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(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
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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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.

 

(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?
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.

 

(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?
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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