### Linear Transformation Problem**Problem Statement:**1. Show the function \( T : \mathbb{R}^2 \rightarrow \mathbb{R}^3 \) given by \( T(x, y) = (x - y, 3x + 2y, 0) \) is linear.**Further Instructions:**Find the matrix \( A \) of \( T \) so that \[ T \left( \begin{bmatrix} x \\ y \end{bmatrix} \right) = A \begin{bmatrix} x \\ y \end{bmatrix}. \]

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Answered: 1. Show the function T : R² → R³ 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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Find the standard matrices A and A' for T= T2 0 T and T' = T1 0 T2. T1: R2 → R2, T1(x, Y) = (x – 3y, 3x + 5y) T2: R2 - R2, T2(x, y) = (0, x) A = A' =
Find the standard matrices A and A' for T = T2 ∘ T1 and T' = T1 ∘ T2.T1: R2 → R2, T1(x, y) = (x − 5y, 2x + 3y)T2: R2 → R2, T2(x, y) = (0, x)
Let T: R² →→ R² where Find the matrix A such that T A = ([₁]) = ^ [*] · A T([7])=[52] and 1([2])-[1] -59 -43 T 469 238
1. Let A = MB (f) be the matrix associated with the linear map f(x,y.z) = (3x-y+4z.y+z-x.x+y+z) relative to the bases B={u=(4,18,6), v= (17.1.0), w=(17,0,-5)} and B'= {u'=(2,2,1); v' = (1,0,0); w'=(3,2,0)) Then the first column of A is: OA-16 OB. 4 17 17 B 0 (3) 13 1 OC. 3 O D./ O E. 7-
Let A = M (f) be the matrix associated with the linear map ƒ((x,y,z)) = (x-2y,y+z,0) relative to the bases B={u= (1,0,3); v = (0,2,1), w = (2,1,0) } and B'= {u'=(0,0,2); v' = (1,0,3); w'=(1,3,1) } Then the first column of A is: O O 1/2 - 1/2 - 1/2 1/2 -1/2 1/2 - 1/2 1/2 -1/2
5. Find the standard matrix A and A' for T = T2° T1 and T' = T1° T2, where T1:R2 → R3, T1(x,y) = (x,x+y,y) and T2:R3 → R², T2(x,y,z) = (0,y). Use standard basis vectors to derive your re- sults.
Consider the function T : R → R², defined by T(r, y, 2) = (x – 2y + 2,1 – 2) (a) Write the matrix for T. (b) For the vectors u = (2,3, – 1), v = (-1,3,5) € R*, verify that T(3u + 4v) = 3T(u) + 4T(v). (c) For the unit vectors i = (1,0, 0), j = (0,1,0), k = (0,0, 1), write the matrix [T] = [T(i) T(j) T(k)]. (i.e. write the matrix [T] whose columns are the vectors T(i), T(j), T(k)) %3D
Let B = {(1, 1, 0), (0, 1, 1), (1, 0, 1)} and B' = {(1, 0, 0), (0, 1, 0), (0, 0, 1)) be bases for R³, and let 1 2 1 2 A = P = 22555T 4 be the matrix for T: R3 R3 relative to B. [V] B -1 a) Find the transition matrix P from B' to B. 2 = 1 (b) Use the matrices P and A to find [v] and [T(v)]B, where [v] [-1 1 0]. [T(V)]B = 1
5. Find the standard matrix for the linear operator T: R3 → R³ that first transform by the formula T(x, y, z) = (2x – y + z, –x + 2y + z,x – z), then rotate the resulting vector anti clockwise about the y-axis through an angle 0 = 120°, and then project the resulting vector about yz-plane. Hence compute T(1, –1,1).
5. Given that G(x, y) = (x² + 1, y²) and F(u, v) = (u + v, v²), compute the Jacobian derivative matrix of F(G(x, y)) at the point (x, y) = (1, 1).
1. Let A = MB (f) be the matrix associated with the linear map f(x,y,z) = (2x − 2y+z, x − 4y+z, x-y-z) relative to the bases B={u=(5,16,1); v= (2, 1, –4); w=(7,17,2)} and B'={u'=(2,0,1); v' = (1,0,1); w'=(3,2,0)) Then the second column of A is: *) 0 O A. (0 O B. "B) °) SOC. O D. (6)
7). Given functions F(x, y) = (x²y x³ – y + 4, ) and G(u, v) = (u² + v, v) compute the Jacobian derivative matrix of function G o F at the point (1,0).

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1. Show the function T : R² → R³ given by T(x, y) = (x − y, 3x + 2y, 0) is linear. Find the matrix A of T so that T ([]) = ₁ [] A