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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 all points 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 the second 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 =

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 all points 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 the second 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 =

Elementary Linear Algebra (MindTap Course List)
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 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 all points 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 the second 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 =
Transcribed Image Text: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 all points 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 the second 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 =
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