4) Consider the following three vector spaces with corresponding basis: V₁ =R³ B₁ = {e₁,e2, e3} B₂ = {1, t, t²} V₂ =R[t]<2 1 V3 = M2x2 (R) B3 = - - ( ) ( ) ( ) ( ) 0 1 Now consider the following two linear transformations: T (₁) T(b)=a+b-ct + at² C S(a + a₁ + a₂t²) = ao-a₂ a1 ao + a₁ a2 (a) Compute [T] (b) Compute [S] (c) Compute [ST] and verify that [ST]B³ = [S]B³₂[T]B³²
4) Consider the following three vector spaces with corresponding basis: V₁ =R³ B₁ = {e₁,e2, e3} B₂ = {1, t, t²} V₂ =R[t]<2 1 V3 = M2x2 (R) B3 = - - ( ) ( ) ( ) ( ) 0 1 Now consider the following two linear transformations: T (₁) T(b)=a+b-ct + at² C S(a + a₁ + a₂t²) = ao-a₂ a1 ao + a₁ a2 (a) Compute [T] (b) Compute [S] (c) Compute [ST] and verify that [ST]B³ = [S]B³₂[T]B³²
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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simple answers with explanations for three parts would do well. Thank you very much
![4) Consider the following three vector spaces with corresponding basis:
V₁ =R³
B₁ = {e₁,e2, e3}
B₂ = {1, t, t²}
V₂ =R[t]<2
1
V3 = M2x2 (R)
B3 =
- - ( ) ( ) ( ) ( )
0
Now consider the following two linear transformations:
T (1)
T(b)=a+b-ct + at²
C
S(a + a₁ + a₂t²) =
00 - 02 01
ao + a₁
a2
(a) Compute [T]²
B₁
(b) Compute [S]2
(c) Compute [ST] and verify that [ST]B³ = [S]B³₂[T]B³²](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdb708fa5-116d-42c3-bb62-31dd00678e29%2Fb6023aaf-39ef-49b9-9bff-6b41c5745026%2Fe2ycv0e_processed.png&w=3840&q=75)
Transcribed Image Text:4) Consider the following three vector spaces with corresponding basis:
V₁ =R³
B₁ = {e₁,e2, e3}
B₂ = {1, t, t²}
V₂ =R[t]<2
1
V3 = M2x2 (R)
B3 =
- - ( ) ( ) ( ) ( )
0
Now consider the following two linear transformations:
T (1)
T(b)=a+b-ct + at²
C
S(a + a₁ + a₂t²) =
00 - 02 01
ao + a₁
a2
(a) Compute [T]²
B₁
(b) Compute [S]2
(c) Compute [ST] and verify that [ST]B³ = [S]B³₂[T]B³²
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