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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0310REJJY8]], let $m, n \\ge 1$, and let $x_1, \\dots, x_m \\in R$.\nThe *Vandermonde matrix* of $x_1, \\dots, x_m$ with $n$ columns is the $m \\times n$ [[matrix|0VD9ZSWWJG]] $V$ with entries\n$$V_{ij} = x_i^{\\,j-1} \\qquad (1 \\le i \\le m,\\ 1 \\le j \\le n),$$\nwhere $x_i^0 = 1$.\n\n## Remarks\n\nExample: for $x_1 = 1$, $x_2 = 2$, $x_3 = 3$ and $n = 3$, the matrix is $\\begin{pmatrix} 1 \u0026 1 \u0026 1 \\\\ 1 \u0026 2 \u0026 4 \\\\ 1 \u0026 3 \u0026 9 \\end{pmatrix}$; row $i$ is $(1, x_i, x_i^2)$.\nNon-example: $\\begin{pmatrix} 2 \u0026 4 \\\\ 3 \u0026 9 \\end{pmatrix}$ is not a Vandermonde matrix, because a Vandermonde matrix has first column $1, \\dots, 1$.\nIt is obtained from the Vandermonde matrix $\\begin{pmatrix} 1 \u0026 2 \\\\ 1 \u0026 3 \\end{pmatrix}$ of $2, 3$ by multiplying row $i$ by $x_i$.\nIn the square case $m = n$, the [[Vandermonde determinant|0P84B554AS]] is $\\det V = \\prod_{1 \\le i \u003c j \\le n} (x_j - x_i)$.\nFor a polynomial $p(t) = c_0 + c_1 t + \\cdots + c_{n-1} t^{n-1}$, the vector $V c$ lists the values $p(x_1), \\dots, p(x_m)$.\nSome authors use the transpose of $V$ instead.\n",
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