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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 subspaces of $V$.\nThe subspaces $U$ and $W$ are *complementary* when $V = U \\oplus W$, that is, when $U + W = V$ and $U \\cap W = \\{0\\}$.\nThen $W$ is a *complement* of $U$ in $V$.\n\n## Remarks\n\nA complement is not unique: in $\\R^2$, each line through $0$ other than the $x$-axis is a complement of the $x$-axis.\nIf $V$ is finite-dimensional and $W$ is a complement of $U$, then $\\dim W = \\dim V - \\dim U$.\nA complement $W$ of $U$ is isomorphic to the [[quotient space|0BR3FY5Q8M]] $V / U$.\nIn an inner product space, the [[orthogonal complement|0XJXTBCB2T]] is one particular complement.\n\n## Explanations\n\n### L1 human\n\nTwo subspaces are complementary when together they fill the space and they share only the zero vector.\nEach vector then splits in a unique way into a part in each subspace.\nA line and a plane through the origin in space are complementary when the line does not lie in the plane.\n",
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