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  "body": "\n## Definition\n\nLet $F$ be a field, let $V$ be a vector space over $F$, and let $U$ and $W$ be [[linear subspaces|00YC1D652K]] of $V$.\nThe *sum* of $U$ and $W$ is\n$$U + W = \\{u + w : u \\in U, \\ w \\in W\\}.$$\nFor subspaces $U_1, \\dots, U_k$, the sum is $U_1 + \\cdots + U_k = \\{u_1 + \\cdots + u_k : u_i \\in U_i\\}$.\n\n## Remarks\n\nThe sum $U + W$ is a subspace, and it is the smallest subspace of $V$ that contains $U \\cup W$.\nThe sum is the [[span|0DTY8B9HMS]] of $U \\cup W$.\nWhen $U \\cap W = \\{0\\}$, the sum is a [[direct sum|0XYYT5KNTX]].\nFor finite-dimensional subspaces, [[Grassmann's formula|06HTJNCMPA]] gives $\\dim(U + W) = \\dim U + \\dim W - \\dim(U \\cap W)$.\n\n## Explanations\n\n### L1 human\n\nThe sum of two subspaces collects every vector that we can build from one piece in each subspace.\nIn space, the sum of two different lines through the origin is the plane that contains both lines.\nIt is the smallest subspace that holds both subspaces.\n",
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