1. (a) Use definition of linear transformation to determine whether the transformationT: R? → R³ defined as follows is linear.-yx + 2y2x – 4yT(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 appropriatespaces? 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 asaTdi. If T rotates the 2D space 150° counter-clockwise, and then reflects the resultacross 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?

(Solving part (a) as per our guidelines part (b) and (c) are independent of part (a) so can not be…

Answered: (a) Use definition of linear…

(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

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.5: The Kernel And Range Of A Linear Transformation

Problem 3EQ

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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?

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