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  "body": "\n## Definition\n\nLet $F$ be a field, let $V$ be a vector space over $F$, and let $U_1, \\dots, U_k$ be subspaces of $V$.\nThe space $V$ is the *direct sum* of $U_1, \\dots, U_k$, written $V = U_1 \\oplus \\cdots \\oplus U_k$, when each $v \\in V$ can be written as\n$$v = u_1 + \\cdots + u_k, \\qquad u_i \\in U_i,$$\nin exactly one way.\nEquivalently, $V = U_1 + \\cdots + U_k$, and $u_1 + \\cdots + u_k = 0$ with $u_i \\in U_i$ implies $u_1 = \\cdots = u_k = 0$.\n\n## Remarks\n\nFor two subspaces, $V = U \\oplus W$ if and only if $V = U + W$ and $U \\cap W = \\{0\\}$.\nFor three or more subspaces, pairwise zero intersections are not enough: three different lines through $0$ in $\\R^2$ are not a direct sum.\nIf $V = U_1 \\oplus \\cdots \\oplus U_k$ and $V$ is finite-dimensional, then $\\dim V = \\dim U_1 + \\cdots + \\dim U_k$.\n\n## Explanations\n\n### L1 human\n\nA space is the direct sum of some subspaces when we can split each vector into pieces, one piece in each subspace.\nThe split must be unique.\nThe plane is the direct sum of the $x$-axis and the $y$-axis.\nThe pieces of a vector are its components along the subspaces.\n",
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