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6. For each of the following linear transformations T, determine whether T is invertible, and compute T-1 if it exists. (a) T: P2(R) P2(R) defined by T(f(x)) = ƒ"(x)+2f'(x) = f(x). (b) T: P2(R) → P2(R) defined by T(f(x)) = (x + 1)f'(x). (c) T: R3 → R³ defined by T(a1, a2, a3) = (a1 + 2a2+ a3, -a1 + a2 + 2a3, a1 + a3). (d) T: R3 P2(R) defined by T(a1, a2, a3) = (a1 + a2+ a3) + (a1a2+ a3)x + a₁x². (e) T: P2(R) R3 defined by T(f(x)) = (f(-1), f(0), f(1)). (f) T: M2x2(R) → R4 defined by where = T(A) (tr(A), tr(A'), tr(EA), tr(AE)), E- ( ). (3). =

6. For each of the following linear transformations T, determine whether T is invertible, and compute T-1 if it exists. (a) T: P2(R) P2(R) defined by T(f(x)) = ƒ"(x)+2f'(x) = f(x). (b) T: P2(R) → P2(R) defined by T(f(x)) = (x + 1)f'(x). (c) T: R3 → R³ defined by T(a1, a2, a3) = (a1 + 2a2+ a3, -a1 + a2 + 2a3, a1 + a3). (d) T: R3 P2(R) defined by T(a1, a2, a3) = (a1 + a2+ a3) + (a1a2+ a3)x + a₁x². (e) T: P2(R) R3 defined by T(f(x)) = (f(-1), f(0), f(1)). (f) T: M2x2(R) → R4 defined by where = T(A) (tr(A), tr(A'), tr(EA), tr(AE)), E- ( ). (3). =

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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Section 3.2:Number 6(d, f)only

 

6. For each of the following linear transformations T, determine whether T is invertible, and compute T-1 if it exists. (a) T: P2(R) (b) T: P2(R) (c) T: R3 - P2(R) defined by T(f(x)) = f" (x)+2f'(x) − f(x). P2(R) defined by T(f(x)) = (x + 1) f'(x). R³ defined by T(a1, a2, a3) = (a1 + 2a2+ a3, -a1 + a2+2a3, a1 + a3). (d) T: R3 P2(R) defined by T(a1, a2, a3) = (a1 + a2+ a3) + (a1-a2+ a3)x + a₁x². R3 defined by T(f(x)) = (f(-1), f(0), f(1)). R4 defined by (e) T: P2(R) (f) T: M2x2(R) where T(A) = (tr(A), tr(A'), tr(EA), tr(AE)), Dynrega the invertible matnis 0 E- (2). =
Transcribed Image Text:6. For each of the following linear transformations T, determine whether T is invertible, and compute T-1 if it exists. (a) T: P2(R) (b) T: P2(R) (c) T: R3 - P2(R) defined by T(f(x)) = f" (x)+2f'(x) − f(x). P2(R) defined by T(f(x)) = (x + 1) f'(x). R³ defined by T(a1, a2, a3) = (a1 + 2a2+ a3, -a1 + a2+2a3, a1 + a3). (d) T: R3 P2(R) defined by T(a1, a2, a3) = (a1 + a2+ a3) + (a1-a2+ a3)x + a₁x². R3 defined by T(f(x)) = (f(-1), f(0), f(1)). R4 defined by (e) T: P2(R) (f) T: M2x2(R) where T(A) = (tr(A), tr(A'), tr(EA), tr(AE)), Dynrega the invertible matnis 0 E- (2). =
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