Let C (R) be the vector space of "smooth" functions, i.e., real-valued functions f(x) in the variable x that have infinitely many derivatives at allpoints x Є R.Let D C (R) → C∞ (R) and D² : C∞ (R) → C∞ (R) be the linear transformations defined by the first derivative D(f(x)) = f'(x) and thesecond derivative D² (f(x)) = ƒ"(x).a. Determine whether the smooth function g(x) = 8e-3 is an eigenvector of D. If so, give the associated eigenvalue. If not, enter NONE.Eigenvalue =b. Determine whether the smooth function h(x) = sin(5x) is an eigenvector of D2. If so, give the associated eigenvalue. If not, enter NONE.Eigenvalue =

(a) Check ifg(x)=8e−3x is an eigenvector of D: The first derivative of g(x) is:…

Answered: Let C (R) be the vector space of…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.4: Spanning Sets And Linear Independence

Problem 76E: Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are...

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Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are linearly dependent in the vector space C[0,1], but linearly independent in C[1,1].
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