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Let x1, …, xn, be fixed numbers. The matrix below, called a Vandermonde matrix, occurs in applications such as signal processing, error-correcting codes, and polynomial interpolation.
V =
Given y = (y1,…, yn) in ℝn, suppose c = (c0, …, cn−1) in ℝn satisfies Vc = y, and define the polynomial p(t) = c0 +c1t + c2t2 + … + cn−1tn−1.
- a. Show that p(x1) = y1,…, p(xn) = yn. We call p(t) an interpolating polynomial for the points (x1, y1), …, (xn, yn) because the graph of p(t) passes through the points.
- b. Suppose x1,…, xn are distinct numbers. Show that the columns of V are linearly independent [Hint: How many zeros can a polynomial of degree n − 1 have?]
- c. Prove:”If x1,…, xn are distinct numbers, and y1,…,yn are arbitrary numbers, then there is an interpolating polynomial of degree ≤ n − 1 for (x1, y1),…, (xn, yn)”.
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Linear Algebra and Its Applications (5th Edition)
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