(a) Use definition of linear transformation to determine whether the transformation T: R² → R* defined as follows is linear. -y T x + 2y 2x – 4y
(a) Use definition of linear transformation to determine whether the transformation T: R² → R* defined as follows is linear. -y T x + 2y 2x – 4y
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
![1. (a) Use definition of linear transformation to determine whether the transformation
T: R? → R³ defined as follows is linear.
-y
x + 2y
2x – 4y
T
(b) Let T : R" → Rm 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 : R² → R² defined as
a
T
d
i. If T rotates the 2D space 150° counter-clockwise, and then reflects the result
across the new r-axis, find a, b, c, d.
ii. What if T reflects the 2D space across the r-axis first and then rotates 150°
counter-clockwise? Do you expect to get the same a, b, c, ď? Why or why not?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3568304a-1d60-454f-a084-556e94cc6d6c%2Fabe08c92-5260-48b2-9df1-d5715ba6a3e1%2F4v1h7s_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
2x – 4y
T
(b) Let T : R" → Rm 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 : R² → R² defined as
a
T
d
i. If T rotates the 2D space 150° counter-clockwise, and then reflects the result
across the new r-axis, find a, b, c, d.
ii. What if T reflects the 2D space across the r-axis first and then rotates 150°
counter-clockwise? Do you expect to get the same a, b, c, ď? Why or why not?
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