We denote the set of polynomials with real coefficients as R[X] := {ao + a₁X + a₂X² + ... + a₂X": n € N, a; ER}. Consider the map L : R[X] → R[X] defined as follows: L(co + C₁X + c₂X² + ... + G₂X") = coX + ₁X² + ₂X³ + (a) Compute L(fi) for the following polynomials: i. fi = 1 + X² ii. f₂ = 1+X+X² + X³ iii. f3 = 5X + 4X² +3X³ + 2X¹ + X5 •+n+1²₂X^²+1₂
We denote the set of polynomials with real coefficients as R[X] := {ao + a₁X + a₂X² + ... + a₂X": n € N, a; ER}. Consider the map L : R[X] → R[X] defined as follows: L(co + C₁X + c₂X² + ... + G₂X") = coX + ₁X² + ₂X³ + (a) Compute L(fi) for the following polynomials: i. fi = 1 + X² ii. f₂ = 1+X+X² + X³ iii. f3 = 5X + 4X² +3X³ + 2X¹ + X5 •+n+1²₂X^²+1₂
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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![We denote the set of polynomials with real coefficients as
R[X] := {ao + a₁X + @₂X² +…...
· + anX": n € N, a¡ € R}.
Consider the map L : R[X] → R[X] defined as follows:
L(@+cX+oX? +…+GX"):=aX+jqX? + X3 +
(a) Compute L(fi) for the following polynomials:
i. f₁ = 1 + X²
ii. f2 = 1+X+X² + X³
iii. f3 = 5X +4X² +3X³ + 2X¹ + X5
+₂X+1
(b) Solve L(f₂) = 9; for the unknown polynomial f; in the following cases:
i. 9₁ = X² + X³
ii. 92 = 5X-X5
iii. 93 = 0
(c) Prove that L is a lincar operator on R[X].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faf62fff0-2315-44d2-8670-74ab8071a288%2F10686c8f-5a61-45d4-91ac-8e8981dc1b23%2Fm715epj_processed.png&w=3840&q=75)
Transcribed Image Text:We denote the set of polynomials with real coefficients as
R[X] := {ao + a₁X + @₂X² +…...
· + anX": n € N, a¡ € R}.
Consider the map L : R[X] → R[X] defined as follows:
L(@+cX+oX? +…+GX"):=aX+jqX? + X3 +
(a) Compute L(fi) for the following polynomials:
i. f₁ = 1 + X²
ii. f2 = 1+X+X² + X³
iii. f3 = 5X +4X² +3X³ + 2X¹ + X5
+₂X+1
(b) Solve L(f₂) = 9; for the unknown polynomial f; in the following cases:
i. 9₁ = X² + X³
ii. 92 = 5X-X5
iii. 93 = 0
(c) Prove that L is a lincar operator on R[X].
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