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Use definition of linear transformation to determine whether the transformation T :R² → R³ defined as follows is linear. -y I+ 2y 2r – 4y %3D

Use definition of linear transformation to determine whether the transformation T :R² → R³ defined as follows is linear. -y I+ 2y 2r – 4y %3D

Advanced Engineering Mathematics
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 21 — 4у 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? → R? defined as a b T с 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, d? Why or why not?
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 21 — 4у 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? → R? defined as a b T с 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, d? Why or why not?
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