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  "body": "\n## Statement\n\nLet $F$ be a field, let $V$ be a vector space over $F$, and let $U$ and $W$ be subspaces of $V$.\nThen the map\n$$U / (U \\cap W) \\to (U + W) / W, \\qquad u + (U \\cap W) \\mapsto u + W,$$\nis well defined, and it is a linear isomorphism.\nHere $U + W$ is the [[sum of the subspaces|0JMTY8MF54]], and both sides are [[quotient spaces|0BR3FY5Q8M]].\n\n## Remarks\n\nIn finite dimensions, it gives [[Grassmann's formula|06HTJNCMPA]]: $\\dim U - \\dim(U \\cap W) = \\dim(U + W) - \\dim W$.\nThe proof applies the [[first isomorphism theorem|0MTBGVZ4XJ]] to the map $U \\to (U + W)/W$, $u \\mapsto u + W$, which is surjective with kernel $U \\cap W$.\n\n## Explanations\n\n### L1 human\n\nAdding $U$ to $W$ adds the same new directions as $U$ has beyond its common part with $W$.\nThe theorem makes this precise with quotient spaces.\nIt is the linear algebra form of a general algebraic theorem.\n",
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