Let V be the vector space of polynomials in a over R of degree <3 and define an inner product f. g = f¹ f(x)g(x) dx for all f, g € V. You may assume that this makes V into an inner product space. Define the linear mapa : V→ V by a(f)(x) = f(x) + x f'(x) where the f' denotes differentiation. Which of the following holds with respect to this inner product?
Let V be the vector space of polynomials in a over R of degree <3 and define an inner product f. g = f¹ f(x)g(x) dx for all f, g € V. You may assume that this makes V into an inner product space. Define the linear mapa : V→ V by a(f)(x) = f(x) + x f'(x) where the f' denotes differentiation. Which of the following holds with respect to this inner product?
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:Let V be the vector space of polynomials in & over R of degree <3 and define an inner product f. g = f₁¹ f(x)g(x)dxæ for all f, g € V. You may assume that this
makes V into an inner product space. Define the linear map a: V→ V by a(f) (x) = f(x)+xƒ'(x) where the f' denotes differentiation. Which of the
following holds with respect to this inner product?
Select one:
O
a is self-adjoint
O None of the others apply
Ⓒa* = I-a, where I denotes the identity map
O
a is `anti-self adjoint' meaning a* = -a
a is orthogonal
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