4) Suppose V is a vector space and L is an operator that maps V to V, that is for any ve V, L(v) € V. (a) What properties must L have to be a linear operator? (b) Suppose L is a linear operator, and L(v₁) = f and L(v₂) = g. What is the value of L(3v1 + 2v₂)? (c) Suppose V is the set of polynomials of degree 10 or less. Give an example of a linear operator on V. (d) For linear operator L on vector space V, suppose up € V satisfies L(up) = f. Show that any solution v € V can be written as v = Up + Uh, where Uh is the general solution to the problem: Find ve V such that L(v) = 0.
4) Suppose V is a vector space and L is an operator that maps V to V, that is for any ve V, L(v) € V. (a) What properties must L have to be a linear operator? (b) Suppose L is a linear operator, and L(v₁) = f and L(v₂) = g. What is the value of L(3v1 + 2v₂)? (c) Suppose V is the set of polynomials of degree 10 or less. Give an example of a linear operator on V. (d) For linear operator L on vector space V, suppose up € V satisfies L(up) = f. Show that any solution v € V can be written as v = Up + Uh, where Uh is the general solution to the problem: Find ve V such that L(v) = 0.
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:4) Suppose V is a vector space and L is an operator that maps V to V, that is for any
ve V, L(v) € V.
(a) What properties must L have to be a linear operator?
(b) Suppose L is a linear operator, and L(v₁) = f and L(v₂) = g. What is the value of
L(3v1 + 2v₂)?
(c) Suppose V is the set of polynomials of degree 10 or less. Give an example of a linear
operator on V.
(d) For linear operator L on vector space V, suppose up € V satisfies L(up) = f. Show that
any solution v € V can be written as v = Up + Uh, where Uh is the general solution to the
problem: Find ve V such that L(v) = 0.
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