Let V and W be vector spaces over a field K having bases α = {V₁, V₂} and B = {w₁, W2, W3} respectively. Recall that the set of linear transformations (V, W) = {T: V → W | T is linear} is a vector space. For i = 1,2 and j = 1,2,3 define Tij : V → W by if k = i, if k i Recall that by Theorem 34, the above information is sufficient to define each Tij as a linear transformation since it specifies the value on the basis {V₁, V₂}. (a) Let S = 2T12 - 3T23 + 4T₁1. Evaluate S(2v₁ - 30₂) and compute the matrix [S]. (b) (c) ] Prove that A = {T11, T12, T13, T21, T22, T23} is a basis of (V, W). 0 (36) 0 1 Let B = Tij (Uk) = { W ₁ 0 :(V, W). → M 3x2 (K) by 0 be the standard basis of M 3x2 (K). Define $(T) = [T] and recall from Theorem 46 that is an isomorphism (in particular, is a linear transformation). Compute its matrix [$13.
Let V and W be vector spaces over a field K having bases α = {V₁, V₂} and B = {w₁, W2, W3} respectively. Recall that the set of linear transformations (V, W) = {T: V → W | T is linear} is a vector space. For i = 1,2 and j = 1,2,3 define Tij : V → W by if k = i, if k i Recall that by Theorem 34, the above information is sufficient to define each Tij as a linear transformation since it specifies the value on the basis {V₁, V₂}. (a) Let S = 2T12 - 3T23 + 4T₁1. Evaluate S(2v₁ - 30₂) and compute the matrix [S]. (b) (c) ] Prove that A = {T11, T12, T13, T21, T22, T23} is a basis of (V, W). 0 (36) 0 1 Let B = Tij (Uk) = { W ₁ 0 :(V, W). → M 3x2 (K) by 0 be the standard basis of M 3x2 (K). Define $(T) = [T] and recall from Theorem 46 that is an isomorphism (in particular, is a linear transformation). Compute its matrix [$13.
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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Question
only answer part b no need to solve other parts just part b required
![Let V and W be vector spaces over a field K having bases α = {V₁, V₂} and B = {w₁, W2, W3} respectively.
Recall that the set of linear transformations
(V, W) = {T: V → W | T is linear}
is a vector space. For i = 1,2 and j = 1,2,3 define Tij : V → W by
if k = i,
Tij (Uk) = { W ₁ if k i
Recall that by Theorem 34, the above information is sufficient to define each Tij as a linear transformation since
it specifies the value on the basis {V₁, V₂}.
(a)
Let S = 2T12 - 3T23 + 4T₁1. Evaluate S(2v₁ - 30₂) and compute the matrix [S].
(b)
(c)
] Prove that A = {T11, T12, T13, T21, T22, T23} is a basis of (V, W).
0
(36)
0 1
Let B =
0
:(V, W). → M 3x2 (K) by
0 be the standard basis of M 3x2 (K). Define
$(T) = [T]
and recall from Theorem 46 that is an isomorphism (in particular, is a linear transformation). Compute its matrix
[$13.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6262eb3c-a93b-428b-931a-57a08b3f7c7d%2Fd81550fa-0275-4e44-85d2-52a67d95bc16%2Fkh2uigy_processed.png&w=3840&q=75)
Transcribed Image Text:Let V and W be vector spaces over a field K having bases α = {V₁, V₂} and B = {w₁, W2, W3} respectively.
Recall that the set of linear transformations
(V, W) = {T: V → W | T is linear}
is a vector space. For i = 1,2 and j = 1,2,3 define Tij : V → W by
if k = i,
Tij (Uk) = { W ₁ if k i
Recall that by Theorem 34, the above information is sufficient to define each Tij as a linear transformation since
it specifies the value on the basis {V₁, V₂}.
(a)
Let S = 2T12 - 3T23 + 4T₁1. Evaluate S(2v₁ - 30₂) and compute the matrix [S].
(b)
(c)
] Prove that A = {T11, T12, T13, T21, T22, T23} is a basis of (V, W).
0
(36)
0 1
Let B =
0
:(V, W). → M 3x2 (K) by
0 be the standard basis of M 3x2 (K). Define
$(T) = [T]
and recall from Theorem 46 that is an isomorphism (in particular, is a linear transformation). Compute its matrix
[$13.
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