can you please solve#28 and show all of your work please, explain each step you make and why is the reason you did it the way. For Problems 27-30, assume that T defines a linear trans-formation and use the given information to find the matrixof T.27. T : R² → Rª such that T(−1, 1) = (1, 0, −2, 2) andT(1, 2) = (–3, 1, 1, 1).28. TR4 → R² such that T(1, 0, 0, 0) = (3, −2),T(1, 1, 0, 0) = (5, 1), T(1, 1, 1, 0) = (−1,0), andT(1, 1, 1, 1) = (2, 2).29. T : R³ → R³ such that T(1, 2, 0)=(2, 1, 1),T (0, 1, 1) = (3, −1, −1) and T (0, 2, 3) = (6, −5, 4).30. T R³ R4 such that T(0,−1, 4) = (2, 5, -2, 1),T (0, 3, 3) (-1, 0, 0, 5), and T(4,4, -1)(-3, 1, 1, 3).==31. Let T: P₂ (R) → P₂(R) be the linear transformationsatisfyingT(1) = x+1, T(x)=x²-1, T(x²) = 3x +2.Determine T (ax² + bx + c), where a, b, and c arearbitrary real numbers.32. Let TV → V be a linear transformation, and sup-pose thatT(2v₁ + 3v₂) = V₁ + V₂,then T = S; that is, T(v) = S(v) for each v E V.36. Let V be a vector space with basis {V₁, V2, . . . , Vk} andsuppose TV → W is a linear transformation suchthat T (vi) = 0 for each i = 1, 2, ..., k. Prove that Tis the zero transformation; that is, 7 (v) = 0 for eachVE V.Let T₁ V W and T₂: V → W be linear transforma-tions, and let c be a scalar. We define the sum T₁+T₂ andthe scalar product cT₁ by(T₁+T₂) (v) = T₁ (v) + T₂(v)and(cT₁)(v) = cT₁ (v)for all v € V. The remaining problems in this section con-sider the properties of these mappings.37. Verify that T₁+T₂ and cT₁ are linear transformations.38. Let T₁ : R² → R² and T₂ : R² → R² be the lineartransformations with matrices3 12^-[-2] *-B ]A =в-139

Answered: T: TR4 R² such that T(1, 0, 0, 0) =…

T: TR4 R² such that T(1, 0, 0, 0) = (3,-2), T(1, 1, 0, 0) (5, 1), 7(1, 1, 1, 0) = (-1,0), and T(1, 1, 1, 1) = (2, 2). =

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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