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a. Let U : P3(R) P2(R) and T: P2(R) P(R) be the linear transformations defined by U(S(x)) = 3f'(x) and T(f(x)) = 6 f S(1)dt, respectively. Let B= {1, r, a"} and y = {1, x, x , r" } be the standard ordered bases for P2(R) and P3(R), respectively Compute the representation matrix for their composition [UT|A. b. Let V be a vector space and let T': V V, U, : V →V and U, : V - V be linear transformations. Prove that T(U +U2) = TU, + TU2.

a. Let U : P3(R) P2(R) and T: P2(R) P(R) be the linear transformations defined by U(S(x)) = 3f'(x) and T(f(x)) = 6 f S(1)dt, respectively. Let B= {1, r, a"} and y = {1, x, x , r" } be the standard ordered bases for P2(R) and P3(R), respectively Compute the representation matrix for their composition [UT|A. b. Let V be a vector space and let T': V V, U, : V →V and U, : V - V be linear transformations. Prove that T(U +U2) = TU, + TU2.

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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Problem 4. a. Let U : P3(R) → P2(R) and T: P2(R) - P3(R) be the linear transformations defined by U(f(x)) = 3f'(r) and T(f(x)) 6 f" S(t)dt, respectively. Let B= {1, x, x2} and y = {1, x, x" , x" } be the standard ordered bases for P2(R) and P3(R), respectively. Compute the representation matrix for their composition UTA. b. Let V be a vector space and let T': V → V, U, : V → V and U, : V –→ V be linear transformations. Prove that T(U1 + U2) = TU, + TU2. CS Scanned with CamScanner
Transcribed Image Text:Problem 4. a. Let U : P3(R) → P2(R) and T: P2(R) - P3(R) be the linear transformations defined by U(f(x)) = 3f'(r) and T(f(x)) 6 f" S(t)dt, respectively. Let B= {1, x, x2} and y = {1, x, x" , x" } be the standard ordered bases for P2(R) and P3(R), respectively. Compute the representation matrix for their composition UTA. b. Let V be a vector space and let T': V → V, U, : V → V and U, : V –→ V be linear transformations. Prove that T(U1 + U2) = TU, + TU2. CS Scanned with CamScanner
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