4. Consider the map T : P1 → R² given by T(a+bt) –Show that T is a linear transformation.Determine ker(T'). Justify your answer.(a)(b)(c)Determine Im(T).(d)What do the previous two parts imply about the map T?

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Answered: 4. Consider the map T : P1 → R² given…

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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II. Let V =span- and define the transformations [a b] c d [a² – 6? c? - d²1 S(A) = adj(A) d Determine which transformations are linear. i. T: M2x2(R) → M2x2(R) ii. S: M2x2(R) → M2x2(R) iii. T :V → M2x2(R)
2 [1], 0₂ = [1] · ₁ = [3], and y₂ = [¯ ] - 5 6 R² be a linear transformation that maps e₁ into y₁ and 19. Let e₁ = T: R² maps e2 into y2. Find the images of [3] -3 and [2]. x2 and let
Define the linear transformation T: R² → R² by T(x) = A. Find the images of u = 1 and 7 T(u) T(v) = = 0 A 1=[22₂ 3]. -2 under T. 5 [+]
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38. a. Consider a linear transformation T from R³ to R. What are the possible values of dim(ker T)? Explain. b. Consider a linear transformation T from R4 to R'. What are the possible values of dim(imT)? Explain.
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Q3. Let A = -1 1 2 0 3 -2 0-6 4 0 12 (a) Find the null space of A. (b) Let a linear transformation T: R → R be defined by T() = Au. Find n and m. (c) For the linear transformation T defined in part (b) find dim(Ker(T)) and dim(Im(T)).
10. Let L: R R' be a map defined as follows: (4x1 +4 x2 - 2 x3 -9 x4 5x1 +2 x2 - 3 x3-9 x4 6x) – 4 x3 - 9 x4 7x1-2x2-5x3-9 x4) X2 X3 X4, (a) Show that L is a linear transformation on R'. (b) Write out the matrix S which represents L with respect to the standard basis. (c) Find a basis for Range S and Ker S. (d) State the "rank-nullity theorem" and verify explicitly that the result obtained in part (c) matches the statement of the theorem.
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4. Consider the map T : P1 → R² given by T(a+bt) – Show that T is a lincar transformation. Determine ker(T'). Justify your answer. (a) (b) (c) Determine Im(T). (d) What do the previous two parts imply about the map T?