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  "body": "\n## Statement\n\nLet $K$ be a commutative ring, let $n \\ge 1$, and let $x_1, \\dots, x_n \\in K$.\nLet $V$ be the $n \\times n$ *Vandermonde matrix* with entries $V_{ij} = x_i^{\\,j-1}$.\nThen\n$$\\det V = \\prod_{1 \\le i \u003c j \\le n} (x_j - x_i),$$\nwhere $\\det$ is the [[determinant|00KDZRW4C0]].\n\n## Remarks\n\nOver a field, $\\det V \\ne 0$ if and only if the $x_i$ are pairwise distinct.\nThen, for all $y_1, \\dots, y_n$, there is exactly one polynomial $p$ of degree less than $n$ with $p(x_i) = y_i$ for all $i$.\nFor $n = 2$, the matrix is $\\begin{pmatrix} 1 \u0026 x_1 \\\\ 1 \u0026 x_2 \\end{pmatrix}$ with determinant $x_2 - x_1$.\n\n## Explanations\n\n### L1 human\n\nA Vandermonde matrix lists the powers $1, x, x^2, \\dots$ of some numbers, one row for each number.\nIts determinant is the product of all differences of the numbers.\nSo it is nonzero exactly when the numbers are distinct.\n",
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