2. Let C be the vector space of continuous functions and let V be the subspace spanned by the set E={cos(x), sin(x), cos(2x), sin(2x)}. Consider the linear transformation T : VV defined by T(f(x)) = f'(x), sending a function f(x) = V to its derivative. (a) Show that Ɛ is a basis for V. Hint: Plug in x = 0, π/2, π. (b) Find the matrix [T] of the transformation T with respect to the basis ε. (c) Show that [T] is a diagonal matrix. (d) Describe the linear transformation represented by the matrix [T]. That is, give L(f(x)) for the linear transformation LVV with [L] = [T]. [1234] -5 (e) Suppose f(x) = V has coordinates [f(x)] = with respect to the basis Ɛ. Compute the 100th π derivative of f. Your answer should be an element of V.
2. Let C be the vector space of continuous functions and let V be the subspace spanned by the set E={cos(x), sin(x), cos(2x), sin(2x)}. Consider the linear transformation T : VV defined by T(f(x)) = f'(x), sending a function f(x) = V to its derivative. (a) Show that Ɛ is a basis for V. Hint: Plug in x = 0, π/2, π. (b) Find the matrix [T] of the transformation T with respect to the basis ε. (c) Show that [T] is a diagonal matrix. (d) Describe the linear transformation represented by the matrix [T]. That is, give L(f(x)) for the linear transformation LVV with [L] = [T]. [1234] -5 (e) Suppose f(x) = V has coordinates [f(x)] = with respect to the basis Ɛ. Compute the 100th π derivative of f. Your answer should be an element of V.
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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![2. Let C be the vector space of continuous functions and let V be the subspace spanned by the set
E={cos(x), sin(x), cos(2x), sin(2x)}.
Consider the linear transformation T : VV defined by T(f(x)) = f'(x), sending a function f(x) = V to
its derivative.
(a) Show that Ɛ is a basis for V.
Hint: Plug in x = 0, π/2, π.
(b) Find the matrix [T] of the transformation T with respect to the basis ε.
(c) Show that [T] is a diagonal matrix.
(d) Describe the linear transformation represented by the matrix [T]. That is, give L(f(x)) for the linear
transformation LVV with [L] = [T].
[1234]
-5
(e) Suppose f(x) = V has coordinates [f(x)] =
with respect to the basis Ɛ. Compute the 100th
π
derivative of f. Your answer should be an element of V.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb15d15d9-e398-4283-aed7-d6fa30a19f66%2F3fe3603e-7a11-49c0-a636-784e15f11cda%2F2anmtcf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Let C be the vector space of continuous functions and let V be the subspace spanned by the set
E={cos(x), sin(x), cos(2x), sin(2x)}.
Consider the linear transformation T : VV defined by T(f(x)) = f'(x), sending a function f(x) = V to
its derivative.
(a) Show that Ɛ is a basis for V.
Hint: Plug in x = 0, π/2, π.
(b) Find the matrix [T] of the transformation T with respect to the basis ε.
(c) Show that [T] is a diagonal matrix.
(d) Describe the linear transformation represented by the matrix [T]. That is, give L(f(x)) for the linear
transformation LVV with [L] = [T].
[1234]
-5
(e) Suppose f(x) = V has coordinates [f(x)] =
with respect to the basis Ɛ. Compute the 100th
π
derivative of f. Your answer should be an element of V.
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