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Consider the following linear map T: P3 → P3 defined as follows: T(ao + a1x+ azx² + a3x³) = ao + a1(r+ 2) + a2(x+ 2)² + a3(x+ 3)³ = (ao + 2a1 + 4a2 + 8a3) + (a1 + 4a2 + 12a3)x + (a2 + 6a3)x² + a3x³ %3D a) Determine if it is an isomorphism. b) Determine if x + 2x² is in the image of T.

Consider the following linear map T: P3 → P3 defined as follows: T(ao + a1x+ azx² + a3x³) = ao + a1(r+ 2) + a2(x+ 2)² + a3(x+ 3)³ = (ao + 2a1 + 4a2 + 8a3) + (a1 + 4a2 + 12a3)x + (a2 + 6a3)x² + a3x³ %3D a) Determine if it is an isomorphism. b) Determine if x + 2x² is in the image of T.

Advanced Engineering Mathematics
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. Consider the following linear map T: P3 → P3 defined as follows: T(ao + a1x + a2r² + a3x*) = ao + a1(r+ 2) + a2(x+ 2)² + a3(x+3)* (ao + 2a1 + 4a2 + 8a3) + (a1 + 4az + 12a3)r + (a2 + 6a3)r² + azr %3D a) Determine if it is an isomorphism. b) Determine if r + 2x? is in the image of T.
Transcribed Image Text:4. Consider the following linear map T: P3 → P3 defined as follows: T(ao + a1x + a2r² + a3x*) = ao + a1(r+ 2) + a2(x+ 2)² + a3(x+3)* (ao + 2a1 + 4a2 + 8a3) + (a1 + 4az + 12a3)r + (a2 + 6a3)r² + azr %3D a) Determine if it is an isomorphism. b) Determine if r + 2x? is in the image of T.
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