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  "body": "\n## Summary\n\nThe Lagrange basis polynomials of distinct nodes $x_0, \\dots, x_n$ are $n + 1$ polynomials of degree $n$.\nEach one equals $1$ at its own node and $0$ at the other nodes.\nThey form a basis of the polynomials of degree at most $n$.\nIn this basis the coordinates of a polynomial are its values at the nodes.\nSo the [[Lagrange polynomial|Y7KSGE97GK]] of given data is simply the sum of the basis polynomials weighted by the data values.\n\n## Explanations\n\n### L1 human\n\nThink of each basis polynomial as a switch attached to one node: it is on, with value $1$, at its node and off, with value $0$, at all the others.\nTo get a polynomial with prescribed values, turn on each switch with the strength you want and add them up.\nFor the nodes $0$ and $1$ the two basis polynomials are $1 - x$ and $x$.\nThen the line with value $a$ at $0$ and $b$ at $1$ is $a(1 - x) + bx$.\nThe [[definition|ZCVB4ZQ0SA]] gives the general formula.\n",
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