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(b) Let ƒ and g be linear transformations represented by the matrices A = [¯7² º] and B = [33], respectively, relative to the standard basis {i,j}. Determine the matrices that represent the composite transformations (i) gof: R² R² (ii) fog: R2 R² Hence find a rule for each composite transformation in the form F(x, y) = (ax +by, cx + dy).

(b) Let ƒ and g be linear transformations represented by the matrices A = [¯7² º] and B = [33], respectively, relative to the standard basis {i,j}. Determine the matrices that represent the composite transformations (i) gof: R² R² (ii) fog: R2 R² Hence find a rule for each composite transformation in the form F(x, y) = (ax +by, cx + dy).

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.6: Introduction To Linear Transformations
Problem 55EQ
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Just solve 1b, thanks

1 (a) For each of the following transformations f: R² → R², say whether f is linear or not. (i) f(x, y) f(x, y) = (0,y-3x) (ii) f(x, y) = (x-y-1,2x+y) (iii) f(x, y) = (x, y) If it is, write down the matrix that represents f relative to the standard basis {i,j}. (b) Let f and g be linear transformations represented by the matrices A [3] and B = [38], respectively, relative to the standard basis {i, j}. Determine the matrices that represent the composite transformations 4-3 (i) gof: R² → R² (ii) fog: R2 R² Hence find a rule for each composite transformation in the form F(x, y) = (ax + by, cx + dy). {[1], [1]• [A]}. 2 Let (a) Show that T is a basis for R³. (b) T = = = Calculate the change of basis matrix Q that converts from this basis T to the standard basis S and its inverse Q-1 that converts from S to T. (c) Hence find the coordinates of the vector v = i - 2j + 3k relative to this new basis T. 3 The equation of the unit circle C is given by x² + y² = 1. Let ƒ be the linear transformation represented by the matrix A = [22]. (a) Find an equation for the image D = f(C) of the unit circle under the action of this transformation. (b) Calculate the area enclosed by D. {i, j, k},
Transcribed Image Text:1 (a) For each of the following transformations f: R² → R², say whether f is linear or not. (i) f(x, y) f(x, y) = (0,y-3x) (ii) f(x, y) = (x-y-1,2x+y) (iii) f(x, y) = (x, y) If it is, write down the matrix that represents f relative to the standard basis {i,j}. (b) Let f and g be linear transformations represented by the matrices A [3] and B = [38], respectively, relative to the standard basis {i, j}. Determine the matrices that represent the composite transformations 4-3 (i) gof: R² → R² (ii) fog: R2 R² Hence find a rule for each composite transformation in the form F(x, y) = (ax + by, cx + dy). {[1], [1]• [A]}. 2 Let (a) Show that T is a basis for R³. (b) T = = = Calculate the change of basis matrix Q that converts from this basis T to the standard basis S and its inverse Q-1 that converts from S to T. (c) Hence find the coordinates of the vector v = i - 2j + 3k relative to this new basis T. 3 The equation of the unit circle C is given by x² + y² = 1. Let ƒ be the linear transformation represented by the matrix A = [22]. (a) Find an equation for the image D = f(C) of the unit circle under the action of this transformation. (b) Calculate the area enclosed by D. {i, j, k},
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