3. Let V = P₂[t] be the space of polynomials in t of degree ≤ 2. Let 01, 02, 03 be the linear functionals on V defined by $1(f(t) = f* f(t) dt, $2(f(t)) = f'(1), ¢s(ƒ(t)) = ƒ(0) where f(t) = ao+a₁t+ a2t² € P2[t] and f' denotes the derivative of f. Find a basis {f1, f2, f3} of P2[t] that is dual to {01, 02, 03}.
3. Let V = P₂[t] be the space of polynomials in t of degree ≤ 2. Let 01, 02, 03 be the linear functionals on V defined by $1(f(t) = f* f(t) dt, $2(f(t)) = f'(1), ¢s(ƒ(t)) = ƒ(0) where f(t) = ao+a₁t+ a2t² € P2[t] and f' denotes the derivative of f. Find a basis {f1, f2, f3} of P2[t] that is dual to {01, 02, 03}.
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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![3. Let V = P₂[t] be the space of polynomials in t of degree ≤ 2. Let 01, 02, 03 be the linear
functionals on V defined by
1
$1(f(t)) = f* f(t)dt, $2(f(t)) = f'(1), øs(ƒ(t)) = ƒ(0)
0
where f(t) = ao+a₁t+a₂t² € P₂[t] and f' denotes the derivative of f. Find a basis {f1, f2, f3}
of P2[t] that is dual to {1, 02, 03}.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fae926860-316a-42c8-9528-57a4a819a4f8%2F26c29348-d994-492e-8923-fe90d77d4d22%2Fcch1dct_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. Let V = P₂[t] be the space of polynomials in t of degree ≤ 2. Let 01, 02, 03 be the linear
functionals on V defined by
1
$1(f(t)) = f* f(t)dt, $2(f(t)) = f'(1), øs(ƒ(t)) = ƒ(0)
0
where f(t) = ao+a₁t+a₂t² € P₂[t] and f' denotes the derivative of f. Find a basis {f1, f2, f3}
of P2[t] that is dual to {1, 02, 03}.
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