Vandermonde determinants (introduced by Alexandre-Théophile Vandermonde). Consider distinct real numbers . We define the matrix
Vandermonde showed that
the product of all differences , where exceeds j . a. Verify this formula in the case of . b. Suppose the Vandermonde formula holds for . You are asked to demonstrate it for n. Consider the function
Explain why f(t) is a polynomial of degree. Find the coefficient k of using Vandermonde's formula for . Explain why
for the scalar k you found above. Substitute to demonstrate Vandermonde's formula.
Therefore, the being the Vandermonde's determinant for , we have exactly .
For , we have a matrix
Using Vandermonde's formula, we have
By the Laplace expansion along the -th column, we see that f is a polynomial of n -th degree, the coefficient of being in fact the Vandermonde's determinant for , which is .
For , the -th and the -th column will be the same, thus the determinant will be 0 . So,
For k being the Vandermonde's determinant for , we have exactly .
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