(a) linear map Suppose we are given linear maps Tij: Vj W; for each i,j. Construct a T: V₁0 V₂ W₁ ... Wm that "behaves" like the mx n matrix of linear maps T11 T12 … T21 T22 Tm1 Tm2 Tin T2n Tmn
(a) linear map Suppose we are given linear maps Tij: Vj W; for each i,j. Construct a T: V₁0 V₂ W₁ ... Wm that "behaves" like the mx n matrix of linear maps T11 T12 … T21 T22 Tm1 Tm2 Tin T2n Tmn
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
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Please just answer (a,b &c)
Question
This question here is to show that linear maps out of/into direct sums "behave" like matrices.
Let V1,...,Vn, W1,..,Wm be vector spaces over F.
Transcribed Image Text:(a)
(b)
Suppose we are given linear maps Tij: Vj
T: V₁0
V₂ → W₁ ...
Vn
that "behaves" like the mx n matrix of linear maps
linear map
T₁1
T12
T21 T22
Tm1 Tm2
Show that every linear map
T: V₁0 Vn
with coefficients Tij € L(V₁, W₁)
→
Conclude that the set of linear maps
T: V₁0 V₂
can be identified with the set of matrices
T11
T12
T21 T22
Tm1 Tm2
is of this form for linear maps Tij: V; → W; (hint: inclusion into direct sum/projection
onto subspace)
(c)
Tin
T2n
W₁ ... Wm
Tmn
...
...
W; for each i,j. Construct a
Wm
W₁ ... Wm
Tin
T2n
Tmn
(d)
Does anything familiar happen when V₁
Vn = W₁
==
(Hint: when Vj = W₁ = F what is L(V₁, W;) isomorphic to? Think dimensions....)
=...
Wm= F?
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Ok can you please work on parts (b &c)
Question
This question here is to show that linear maps out of/into direct sums "behave" like matrices.
Let V1,...,Vn, W1,..,Wm be vector spaces over F.
Transcribed Image Text:(a)
(b)
Suppose we are given linear maps Tij: Vj
T: V₁0
V₂ → W₁ ...
Vn
that "behaves" like the mx n matrix of linear maps
linear map
T₁1
T12
T21 T22
Tm1 Tm2
Show that every linear map
T: V₁0 Vn
with coefficients Tij € L(V₁, W₁)
→
Conclude that the set of linear maps
T: V₁0 V₂
can be identified with the set of matrices
T11
T12
T21 T22
Tm1 Tm2
is of this form for linear maps Tij: V; → W; (hint: inclusion into direct sum/projection
onto subspace)
(c)
Tin
T2n
W₁ ... Wm
Tmn
...
...
W; for each i,j. Construct a
Wm
W₁ ... Wm
Tin
T2n
Tmn
(d)
Does anything familiar happen when V₁
Vn = W₁
==
(Hint: when Vj = W₁ = F what is L(V₁, W;) isomorphic to? Think dimensions....)
=...
Wm= F?
Solution
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