1. Suppose that T : R³ → R' is a linear transformation and that123Tand T23Compute T 2and TT633

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Answered: 1. Suppose that T : R' → R² is a linear…

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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1. Find the null-space of the linear transformation from R³to R² given by: ´1 1 2 A 1 3. Hint: Solve Ax = 0
3. Consider the linear transformations T: R² → R² and S: R² → R² defined by X1 X1 X2 ([2₂]) S ([21]) = [₁ ² ³37 X2 x1 - 3x2 T 2x2 3x1 1 Find T-¹ and S-1. Using your answers, write a formula for (TS)-¹.
5. Let T : R3 → R³ be a linear transformation, given by 0 1 1 1 0 1 1 1 0 T(v) = V. (a) Show that T is one-to-one. (b) Show that T is onto. 1 (c) Find the inverse linear transformation of T.
4. Let L: R5 R¹ be a linear transformation, and let S = (v₁, U2, U3) be an indexed subset of R5. Suppose that {L(v₁), L(v₂), L(v3)} is linearly independent. Show that S = {₁, 2, 3} is linearly independent.
Suppose T : R? → R² is a linear transformation such that -1 T and T -4 ()- Then T is (A) (B) (C) 3/2 (D) 11 2
1. (a) Use definition of linear transformation to determine whether the transformation T: R? → R3 defined as follows is linear. -y x + 2y 21 — 4у T = (b) Let T : R" → R" be a linear transformation. i. Is it always true that T(0) = 0 where the zero vectors live in the appropriate spaces? Justify. ii. Does S(0) = 0 mean that S is linear? If not, can you give a counter-example? (c) Consider the following matrix transformation T : R? → R? defined as a b T с d i. If T rotates the 2D space 150° counter-clockwise, and then reflects the result across the new r-axis, find a, b, c, d. ii. What if T reflects the 2D space across the r-axis first and then rotates 150° counter-clockwise? Do you expect to get the same a, b, c, d? Why or why not?
If T:V → W is a linear transformation. If T(v) = T(w) then T(v – w) = Ow. O True O False
21
If T: P₁ → P₁ is a linear transformation such that T(1 + 5x) = 2 + 2x and T(4+19x) = 1 + 2x, then T(4- 4x) =
2. Suppose T: R¹4 → R¹4 is a linear transformation and the rank of T is 10. (a) Determine whether T is injective. (b) Determine whether T is surjective. (c) Determine whether T is invertible.

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1. Suppose that T : R' → R² is a linear transformation and that T 2 and T 3 Compute T 2 and T 6 3 3