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I am reading Segei Winitzki's book: Linear Algebra via Exterior Products ...
I am currently focused on Section 1.7.3 Dimension of a Tensor Product is the Product of the Dimensions ... ...
I need help in order to get a clear understanding of an aspect of the proof of Lemma 3 in Section 1.7.3 ...
The relevant part of Winitzki's text reads as follows:
In the above quotation from Winitzki we read the following:
" ... ... By the result of Exercise 1 in Sec. 6.3 there exists a covector [itex]f^* \in V^*[/itex] such that
[itex]f^* ( v_j ) = \delta_{ j_1 j }[/itex] for [itex]j = 1, \ ... \ ... \ , \ n[/itex] ... ... "I cannot see how to show that there exists a covector [itex]f^* \in V^*[/itex] such that
[itex]f^* ( v_j ) = \delta_{ j_1 j }[/itex] for [itex]j = 1, \ ... \ ... \ , \ n[/itex] ... ...Can someone help me to show this from first principles ... ?It may be irrelevant to my problem ... but I cannot see the relevance of Exercise 1 in Section 6 which reads as follows:
Exercise 1 refers to Example 2 which reads as follows:
BUT ... since I wish to show the result:
... ... ... "there exists a covector [itex]f^* \in V^*[/itex] such that
[itex]f^* ( v_j ) = \delta_{ j_1 j }[/itex] for [itex]j = 1, \ ... \ ... \ , \ n[/itex] ... ..."... from first principles the above example is irrelevant ... BUT then ... I cannot see its relevance anyway!Hope someone can help ... ...
Peter
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*** NOTE ***
To help readers understand Winitzki's approach and notation for tensors I am providing Winitzki's introduction to Section 1.7 ... ... as follows ... ... :
I am currently focused on Section 1.7.3 Dimension of a Tensor Product is the Product of the Dimensions ... ...
I need help in order to get a clear understanding of an aspect of the proof of Lemma 3 in Section 1.7.3 ...
The relevant part of Winitzki's text reads as follows:
In the above quotation from Winitzki we read the following:
" ... ... By the result of Exercise 1 in Sec. 6.3 there exists a covector [itex]f^* \in V^*[/itex] such that
[itex]f^* ( v_j ) = \delta_{ j_1 j }[/itex] for [itex]j = 1, \ ... \ ... \ , \ n[/itex] ... ... "I cannot see how to show that there exists a covector [itex]f^* \in V^*[/itex] such that
[itex]f^* ( v_j ) = \delta_{ j_1 j }[/itex] for [itex]j = 1, \ ... \ ... \ , \ n[/itex] ... ...Can someone help me to show this from first principles ... ?It may be irrelevant to my problem ... but I cannot see the relevance of Exercise 1 in Section 6 which reads as follows:
BUT ... since I wish to show the result:
... ... ... "there exists a covector [itex]f^* \in V^*[/itex] such that
[itex]f^* ( v_j ) = \delta_{ j_1 j }[/itex] for [itex]j = 1, \ ... \ ... \ , \ n[/itex] ... ..."... from first principles the above example is irrelevant ... BUT then ... I cannot see its relevance anyway!Hope someone can help ... ...
Peter
===========================================================
*** NOTE ***
To help readers understand Winitzki's approach and notation for tensors I am providing Winitzki's introduction to Section 1.7 ... ... as follows ... ... :
Attachments
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Winitzki - 1 - Lemma 3 - Section 1.7.3 - PART 1 ....png17.9 KB · Views: 610
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Winitzki - 2 - Lemma 3 - Section 1.7.3 - PART 2 ....png87.1 KB · Views: 739
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Winitzki - Exercise 1 - Section 1.6 ....png5.2 KB · Views: 511
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Winitzki - 1 - Example 2 - Section 1.6 - PART 1 ....png30.1 KB · Views: 654
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Winitzki - 2 - Example 2 - Section 1.6 - PART 2 ....png4.7 KB · Views: 504
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Winitzki - 1 - Section 1.7 - PART 1 ....png50.4 KB · Views: 617
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Winitzki - 2 - Section 1.7 - PART 2 ....png44.9 KB · Views: 630
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Winitzki - 3 - Section 1.7 - PART 3 ....png47.3 KB · Views: 618
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