nWhat remains has the dimension of the space minus the dimension of the subspace.\nOrdinary space divided by a plane leaves one dimension.\n",
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