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Problem 8. Determine which of the following are linear maps. Prove your answers. (a) T₁ : R² → R2 defined by T(x, y) = (2x − y, x+y). (b) T₂: R² → R² defined by T(x, y) = (0, xy). (c) T3 : P(R) → R defined by T3(p) = |p'(0)|. (d) T₁ : R³ → P2 (R) defined by [T₁(a, b, c)](x) = a +bx+cx² for all x E R.

Problem 8. Determine which of the following are linear maps. Prove your answers. (a) T₁ : R² → R2 defined by T(x, y) = (2x − y, x+y). (b) T₂: R² → R² defined by T(x, y) = (0, xy). (c) T3 : P(R) → R defined by T3(p) = |p'(0)|. (d) T₁ : R³ → P2 (R) defined by [T₁(a, b, c)](x) = a +bx+cx² for all x E R.

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 8. Determine which of the following are linear maps. Prove your answers. 2 (a) T₁ R² R2 defined by T(x, y) = (2x - y, x + y). : (b) T₂ : R² → R² defined by T(x, y) = (0, xy). (c) T3: P(R) → R defined by T3(p) = [p'(0)|. (d) T₁ : R³ → P2 (R) defined by [T₁(a, b, c)](x) = a + bx + cx² for all x ER.
Transcribed Image Text:Problem 8. Determine which of the following are linear maps. Prove your answers. 2 (a) T₁ R² R2 defined by T(x, y) = (2x - y, x + y). : (b) T₂ : R² → R² defined by T(x, y) = (0, xy). (c) T3: P(R) → R defined by T3(p) = [p'(0)|. (d) T₁ : R³ → P2 (R) defined by [T₁(a, b, c)](x) = a + bx + cx² for all x ER.
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