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  "body": "\n## Definition\n\nLet $F$ be a field and let $V$ be a vector space over $F$.\nA *projection* on $V$ is an [[endomorphism|0J7H0YBFDJ]] $P \\colon V \\to V$ with $P \\circ P = P$.\n\n## Remarks\n\nIf $P$ is a projection, then $V = \\im P \\oplus \\ker P$, and $P$ acts as the identity on $\\im P$.\nConversely, for each [[direct sum|0XYYT5KNTX]] $V = U \\oplus W$, the map $u + w \\mapsto u$ is a projection with image $U$ and kernel $W$.\nIf $P$ is a projection, then $I - P$ is the projection with image $\\ker P$ and kernel $\\im P$.\nIn an inner product space, an [[orthogonal projection|0B8KZRT28M]] is a projection whose kernel is orthogonal to its image.\n\n## Explanations\n\n### L1 human\n\nA projection maps space onto a subspace and leaves the vectors of that subspace where they are.\nProjecting a second time changes nothing.\nThe shadow of an object on the floor, in light from a fixed direction, is a picture of a projection.\n",
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