Determine whether the following transformations are linear. If a transforma- tion is linear, provide a proof. If the transformation is not linear, provide an example (or other reason) showing how it violates a property of a lincar transformation: i. T: R³ ii. T: R³ iii. T: R³ R³, defined by T([x, y, z]) = [y-z, z- x, xz]; R³, defined by T([x, y, z]) = [2x + y, y-z+1, 2]; R³, defined by T([x, y, z]) = [x − y + ln(e³)z, y + z, x − z];
Determine whether the following transformations are linear. If a transforma- tion is linear, provide a proof. If the transformation is not linear, provide an example (or other reason) showing how it violates a property of a lincar transformation: i. T: R³ ii. T: R³ iii. T: R³ R³, defined by T([x, y, z]) = [y-z, z- x, xz]; R³, defined by T([x, y, z]) = [2x + y, y-z+1, 2]; R³, defined by T([x, y, z]) = [x − y + ln(e³)z, y + z, x − z];
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![3.
(a)
Determine whether the following transformations are linear. If a transforma-
tion is linear, provide a proof. If the transformation is not linear, provide an example(or
other reason) showing how it violates a property of a linear transformation:
(b)
i. T: R³
ii. T: R³
iii. T: R³
R³, defined by T([x, y, z]) = [y-z, z — x, xz];
R³, defined by T([x, y, z]) = [2x + y, y-z+1, 2];
R³, defined by T([r, y, z]) = [x − y + ln(e³)z, y + z, x − z];
Let T: R³ → R³ be a linear transformation such that:
1
2
1
2
(D)-[] (GD)-[²³] GD-G}
=
=
=
3
6
Find the standard matrix [T] for this transformation.
T
T
T](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe9ff5cc5-ffa0-4d62-80cc-4b5d73842eb0%2F32c9a812-3454-46dd-bab7-2e024828fbb1%2Fe0hnylf_processed.png&w=3840&q=75)
Transcribed Image Text:3.
(a)
Determine whether the following transformations are linear. If a transforma-
tion is linear, provide a proof. If the transformation is not linear, provide an example(or
other reason) showing how it violates a property of a linear transformation:
(b)
i. T: R³
ii. T: R³
iii. T: R³
R³, defined by T([x, y, z]) = [y-z, z — x, xz];
R³, defined by T([x, y, z]) = [2x + y, y-z+1, 2];
R³, defined by T([r, y, z]) = [x − y + ln(e³)z, y + z, x − z];
Let T: R³ → R³ be a linear transformation such that:
1
2
1
2
(D)-[] (GD)-[²³] GD-G}
=
=
=
3
6
Find the standard matrix [T] for this transformation.
T
T
T
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