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  "body": "\n## Definition\n\nLet $K$ be a [[field|04C5ZBG1WT]], let $n \\ge 0$, and let $x_0, \\dots, x_n \\in K$ be pairwise distinct.\nFor $j = 0, \\dots, n$, the *Lagrange basis polynomial* $\\ell_j$ is the polynomial in the [[polynomial ring|0SDC934R52]] $K[x]$ given by\n$$\\ell_j(x) = \\prod_{\\substack{0 \\le k \\le n \\\\ k \\ne j}} \\frac{x - x_k}{x_j - x_k}.$$\n\n## Remarks\n\nEach $\\ell_j$ has [[degree|0P4JRMSWNH]] $n$, and $\\ell_j(x_i) = 1$ if $i = j$ and $\\ell_j(x_i) = 0$ if $i \\ne j$.\nThe $\\ell_j$ form a [[basis|0KCQ9Y3QKA]] of the space of polynomials of degree at most $n$, and every such $p$ satisfies $p = \\sum_{j} p(x_j) \\, \\ell_j$.\nThey also satisfy $\\sum_j \\ell_j = 1$, the case $p = 1$.\nWith $\\ell(x) = \\prod_{k=0}^{n} (x - x_k)$, one has $\\ell_j(x) = \\dfrac{\\ell(x)}{(x - x_j) \\, \\ell'(x_j)}$.\n\nExample: for the nodes $0, 1, 2$ in $\\R$, $\\ell_1(x) = \\frac{(x - 0)(x - 2)}{(1 - 0)(1 - 2)} = -x(x - 2)$.\nCheck: $\\ell_1(0) = 0$, $\\ell_1(1) = 1$ and $\\ell_1(2) = 0$.\n\nNon-example: for the nodes $0, 0, 1$, the nodes are not distinct, and the factor $\\frac{x - x_1}{x_0 - x_1}$ in $\\ell_0$ has denominator $0$, so $\\ell_0$ is not defined.\n",
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