6) Consider the following vector spaces with corresponding basis: V₁ =R³ Bv₁ = {e₁,e2, е3} Bw₁ = {1, t, t²} W₁ =R[t]<2 V₂ = M2x2 (R) (a) (b) W2 = M2x2 (R) Now consider the following two linear transformations T₁: V₁ W₁ and T₂: V2 → W₂ given by Bv₂ T₂ a1 az = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Bw₂ = { T√(-) -a |) = a +b-ct+at² C :)) - (202 Write the corresponding basis for V₁ V₂ and W₁ W₂ Recall we get the linear map a2 a4 a4 3a3 T₁ T₂: V₁0 V₂ W₁ ⓇW₂ (T₁ ® T₂) (V1 ® V₂) = T1 (V₁) ® T₂(V₂) Compute the matrix of this map with respect to the two basis you found in part a Remark: This is an example of what is called the Kronecker-Product of ma- trices. It is an operation that takes an m × n and a k × 1 matrix and produces an mk × nl matrix. This matrix is precisely the matrix of the tensor product of linear maps we defined
6) Consider the following vector spaces with corresponding basis: V₁ =R³ Bv₁ = {e₁,e2, е3} Bw₁ = {1, t, t²} W₁ =R[t]<2 V₂ = M2x2 (R) (a) (b) W2 = M2x2 (R) Now consider the following two linear transformations T₁: V₁ W₁ and T₂: V2 → W₂ given by Bv₂ T₂ a1 az = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Bw₂ = { T√(-) -a |) = a +b-ct+at² C :)) - (202 Write the corresponding basis for V₁ V₂ and W₁ W₂ Recall we get the linear map a2 a4 a4 3a3 T₁ T₂: V₁0 V₂ W₁ ⓇW₂ (T₁ ® T₂) (V1 ® V₂) = T1 (V₁) ® T₂(V₂) Compute the matrix of this map with respect to the two basis you found in part a Remark: This is an example of what is called the Kronecker-Product of ma- trices. It is an operation that takes an m × n and a k × 1 matrix and produces an mk × nl matrix. This matrix is precisely the matrix of the tensor product of linear maps we defined
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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can u please answer a and b
![6) Consider the following vector spaces with corresponding basis:
V₁ =R³
Bv₁ = {e₁,e2, е3}
Bw₁ = {1,t, t²}
W₁ =R[t]<2
V₂ = M2x2 (R)
Bv₂ = ( (
(a)
(b)
9 )· ( )· (i
BW₂
W₂ = M2x2 (R)
Now consider the following two linear transformations T₁ : V₁ → W₁ and T₂ : V₂ → W₂ given
by
=
- 6 ) 6 ) ( ) ( )
0
1
a
T())-
T₁(b)= a + b = ct + at²
с
) · ( 2 )
2a2
T₂
T-|(21 )) = (²
a1 a2
a3 a s
Write the corresponding basis for V₁ V₂ and W₁ W₂
Recall we get the linear map
a1
as
3a3
T₁ T₂: V₁0 V₂ → W₁ W₂
1
(T₁ ® T2) (V1 0 V₂) = T1(v₁) • T₂(V₂)
Compute the matrix of this map with respect to the two basis you found in part a
Remark: This is an example of what is called the Kronecker-Product of ma-
trices. It is an operation that takes an m × n and a k × 1 matrix and produces an mk x nl
matrix. This matrix precisely the matrix of the tensor product of linear maps we defined](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6214fda8-f992-4e88-b320-5195339361f5%2F3a054d18-e7e4-4743-b095-eff246b9e555%2Fuy27fwk_processed.png&w=3840&q=75)
Transcribed Image Text:6) Consider the following vector spaces with corresponding basis:
V₁ =R³
Bv₁ = {e₁,e2, е3}
Bw₁ = {1,t, t²}
W₁ =R[t]<2
V₂ = M2x2 (R)
Bv₂ = ( (
(a)
(b)
9 )· ( )· (i
BW₂
W₂ = M2x2 (R)
Now consider the following two linear transformations T₁ : V₁ → W₁ and T₂ : V₂ → W₂ given
by
=
- 6 ) 6 ) ( ) ( )
0
1
a
T())-
T₁(b)= a + b = ct + at²
с
) · ( 2 )
2a2
T₂
T-|(21 )) = (²
a1 a2
a3 a s
Write the corresponding basis for V₁ V₂ and W₁ W₂
Recall we get the linear map
a1
as
3a3
T₁ T₂: V₁0 V₂ → W₁ W₂
1
(T₁ ® T2) (V1 0 V₂) = T1(v₁) • T₂(V₂)
Compute the matrix of this map with respect to the two basis you found in part a
Remark: This is an example of what is called the Kronecker-Product of ma-
trices. It is an operation that takes an m × n and a k × 1 matrix and produces an mk x nl
matrix. This matrix precisely the matrix of the tensor product of linear maps we defined
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