6. Determine whether the following linear transformations areļ one-to-one. Are they onto?Justify your answer.(a) S1 : M2(R) → R defined by S1(A) = a11 + a22.(b) S2 : M2(R) → M2(R) defined by S2(A) = A + A" .(c) S3 : Po(R) –→ P1 (R) defined by S3(p(x))p(t)dt.

Answered: Image /qna-images/answer/29f64a2e-aeea-40bd-804e-5a1553b0b84d.jpg

Answered: 6. Determine whether the following…

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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Which of the following transformations are linear? Select all of the linear transformations. There may be more than one correct answer. Be sure you can justify your answers. □A. T(f(t)) = (f(t))³ + 6(f(t))² + 5f (t) from C to C B. T(x0, X₁, X2, ...) = (1, X0, X₁, X₂, ...) from the space of infinite sequences into itself C. T(f(t)) = f(9) from P7 to R D. T(f(t)) = f(-t) from P5 to P5 □E. T(f(t)) = f'(t) + 6f (t) from C* to C* = [₁₁ □ F. T(f(t)) = f(t)dt from P3 to R
IV. Explain why the following statements are false. 1. The function f: M₁ (F) → F defined by f(A) = det(A) is a linear transformation 2. If A is equivalent to B then det (B) = det(A).
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Let f: R²R be defined by f((x, y)) = 8y - 4x + 1. Is ƒ a linear transformation? a. f((x1, y₁) + (x2, Y2)) (Enter x₁ as x1, etc.) f((x₁, y₁)) + f((x2, y₂)) = + Does f((x1, y₁) + (x2, 2)) = f((x₁, y₁)) + f((x2, ₂)) for all (x1, y₁), (x2, Y₂) € R²? choose b. f(c(x, y)) = c(f((x, y))) = Does f(c(x, y)) = c(f((x, y))) for all CER and all (x, y) = R²? choose c. Is f a linear transformation? choose
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Let T1 : P2 → R² and T2 : R? → R2x2 be linear transformations defined as follows. a +b T1(ax? + bx + c) = b — с 4x1 2x1 T2 %3D -8x2 -2x2 x2 Ex: 42 (Т, о Ti) (-За? + 4г + 3)
Which of the following transformations are linear? Select all of the linear transformations. There may be more than one correct answer. Be su you can justify your answers. A. T(A) det(A) from R³x³ to R B. T(A) 1A from R4x3 to R4x3 OC. T(A) = AT from R5x3 to R³x5 1 D. T(A) = A 1 from IR to R2x2 6 8 JE. T(A)= A2 from R4x4 to R4x4 6 5 R.T(A)=[74 -97 A from R2x5 to R2x5
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6. Determine whether the following linear transformations are one-to-one. Are they onto? Justify your answer. (a) S1 : M2(R) → R defined by S1(A) = a11 + a22. (b) S2 : M2(R) → M2(R) defined by S2(A) = A+ A". (c) S3 : Po(R) → P; (R) defined by S3(p(x)) = p(t)dt.