(9) (?) (3) Let ei = and e2 = Let T be the linear transformation which sends e, to ( ) and sends e2 to (a) Find the image of ( -3 and X2 (b) What is the standard matrix of T? (4) Let T : R² → R² be the linear transformation given by r(:)-(, (:) xi + x2 4x1 + 5x2 T X2 3 Find x such that Tx = 8

(9) (?) (3) Let ei = and e2 = Let T be the linear transformation which sends e, to ( ) and sends e2 to (a) Find the image of ( -3 and X2 (b) What is the standard matrix of T? (4) Let T : R² → R² be the linear transformation given by r(:)-(, (:) xi + x2 4x1 + 5x2 T X2 3 Find x such that Tx = 8

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. (a) Define: invertible linear transformation. (b) T: (1,2) → (6x₁ − 8x2, −5x₁ +7x2) is a linear transformation from R2 to R². Show that T is invertible and find a formula for T-¹ alike to the one for T. Also, state the matrix representations for T and T-¹ in the standard bases.
-2 Use a rectangular coordinate system to plot u = ,v = 1, and their images under the 4 -1 given transformation T(x) = | Describe geometrically what T does to each vector in R2.
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Let L be the linear transformation from R2 → R² given by L(x1, x2) = (2x1 - 3x2,4x1 + 4x2). (i) What is the matrix A of L? A = (ii) What is the determinant of L? Determinant = (iii) Is L shape preserving? (No answer given) (iv) Does L preserve orientation? (No answer given) (v) Is L injective? (No answer given) (vi) Is La clockwise rotation? (No answer given) (vii) Is L an anti-clockwise rotation? (No answer given) (viii) Compute the pre-image of the point (2, –1) under L.
Cross product is a linear transformation from R³ →R³ You can define it as X1 X2 X3 X Yı Y2 Y3 e₁ X1 Y1 x2 Y2 Y3 = = det( e2 → ez X3 Note that this is the determinant of a matrix that has vectors as some of the entries. You can look at section 3.6 in the textbook for alternate definitions. Define the linear transformation Tas T(x) = x x 2 3 a. Compute T(ei), T(₂), Te) b. Use your answer from a. to write the standard matrix of the linear transformation T c. Use your answer from b. to compute T( 4 H 3
Find the matrix of the linear transformation T from R2 - R² where T([1, 0]) = [1, –2] and T([2, 1]) = [2, 3] %3D (Note: matrix A = [T([1, 0]), T([0, 1])] 1 -2 (a) 1 (b) 1 (c) -2 (d) -2 3 7 3 Ob a C
3 1 Let TA: R2 R³ be the matrix transformation corresponding to A = 1 -2 5 ТA(u) 3 18 -1 = TA(V) X -1 4 Find TA(u) and TA(V), where u = 1 3 [2] and v = [-2]
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* (E)-E x1 +x2 Let the transformation T be given by T X2 %3D 2.x1 (a) Show that T is a linear transformation by finding a matrix that implements the mapping. (b) Show that the mapping is not onto by finding a vector in the codomain of T that is not in the range of T. Verify/show that your vector is not in the range of T.
3x1-x2 (E) 1. Let T : R? → R³ be the linear transformation defined as T 2x1 + x2 %3D -X1 - X2 (a) Find its matrix representation. (b) Find T(0). (c) Find T-(0). 1 (d) Find T-
3x₁ - x₂ [-x₁ + x₂] 1. (i) Show that 7 ([x₂]) = [³ T (ii) Find the matrix representation is a linear transformation. of T relative to the standard basis.
Could you explain how to show this in detail?