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  "body": "\n## Definition\n\nLet $V$ be an inner product space over $\\R$ or $\\C$, and let $W$ be a [[subspace|00YC1D652K]] of $V$.\nThe *orthogonal complement* of $W$ is\n$$W^\\perp = \\{v \\in V : \\langle v, w \\rangle = 0 \\text{ for all } w \\in W\\}.$$\n\n## Remarks\n\n$W^\\perp$ is a subspace of $V$, and $W \\cap W^\\perp = 0$.\nIn $\\R^3$ with the dot product, the orthogonal complement of a plane through $0$ is its normal line.\nFor a real $m \\times n$ matrix $A$, the null space of $A$ is the orthogonal complement of the row space of $A$ in $\\R^n$.\nIn finite dimensions, $V = W \\oplus W^\\perp$, see [[the orthogonal decomposition|0SP7P0G7KQ]].\n\n## Explanations\n\n### L1 human\n\nThe orthogonal complement collects all vectors that are perpendicular to a subspace.\nThe complement of a line in $\\R^3$ is a plane, and the complement of a plane is a line.\nA subspace and its complement together fill the space.\n",
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