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# Given a vector x Rn+1, the (n + 1) (n + 1) matrix V defined by vi j = _ 1 ifj = 1 x j1 i

ISBN: 9780136009290 436

## Solution for problem 28 Chapter 1.5

Linear Algebra with Applications | 8th Edition

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Problem 28

Given a vector x Rn+1, the (n + 1) (n + 1) matrix V defined by vi j = _ 1 ifj = 1 x j1 i for j = 2, . . . , n + 1 is called the Vandermonde matrix. (a) Show that if Vc = y and p(x) = c1 + c2x + +cn+1xn then p(xi ) = yi , i = 1, 2, . . . , n + 1 (b) Suppose that x1, x2, . . . , xn+1 are all distinct. Show that if c is a solution to Vx = 0, then the coefficients c1, c2, . . . , cn must all be zero and hence V must be nonsingular. 2

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