menu
bartleby
search
close search
Hit Return to see all results

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 thatP(a4) Bi, 1i

Question

Hello, kindly assist me with the solution to Q3. I will appreciate it if you provide a very detailed solution, thanks

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
help_outline

Image Transcriptionclose

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

fullscreen
check_circleAnswer
Step 1

Given that

(X-a..(X -a_^)(X-a)...(X -a,
L (x)=
( α,-α). . α-α, )(α, - α,μ)...(α -αμ.).
i+1
m+1
-,1si<m+1
i+1
m+1
such that L, (a, ) = 5,1<i,jsm +1
Consider, B -( (x), L (x).(X))
m+1
help_outline

Image Transcriptionclose

(X-a..(X -a_^)(X-a)...(X -a, L (x)= ( α,-α). . α-α, )(α, - α,μ)...(α -αμ.). i+1 m+1 -,1si<m+1 i+1 m+1 such that L, (a, ) = 5,1<i,jsm +1 Consider, B -( (x), L (x).(X)) m+1

fullscreen
Step 2

Then the values are

L(x)-X-a)(X-a,)...(X-a
LX)-α-αα-α)...(ατα..
m+1
_
m+1
(X-a)X-a,)(x-a
L(X)=a,-a, )(a,-a).(a,-)
m+1
m+1
-a(X-a (X-a)...Xa,)
X -а,
_
m-1
m+1
m
1(X) -α-α α.α, )... (α.αμ)
m+1
-a)
m+1
m+1
m+1
m
help_outline

Image Transcriptionclose

L(x)-X-a)(X-a,)...(X-a LX)-α-αα-α)...(ατα.. m+1 _ m+1 (X-a)X-a,)(x-a L(X)=a,-a, )(a,-a).(a,-) m+1 m+1 -a(X-a (X-a)...Xa,) X -а, _ m-1 m+1 m 1(X) -α-α α.α, )... (α.αμ) m+1 -a) m+1 m+1 m+1 m

fullscreen
Step 3

Since, these values can not be written as lin...

aL(x)+a,L (X)+.... +amX)=0
m+1m+1
if and only if a = a2 = a3 =.
Pm+1
(XL (x),. (X) are linearly independent
Hence,
m+1
help_outline

Image Transcriptionclose

aL(x)+a,L (X)+.... +amX)=0 m+1m+1 if and only if a = a2 = a3 =. Pm+1 (XL (x),. (X) are linearly independent Hence, m+1

fullscreen

Want to see the full answer?

See Solution

Check out a sample Q&A here.

Want to see this answer and more?

Our solutions are written by experts, many with advanced degrees, and available 24/7

See Solution
Tagged in

Math

Advanced Math

Related Advanced Math Q&A

Find answers to questions asked by student like you

Show more Q&A add
question_answer

Q: In the following problems, decide if the groups G and G are isomorphic. If they are not, give proper...

A: (a)   We are given that G = GL(2, R), the group of 2 × 2 non-singular matrices under multiplication;...

question_answer

Q: Find n(A)   A= {1/2, -1/2, 1/3,- 1/3,......1/10, -1/10}

A: Compute n(A) as follows.Assume a set B with 20 elements.B = {1,−1, 2, −2, 3, −3,......10, −10}.

question_answer

Q: Derive the uncertainty delta A for the function A(x,y,z) = x^3 * y + ln(z), where x = 3.1 (plus or m...

A: The function is given as

question_answer

Q: Problem 3 part e

A: We are given that p∈ℕ be a natural number.

question_answer

Q: A quadratic function f is given. f(x) = −x2 + 10x (a) Express f in standard form.         f(x)=   ...

A: a)The general form of quadratic function is, a(x – h)2 + k.Rewrite the quadratic function in general...

question_answer

Q: Assume that you have 7 dimes and 5 quarters (all distinct), and you select 4 coins. In how many ways...

A: To count the number of selections  under given conditions

question_answer

Q: Number 4 please

A: Let three consecutive positive integers be x, x + 1, x + 2.Note that, a Pythagorean triplet is a set...

question_answer

Q: I need help for problem (h). Check that the set at (h) is a subspace of Rn or not.

A: (h) To decide if the given subset (expressed as linear combinations of three vectors) is a subspace ...

question_answer

Q: Let S ⊆ R, and recall M = supS is the least upper bound of S. Suppose that S is bounded, so supS ∈ R...

A: Let S ⊆ R and M = sup S which is the least upper bound of S. We are given that S is bounded, so sup ...

Sorry about that. What wasn’t helpful?