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Problem 7.14. Let (æ1,...,am+1) be a sequence of pairwise distinct scalars in R and let (B1,...,Bm+1) be any sequence of scalars in R, not necessarily distinct. (1) Prove that there is a unique polynomial P of degree at most m such that P(a4) Bi, 1i

Problem 7.14. Let (æ1,...,am+1) be a sequence of pairwise distinct scalars in R and let (B1,...,Bm+1) be any sequence of scalars in R, not necessarily distinct. (1) Prove that there is a unique polynomial P of degree at most m such that P(a4) Bi, 1i

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.2: Determinants
Problem 16AEXP
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Problem 7.14. Let (æ1,...,am+1) be a sequence of pairwise distinct scalars in R and let (B1,...,Bm+1) be any sequence of scalars in R, not necessarily distinct. (1) Prove that there is a unique polynomial P of degree at most m such that P(a4) Bi, 1i <m+1. Hint. Remember Vandermonde! (2) Let Li(X) be the polynomial of degree m given by (X - a1)(X- a1-1)(X -a441) (X -am+1) (a-a1)(a- ai-1)(a-ai+1) 1 im L&(X) (a4-am+1) The polynomials L'(X) Lagrange polynomial interpolants. Prove that are known as Li(aj) 6 1 i,j <m+1 Prove that Bm+1 Lm+1(X) Р(X) — BiL1(X) + is the unique polynomial of degree at most m such that P(a4) Bi, 1< i <m+1 (3) Prove that L1(X),..., Lm+1(X) are linearly independent, and that they form a basis of all polynomials of degree at most m How is 1 (the constant polynomial 1) expressed over the basis (L1(X),..., Lm+1(X) )? Give the expression of every polynomial P(X) of degree at most m over the basis (L1(X), ...,m+1(X) (4) Prove that the dual basis (Li, ..., L1)of the basis (L1(X),..., Lm+1(X)) consists of the linear forms L given by L;(P) P(a) for every polynomial P of degree at most m; this is simply evaluation at a
Transcribed Image Text:Problem 7.14. Let (æ1,...,am+1) be a sequence of pairwise distinct scalars in R and let (B1,...,Bm+1) be any sequence of scalars in R, not necessarily distinct. (1) Prove that there is a unique polynomial P of degree at most m such that P(a4) Bi, 1i <m+1. Hint. Remember Vandermonde! (2) Let Li(X) be the polynomial of degree m given by (X - a1)(X- a1-1)(X -a441) (X -am+1) (a-a1)(a- ai-1)(a-ai+1) 1 im L&(X) (a4-am+1) The polynomials L'(X) Lagrange polynomial interpolants. Prove that are known as Li(aj) 6 1 i,j <m+1 Prove that Bm+1 Lm+1(X) Р(X) — BiL1(X) + is the unique polynomial of degree at most m such that P(a4) Bi, 1< i <m+1 (3) Prove that L1(X),..., Lm+1(X) are linearly independent, and that they form a basis of all polynomials of degree at most m How is 1 (the constant polynomial 1) expressed over the basis (L1(X),..., Lm+1(X) )? Give the expression of every polynomial P(X) of degree at most m over the basis (L1(X), ...,m+1(X) (4) Prove that the dual basis (Li, ..., L1)of the basis (L1(X),..., Lm+1(X)) consists of the linear forms L given by L;(P) P(a) for every polynomial P of degree at most m; this is simply evaluation at a
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