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  "body": "\n## Summary\n\nA Vandermonde matrix has rows of the form $1, x, x^2, \\dots, x^{n-1}$, one row for each of the numbers $x_1, \\dots, x_m$.\nMultiplying it by the coefficient vector of a polynomial gives the values of the polynomial at $x_1, \\dots, x_m$.\nIts determinant, the [[Vandermonde determinant|0P84B554AS]], is nonzero exactly when the $x_i$ are distinct.\nThis is why a polynomial of degree less than $n$ is fixed by its values at $n$ distinct points.\n\n## Explanations\n\n### L1 human\n\nTake a few numbers, say $1, 2, 3$, and write each one's powers in a row: $1, x, x^2$.\nThis gives the square table $\\begin{pmatrix} 1 \u0026 1 \u0026 1 \\\\ 1 \u0026 2 \u0026 4 \\\\ 1 \u0026 3 \u0026 9 \\end{pmatrix}$.\nTo find a parabola through three given points, one solves a linear system with this matrix.\nBecause the numbers are distinct, the matrix is invertible, so there is exactly one such parabola.\n",
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