1. True or False. (a) R³ is isomorphic to R³*2. (b) p19 is isomorphic to R4x5 (e) Span (1+x, x-², 2³) is isomorphic to R¹. 2. Define the linear transformation T: R2x2P2 by T T([a &]) = a +(b+d)x+ 2ax². How can you tell without doing any work that T is not an isomorphism? 3. Define the linear trancf D22 3 bu Tlahr a a+b
1. True or False. (a) R³ is isomorphic to R³*2. (b) p19 is isomorphic to R4x5 (e) Span (1+x, x-², 2³) is isomorphic to R¹. 2. Define the linear transformation T: R2x2P2 by T T([a &]) = a +(b+d)x+ 2ax². How can you tell without doing any work that T is not an isomorphism? 3. Define the linear trancf D22 3 bu Tlahr a a+b
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. True or False.
(a) R5 is isomorphic to R³*2.
(b) p19 is isomorphic to R4x5
(c) Span {1+z, z-2², 2³) is isomorphic to R¹.
2. Define the linear transformation T: R2x2 p² by T
T([ ² ]) = a + (b+d)x+ 2ax².
How can you tell without doing any work that T is not an isomorphism?
3. Define the linear transformation T: p2 R³ by T(a + br + cx²)
Prove that T is an isomorphism.
[at
a
a+b
a+b+c](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd9233af7-28fd-489b-a49d-2327960b93f1%2F069a0036-c494-4452-b249-2d3482ab1156%2Fnj0ycfr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. True or False.
(a) R5 is isomorphic to R³*2.
(b) p19 is isomorphic to R4x5
(c) Span {1+z, z-2², 2³) is isomorphic to R¹.
2. Define the linear transformation T: R2x2 p² by T
T([ ² ]) = a + (b+d)x+ 2ax².
How can you tell without doing any work that T is not an isomorphism?
3. Define the linear transformation T: p2 R³ by T(a + br + cx²)
Prove that T is an isomorphism.
[at
a
a+b
a+b+c
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