Let E: P3 R2 be given by [p(0)] E (p(x)) = [22] (a) Find a matrix that induces the transformation E. (b) Find a polynomial p(z) where E(p(x)) = [], or explain why this isn't possible. (c) Find a polynomial q(z) where E(q(z)) = [], or explain why this isn't possible.
Let E: P3 R2 be given by [p(0)] E (p(x)) = [22] (a) Find a matrix that induces the transformation E. (b) Find a polynomial p(z) where E(p(x)) = [], or explain why this isn't possible. (c) Find a polynomial q(z) where E(q(z)) = [], or explain why this isn't possible.
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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![4. Let E: P3 R2 be given by
E (p(x)) = [P(2)]
(a) Find a matrix that induces the transformation E.
TH
(b) Find a polynomial p(z) where E(p(x)) = [], or explain why this isn't possible.
(c) Find a polynomial q(2) where E(q(z)) = [],
or explain why this isn't possible.
8P
(d) Find a nonzero polynomial r(z) where E(r(2)) = []
or explain why this isn't possible.
(e) Describe the kernel of E. What is the dimension of the kernel? Find a basis for the kernel.
(f) Describe the image of E. What is the dimension of the image? Find a basis for the image.
(g) Is the transformation E one to one? Is it onto? Is it an isomorphism? Explain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7c3b590f-1c1f-4ec6-aef5-6a600fdbbf73%2Fed4ceadb-a1cb-4c0a-8e44-dc618dc17c0d%2Fo72ayl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. Let E: P3 R2 be given by
E (p(x)) = [P(2)]
(a) Find a matrix that induces the transformation E.
TH
(b) Find a polynomial p(z) where E(p(x)) = [], or explain why this isn't possible.
(c) Find a polynomial q(2) where E(q(z)) = [],
or explain why this isn't possible.
8P
(d) Find a nonzero polynomial r(z) where E(r(2)) = []
or explain why this isn't possible.
(e) Describe the kernel of E. What is the dimension of the kernel? Find a basis for the kernel.
(f) Describe the image of E. What is the dimension of the image? Find a basis for the image.
(g) Is the transformation E one to one? Is it onto? Is it an isomorphism? Explain.
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