Problem, deal with the Vandermonde determinantThe formulas in Problem are the cases n = 2...

Problem, deal with the Vandermonde determinant

The formulas in Problem are the cases n = 2 and n = 3 of the general formula

The case n = 4 is

V (x1, x2, x3, x4) = (x2 – x1)(x3 – x1)(x3 – x2) × (x4 – x1)(x4 – x2)(x4 – x3)

Prove this as follows. Given x1, x2, and , x3, define the cubic polynomial P(y) to be

Because P(x1) = P(x2) = P(x3) = 0 (why?), the roots of P(y) are x1, x2, and x3. It follows that

P(y) = k(y – x1)(y – x2)(y – x3)

where k is the coefficient of y3 in P(y). Finally, observe that expansion of the 4 × 4 determinant in (26) along its last row gives k = V(x1, x2, x3.) and that V(x1, x2, x3, x4) = P(x4).

Problem 1

Show by direct computation that V (a, b) = b − a and that