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: VjT: V₁0V₂ → W₁ ...Vnthat "behaves" like the mx n matrix of linear mapslinear mapT₁1T12T21 T22Tm1 Tm2Show that every linear mapT: V₁0 Vnwith coefficients Tij € L(V₁, W₁)→Conclude that the set of linear mapsT: V₁0 V₂can be identified with the set of matricesT11T12T21 T22Tm1 Tm2is of this form for linear maps Tij: V; → W; (hint: inclusion into direct sum/projectiononto subspace)(c)TinT2nW₁ ... WmTmn......W; for each i,j. Construct aWmW₁ ... WmTinT2nTmn(d)Does anything familiar happen when V₁Vn = W₁==(Hint: when Vj = W₁ = F what is L(V₁, W;) isomorphic to? Think dimensions....)=...Wm= F?

We shall solve first question only as you have asked more them one different question. For others…

(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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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,...,V_n, W₁,..,W_m 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?

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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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,...,V_n, W₁,..,W_m be vector spaces over F.

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