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  "body": "\n## Definition\n\nLet $V$ be an inner product space over $\\R$ or $\\C$, and let $W$ be a finite-dimensional subspace of $V$.\nBy the [[orthogonal decomposition|0SP7P0G7KQ]], each $v \\in V$ is $v = w + w'$ with unique $w \\in W$ and $w' \\in W^\\perp$.\nThe *orthogonal projection* onto $W$ is the map $P_W \\colon V \\to V$, $P_W(v) = w$.\n\n## Remarks\n\n$P_W$ is a linear [[projection|033DA1WE8Y]] with image $W$ and kernel $W^\\perp$, and it is self-adjoint.\nFor an orthonormal basis $e_1, \\dots, e_k$ of $W$, $P_W(v) = \\sum_{i=1}^{k} \\langle v, e_i \\rangle e_i$.\nFor the line spanned by $u \\ne 0$, $P(v) = \\frac{\\langle v, u \\rangle}{\\langle u, u \\rangle} u$.\n$P_W(v)$ is the point of $W$ [[closest to $v$|0MCDDSCB9Z]].\n\n## Explanations\n\n### L1 human\n\nThe orthogonal projection drops a perpendicular from a vector onto a subspace.\nThe foot of the perpendicular is the projection.\nIn $\\R^3$, projecting onto the $xy$-plane sets the $z$-coordinate to zero.\n",
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