This question examines your understanding of spaces of polynomials and their bases. Please tick all correct statements. (Deduce the statements from facts you know, or find counterexamples.) You gain marks for every correct statement you tick, and you lose marks for every incorrect statement you tick. In total, the lowest number of marks you can score for this question is zero. (If you tick more incorrect than correct statements, your marks for this question will be set to zero.) O a. The polynomial p(x)=-(x+1)(x-1) is a Lagrange polynomial for certain nodes X₁
This question examines your understanding of spaces of polynomials and their bases. Please tick all correct statements. (Deduce the statements from facts you know, or find counterexamples.) You gain marks for every correct statement you tick, and you lose marks for every incorrect statement you tick. In total, the lowest number of marks you can score for this question is zero. (If you tick more incorrect than correct statements, your marks for this question will be set to zero.) O a. The polynomial p(x)=-(x+1)(x-1) is a Lagrange polynomial for certain nodes X₁
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:This question examines your understanding of spaces of polynomials and their bases.
Please tick all correct statements. (Deduce the statements from facts you know, or find counterexamples.)
You gain marks for every correct statement you tick, and you lose marks for every incorrect statement you tick. In total, the lowest
number of marks you can score for this question is zero. (If you tick more incorrect than correct statements, your marks for this question
will be set to zero.)
a. The polynomial p(x)=-(x+1)(x-1) is a Lagrange polynomial for certain nodes xº<×₁<×2.
b. The polynomials P₁(x)=x¹, P₂(x)=x² and P3(x)=x³ form a basis of the space P3 of polynomials of degree at most 3.
C. We have dim(Pn)=n, where PÅ denotes the space of all polynomials of degree at most n.
d. Given pairwise distinct nodes Xo,...‚X‚Xn+1ER, the associated Newton polynomials satisfy the identity Nn+1(x)=(x-x₁)N₁(x) for all
XER.
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