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  "body": "\n## Statement\n\nLet $V$ be an inner product space over $\\R$ or $\\C$, and let $W$ be a finite-dimensional subspace of $V$.\nThen\n$$V = W \\oplus W^\\perp,$$\nwhere $W^\\perp$ is the [[orthogonal complement|0XJXTBCB2T]] of $W$ and $\\oplus$ is the [[direct sum|0XYYT5KNTX]].\n\n## Remarks\n\nFor an orthonormal basis $e_1, \\dots, e_k$ of $W$, each $v \\in V$ is $w + (v - w)$ with $w = \\sum_i \\langle v, e_i \\rangle e_i \\in W$ and $v - w \\in W^\\perp$.\nThe statement holds for each closed subspace of a Hilbert space.\nIt can fail for a subspace that is not closed in an infinite-dimensional space.\n\n## Explanations\n\n### L1 human\n\nEach vector splits in exactly one way into a part in $W$ and a part perpendicular to $W$.\nThe part in $W$ is the shadow of the vector on $W$.\nThe part perpendicular to $W$ is what remains.\n",
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