Recall that Pn is the vector space of polynomials p(x) of degree at most n with real coefficients. Let H: P2 P₂ be given by H(p(x)) = p(x + 1) + √" p(x)da. (i) Show that H is a linear transformation. (ii) Find the matrix A of H with respect to the basis S = {1, x, x²} of P₂. (iii) Show that S'′ = {1, 2x − 3, 3x² – 13} is a basis of P2.
Recall that Pn is the vector space of polynomials p(x) of degree at most n with real coefficients. Let H: P2 P₂ be given by H(p(x)) = p(x + 1) + √" p(x)da. (i) Show that H is a linear transformation. (ii) Find the matrix A of H with respect to the basis S = {1, x, x²} of P₂. (iii) Show that S'′ = {1, 2x − 3, 3x² – 13} is a basis of P2.
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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![Recall that Pn is the vector space of polynomials p(x) of degree at most n with real coefficients.
: P2
P₂ P2 be given by
→
Let H
1
H(p(x)) = p(x + 1) + √" p(x)da.
(i) Show that H is a linear transformation.
(ii) Find the matrix A of H with respect to the basis S = {1, x, x²} of P2.
(iii) Show that S′ = {1, 2x − 3, 3x² – 13} is a basis of P2.
(iv) Find the matrix A' of H with respect to S'.
(v) Find the transition matrix P from S to S'.
(vi) Verify that A' = P-¹AP. [Better to check that PA' = AP.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F986f6c25-9d5d-4e3f-93bc-35b073c4428b%2F6444afb7-3629-4f26-8303-915c1de74a57%2Frd1nmeb_processed.png&w=3840&q=75)
Transcribed Image Text:Recall that Pn is the vector space of polynomials p(x) of degree at most n with real coefficients.
: P2
P₂ P2 be given by
→
Let H
1
H(p(x)) = p(x + 1) + √" p(x)da.
(i) Show that H is a linear transformation.
(ii) Find the matrix A of H with respect to the basis S = {1, x, x²} of P2.
(iii) Show that S′ = {1, 2x − 3, 3x² – 13} is a basis of P2.
(iv) Find the matrix A' of H with respect to S'.
(v) Find the transition matrix P from S to S'.
(vi) Verify that A' = P-¹AP. [Better to check that PA' = AP.]
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