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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 finite-dimensional [[linear subspaces|00YC1D652K]] of $V$.\nThen\n$$\\dim(U + W) + \\dim(U \\cap W) = \\dim U + \\dim W,$$\nwhere $U + W$ is the [[sum of the subspaces|0JMTY8MF54]] and $\\dim$ is the [[dimension|03GSZ8RTZJ]].\n\n## Remarks\n\nThe proof extends a basis of $U \\cap W$ to a basis of $U$ and to a basis of $W$.\nTogether, these vectors form a basis of $U + W$.\nAnother proof applies the [[rank–nullity theorem|0W4FSW1TVB]] to the map $U \\times W \\to V$, $(u, w) \\mapsto u - w$.\nTwo planes through the origin in $\\R^3$ with $U \\ne W$ have $\\dim(U + W) = 3$, so $\\dim(U \\cap W) = 1$.\n\n## Explanations\n\n### L1 human\n\nThe formula counts dimensions like a count of elements in a union of two sets.\nWe add the two dimensions, but then the common part is counted twice, so we subtract it once.\nTwo different planes through the origin in space always share at least a line.\n",
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