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)”.
Want to see the full answer?
Check out a sample textbook solutionChapter 2 Solutions
EBK LINEAR ALGEBRA AND ITS APPLICATIONS
Additional Math Textbook Solutions
Intermediate Algebra
Algebra and Trigonometry
Elementary Algebra: Concepts and Applications (10th Edition)
College Algebra (Collegiate Math)
Elementary Linear Algebra: Applications Version
Prealgebra (7th Edition)
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,