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  "body": "\n## Definition\n\nLet $\\Omega$ be a set and let $V$ be a [[vector space|000AYPA9AJ]] of functions $\\Omega \\to \\R$ of finite [[dimension|03GSZ8RTZJ]] $N$.\nPoints $x_1, \\dots, x_N \\in \\Omega$ are *unisolvent* for $V$ if for every $(y_1, \\dots, y_N) \\in \\R^N$ there is exactly one $f \\in V$ with $f(x_i) = y_i$ for $i = 1, \\dots, N$.\n\n## Remarks\n\nEquivalently, the only $f \\in V$ with $f(x_1) = \\cdots = f(x_N) = 0$ is $f = 0$.\nEquivalently, for a basis $\\varphi_1, \\dots, \\varphi_N$ of $V$, the $N \\times N$ matrix $(\\varphi_j(x_i))$ is invertible.\nThe usual case is $V$ equal to the [[polynomials in $d$ variables|07GNX8GCYP]] of total degree at most $n$, where $N = \\binom{n + d}{d}$.\n\nExample: in one variable, the points $0, 1, 2$ are unisolvent for the polynomials of degree at most $2$.\nA polynomial of degree at most $2$ with three roots is $0$, so the equivalent condition holds.\n\nNon-example: for $V = \\set{a + bx + cy : a, b, c \\in \\R}$ on $\\R^2$, the points $(0, 0)$, $(1, 1)$, $(2, 2)$ are not unisolvent.\nThe nonzero function $x - y \\in V$ vanishes at all three points.\n",
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    "name": "Unisolvent point set",
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  "name": "Unisolvent point set",
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