(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

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Please just answer (a,b &c)

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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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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.

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

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Answer 21

Follow-up Question

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.

Solution

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Answer 22

Follow-up Question

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.

Answer 23

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Answer 24

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Answer 25

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.

Answer 26

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.

Answer 27

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Answer 32

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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.

Solution

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Answer 33

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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.

Solution

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Answer 35

Follow-up Question

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.

Solution

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Answer 36

Follow-up Question

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.

Solution

by Bartleby Expert

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Answer 37

Follow-up Question

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.

Answer 38

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Answer 39

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Answer 40

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.

Answer 41

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.

Answer 42

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Answer 43

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Answer 46

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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

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