2 Let Hence find a rule for each composite transformation in the form F(x, y) = (ax + by, cx + dy). T = { [] [] []} (a) Show that T is a basis for R³. (b) Calculate the change of basis matrix Q that converts from this basis T to the standard basis S = {i, j, k}, 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 f be the linear transformation represented by
2 Let Hence find a rule for each composite transformation in the form F(x, y) = (ax + by, cx + dy). T = { [] [] []} (a) Show that T is a basis for R³. (b) Calculate the change of basis matrix Q that converts from this basis T to the standard basis S = {i, j, k}, 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 f be the linear transformation represented by
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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Question
Just solve for question 2, 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},](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff1ea36b0-99c2-47c7-ac0a-6ed7dd463e6c%2F92f4d25b-a32f-48a3-b54b-b19dc631d1be%2F4et0ia8_processed.png&w=3840&q=75)
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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