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I am reading Bruce N. Coopersteins book: Advanced Linear Algebra (Second Edition) ... ...
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help with another aspect of the proof of Theorem 10.1 regarding the existence of a tensor product ... ...
The relevant part of Theorem 10.1 reads as follows:
In the above text we read the following:
" ... ... Recall that [itex]V_1 \times \ ... \ \times V_m = X[/itex] and that [itex]Z[/itex] is a vector space based on [itex]X[/itex]. Since [itex]W[/itex] is a vector space and [itex]f[/itex] is a map from [itex]X[/itex] to [itex]W[/itex], by the universal property of [itex]Z[/itex] there exists a unique linear transformation [itex]S \ : \ Z \longrightarrow W[/itex] such that [itex]S[/itex] restricted to [itex]X[/itex] is [itex]f[/itex]. ... ..."
Now I have summarised the mappings involved in Theorem 10.1 in Figure 1 below ... ...
My question is as follows:
Why does [itex]Z[/itex] have a universal mapping property ...? ... ... and indeed if [itex]Z[/itex] has one, why doesn't [itex]V[/itex] ... ... giving us the relationship [itex]T \gamma = f[/itex] that we want ... what is special about [itex] Z [/itex]?
Hope someone can help ...
Peter
*** NOTE ***
... ... oh no! ... ... I think I have just realised the answer to my question ... hmm ... embarrassingly simple ... ... I think that [itex] Z [/itex] has a UMP because [itex]( Z, \iota )[/itex] is assumed to be the vector space based on the set [itex] X [/itex]... and vector spaces based on a set have a UMP ... is that right? ... see Cooperstein Definition 10.1 on the first page of Section 10.1 provided below ...
Can someone confirm that this is the reason Z has a Universal Mapping Property ...
Peter
==========================================================
*** NOTE ***
It may help readers of the above post to be able to read Cooperstein's introduction to Section 10.1 where he covers, among other things, the notion of a vector space being based on a set and the idea of the universal mapping problem ... ... so I am providing this text as follows:
I am focused on Section 10.1 Introduction to Tensor Products ... ...
I need help with another aspect of the proof of Theorem 10.1 regarding the existence of a tensor product ... ...
The relevant part of Theorem 10.1 reads as follows:
In the above text we read the following:
" ... ... Recall that [itex]V_1 \times \ ... \ \times V_m = X[/itex] and that [itex]Z[/itex] is a vector space based on [itex]X[/itex]. Since [itex]W[/itex] is a vector space and [itex]f[/itex] is a map from [itex]X[/itex] to [itex]W[/itex], by the universal property of [itex]Z[/itex] there exists a unique linear transformation [itex]S \ : \ Z \longrightarrow W[/itex] such that [itex]S[/itex] restricted to [itex]X[/itex] is [itex]f[/itex]. ... ..."
Now I have summarised the mappings involved in Theorem 10.1 in Figure 1 below ... ...
My question is as follows:
Why does [itex]Z[/itex] have a universal mapping property ...? ... ... and indeed if [itex]Z[/itex] has one, why doesn't [itex]V[/itex] ... ... giving us the relationship [itex]T \gamma = f[/itex] that we want ... what is special about [itex] Z [/itex]?
Hope someone can help ...
Peter
*** NOTE ***
... ... oh no! ... ... I think I have just realised the answer to my question ... hmm ... embarrassingly simple ... ... I think that [itex] Z [/itex] has a UMP because [itex]( Z, \iota )[/itex] is assumed to be the vector space based on the set [itex] X [/itex]... and vector spaces based on a set have a UMP ... is that right? ... see Cooperstein Definition 10.1 on the first page of Section 10.1 provided below ...
Can someone confirm that this is the reason Z has a Universal Mapping Property ...
Peter
==========================================================
*** NOTE ***
It may help readers of the above post to be able to read Cooperstein's introduction to Section 10.1 where he covers, among other things, the notion of a vector space being based on a set and the idea of the universal mapping problem ... ... so I am providing this text as follows:
Attachments

Cooperstein  1  Theorem 10.1  PART 1 ....png96.2 KB · Views: 792

Cooperstein  2  Theorem 10.1  PART 2 ....png60.7 KB · Views: 594

Cooperstein  3  Theorem 10.1  PART 3 ....png71.7 KB · Views: 602

Figure 1  Cooperstein  Theorem 10.1  Mappings.png64.5 KB · Views: 690

Cooperstein  1  Section 10.1  PART 1 ....png95.2 KB · Views: 497

Cooperstein  2  Section 10.1  PART 2 ....png89.9 KB · Views: 435

Cooperstein  3  Section 10.1  PART 3 ....png63 KB · Views: 524

Cooperstein  4  Section 10.1  PART 4 ....png81.7 KB · Views: 465
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