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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 [[linear subspaces|00YC1D652K]] of $V$.\nThen $U \\cap W$ is a linear subspace of $V$.\nIt is the largest subspace of $V$ that is contained in both $U$ and $W$.\nMore generally, the intersection of each nonempty family of subspaces of $V$ is a subspace of $V$.\n\n## Remarks\n\nThe union $U \\cup W$ is a subspace only when $U \\subseteq W$ or $W \\subseteq U$.\nThe smallest subspace that contains $U \\cup W$ is the [[sum|0JMTY8MF54]] $U + W$.\nIn $\\R^3$, two different planes through the origin meet in a line through the origin.\n\n## Explanations\n\n### L1 human\n\nA vector that lies in two subspaces stays in both of them when we add or scale.\nSo the common part of two subspaces is again a subspace.\nTwo planes through the origin in space cross in a line, which is a subspace.\n",
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