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  "body": "\n## Statement\n\nLet $V$ be a finite-dimensional inner product space over $\\R$ or $\\C$, and let $W$ be a subspace of $V$.\nThen\n$$\\dim W + \\dim W^\\perp = \\dim V,$$\nwhere $W^\\perp$ is the [[orthogonal complement|0XJXTBCB2T]] of $W$.\n\n## Remarks\n\nIt follows from $V = W \\oplus W^\\perp$.\nFor example, in $\\R^3$, a line has a plane as orthogonal complement: $1 + 2 = 3$.\nFor a real matrix, it gives $\\rank A + \\dim \\ker A = n$ once more, because $\\ker A$ is the orthogonal complement of the row space.\n\n## Explanations\n\n### L1 human\n\nA subspace and its orthogonal complement share the dimensions of the space.\nIf $W$ uses $k$ directions, then $W^\\perp$ uses the remaining $n - k$ directions.\n",
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