4. Let the linear transformation T: R² → R³ be defined by T(x, y) = (x, 2y, x − y). Find the matrix for Trelative to the basesB = { (4, 2), (3, −1) } and B' = {(1, 0, 0), (0, 1, 0), (0, 0, 1) }, and use it to find T(1, 3).

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Answered: 4. Let the linear transformation T: R²…

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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- R? defined by T(x, y) = (3x + 6y, 4x – y) and the base (24) Consider the linear transformation T: R² B = {(1,2), (3,4)} and B' = {(3,0), (0, –2)}. (a) Find As the standard matrix for T. (b) Find transition matrices PRB' and Pg'B- (c) Find Ags the matrix for T relative to B and the standard basis. (d) Find ARR! the matrix for T relative to B and B'. %3D (e) Suppose [ =La Find [T(5)]g,. (f) Find v and TT).
Let B = {(1, 3), (-2,-2)} and B' = {(-12, 0), (-4,4)} be bases for R2, and let 3 2 40 be the matrix for T: R2 R2 relative to B. 56°F A = (a) Find the transition matrix P from B' to B. P = (b) Use the matrices P and A to find [v] and [T(v)]B, where [v]B = [-5 4]T. [V] B [T(v)]B = (c) Find P-1 and A' (the matrix for T relative to B'). p-1 = 4
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-
Find the matrix A' for T relative to the basis B'. T: R² A' = X A' = 3 11/3 R², T(x, y) = (x - y, y - 2x), B' = {(1, -2), (0, 3)} -3 48 -16 -4 Find the matrix A' for T relative to the basis B'. ↓ 1 T: R² → R², T(x, y) = (-7x + y, 7x - y), B' = {(1, −1), (-1,5)} -72 24
Find the standard matrices A and A' for T = T₂ o T₁ and T' = T₁0 T₂. T₁: R² R², T₁(x, y) = (x - 3y, 3x + 3y) T₂: R² → R², T₂(x, y) = (y, 0) A = A' = ↓ ↑ 00
Let f(u, v) = (u - v, 1, u + v) and g(x, y, z) = xyz. Find the entry a12 (i.e. the entry in the first row and second column) of the derivative matrices Df(u, v), Dg(x, y, z) and D(gof)(0, 1). (a) a 12 of Df(u, v) is (b) a12 of Dg(x, y, z) is (c) a12 of D(gof)(0, 1) is (d) Select the correct answer about D(g. f)(u, v) D(gof)(u, v) is a 1 x 2 matrix. D(gof)(u, v) is a 2 x 2 matrix. D(gof)(u, v) is a 3 x 2 matrix. D(gof)(u, v) is a 2 x 3 matrix. D(gof)(u, v) is a real-valued function of u and v. (e) Select the correct answer about D(f g)(x, y, z) OD(fog)(x, y, z) is not defined. D(fog)(x, y, z) is a real-valued function of x and y. D(fog)(x, y, z) is a 3 x 2 matrix. D(fog)(x, y, z) is a 2 x 2 matrix. D(fog)(x, y, z) is a 2 x 1 matrix.
Let f(u, v) = (u - v, 1,u + v) and g(x, y, z) = xyz. Find the entry a12(i.e. the entry in the first row and second column) of the derivative matrices Df(u, v), Dg(x, y) and D(g. f)(0, 1). (a) a12 of Df(u, v) is (b) a12 of Dg(x, y, z) is (c) a12 of D(g)(0, 1) is (d) Select the correct answer about D(g. f)(u, v) OD(g. f)(u, v) is a real-valued function of u and v. D(g. f)(u, v) is a 2 x 2 matrix. D(g. f)(u, v) is a 2 x 3 matrix. D(g f)(u, v) is a 3 x 2 matrix. 0 D(g f)(u, v) is a 1 x 2 matrix. (e) Select the correct answer about D(f g)(x, y, z) OD(fog)(x, y, z) is a real-valued function of x and y. D(fog)(x, y, z) is a 2 x 1 matrix. D(f g)(x, y, z) is not defined. 0 D(f g)(x, y, z) is a 3 x 2 matrix. D(f g)(x, y, z) is a 2 x 2 matrix.
Determine if the linear transformation T: R R3 defined as T(r, y, z) = (x+y+ 3z, 3a+ 4y + 7z, +y+2z) is invertible or not. For another transformation defined by the formula T'(x, y, z) = (x +y+ z, y-3z, z) write the formula for both ToT' and T' oT.
Find the standard matrices A and A' for T = T₂ o T₁ and T' = T₁ 0 T₂. T₁: R² → R², T₁(x, y) = (x - 5y, 2x + 3y) T₂: R² R², T₂(x, y) = (0, x) A = Juma A' = SHE
Consider the following linear transformation and basis. Find the standard matrix A for the linear transformation. A = Find the transition matrix P from B' to the standard basis B and then find its inverse. P = T: R² R², T(x, y) = (x - 5y, yx), B' = {(1, -2), (0, 3)) p-1 = A' = ↓1 ↓ 1 Find the matrix A' for T relative to the basis B'.
Let B= ((1, 3), (-2,-2)) and 8' = ((-12, 0), (-4, 4)) be bases for R2, and let be the matrix for T: R2 R2 relative to B. (a) Find the transition matrix P from B' to B. p= (b) Use the matrices P and A to find [v] and [7(v)le, where [vla [-1 31. [v] = [T(v)18- 11 p-1 11 (c) Find pand A' (the matrix for 7 relative to 8). 41 41 (d) Find [7(vil bwo ways. (7)=P(7) [vila Al- 11 11
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 Then the first B'= {u'=(2,2,1); v' = (1,0,0); w'=(3,2,0) } column of A is: O A. 2 "B 1 3 B. - - 16 On () 13 1 C. D. E. 7 -5 3 4 17 17 3 -

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4. Let the linear transformation T: R² → R³ be defined by T(x, y) = (x, 2y, x - y). Find the matrix for Trelative to the bases B = {(4,2), (3, -1)} and B' = {(1, 0, 0), (0, 1, 0), (0, 0, 1) }, and use it to find T(1, 3).