1. (a) Use definition of linear transformation to determine whether the transformation T : R² → R³ defined as follows is linear. -y T x + 2y 2.т — 4у
1. (a) Use definition of linear transformation to determine whether the transformation T : R² → R³ defined as follows is linear. -y T x + 2y 2.т — 4у
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) Use definition of linear transformation to determine whether the transformation
T: R? → R³ defined as follows is linear.
-y
x + 2y
2.x – 4y
T
-
(b) Let T : R" → R™ be a linear transformation.
i. Is it always true that T(0) = 0 where the zero vectors live in the appropriate
spaces? Justify.
ii. Does S(0) = 0 mean that S is linear? If not, can you give a counter-example?
(c) Consider the following matrix transformation T: R? → R² defined as
r[;)-[: :][:]
Y
C
d
i. If T rotates the 2D space 150° counter-clockwise, and then reflects the result
across the new x-axis, find a, b, c, d.
ii. What if T reflects the 2D space across the x-axis first and then rotates 150°
counter-clockwise? Do you expect to get the same a, b, c, d? Why or why not?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8f692103-7ac9-4997-b88f-f3e1801f1e0c%2F746d0b73-75ec-451d-babb-abfb8e9532db%2Fb3pjvwj_processed.png&w=3840&q=75)
Transcribed Image Text:1. (a) Use definition of linear transformation to determine whether the transformation
T: R? → R³ defined as follows is linear.
-y
x + 2y
2.x – 4y
T
-
(b) Let T : R" → R™ be a linear transformation.
i. Is it always true that T(0) = 0 where the zero vectors live in the appropriate
spaces? Justify.
ii. Does S(0) = 0 mean that S is linear? If not, can you give a counter-example?
(c) Consider the following matrix transformation T: R? → R² defined as
r[;)-[: :][:]
Y
C
d
i. If T rotates the 2D space 150° counter-clockwise, and then reflects the result
across the new x-axis, find a, b, c, d.
ii. What if T reflects the 2D space across the x-axis first and then rotates 150°
counter-clockwise? Do you expect to get the same a, b, c, d? Why or why not?
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