Let P3 (R) be the space of polynomials of degree at most 3, with real coefficients. Let B = {1, x, x², x³} be the standard basis for P3 (R). Let T : P3 (R) → P3 (IR) be the linear transformation defined by df Tf = dx (E.g. T(x + 2x) = 3x2 + 2.) You may use without proof the fact that T is indeed a linear transformation. A. Compute the matrix [T]B. B. Compute the characteristic polynomial of [TB. C. What are the eigenvectors of T? (Hint: The eigenvectors of T are related to, but not quite the same as, the eigenvectors of TB.) D. Is T diagonalizable? Explain your reasoning.
Let P3 (R) be the space of polynomials of degree at most 3, with real coefficients. Let B = {1, x, x², x³} be the standard basis for P3 (R). Let T : P3 (R) → P3 (IR) be the linear transformation defined by df Tf = dx (E.g. T(x + 2x) = 3x2 + 2.) You may use without proof the fact that T is indeed a linear transformation. A. Compute the matrix [T]B. B. Compute the characteristic polynomial of [TB. C. What are the eigenvectors of T? (Hint: The eigenvectors of T are related to, but not quite the same as, the eigenvectors of TB.) D. Is T diagonalizable? Explain your reasoning.
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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![Let P3 (R) be the space of polynomials of degree at most 3, with real
coefficients. Let B = {1, x, x² , x³} be the standard basis for P3 (R).
Let T : P3 (R) → P3 (R) be the linear transformation defined by
df
Tf =
dx
(E.g. T(x³ + 2x)
3x2 + 2.) You may use without proof the fact that T is
indeed a linear transformation.
A. Compute the matrix [T]B.
B. Compute the characteristic polynomial of [T]B.
C. What are the eigenvectors of T? (Hint: The eigenvectors of T are related
to, but not quite the same as, the eigenvectors of [T]B.)
D. Is T diagonalizable? Explain your reasoning.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7297a90f-0927-4369-b782-4368eb3beb88%2Fb755cf00-f9ec-45ae-8305-ad42ee28c161%2Fg1qb63p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let P3 (R) be the space of polynomials of degree at most 3, with real
coefficients. Let B = {1, x, x² , x³} be the standard basis for P3 (R).
Let T : P3 (R) → P3 (R) be the linear transformation defined by
df
Tf =
dx
(E.g. T(x³ + 2x)
3x2 + 2.) You may use without proof the fact that T is
indeed a linear transformation.
A. Compute the matrix [T]B.
B. Compute the characteristic polynomial of [T]B.
C. What are the eigenvectors of T? (Hint: The eigenvectors of T are related
to, but not quite the same as, the eigenvectors of [T]B.)
D. Is T diagonalizable? Explain your reasoning.
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