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Space of Alternating Tensors of Rank r ... ... Browder, Proposition 12.22 ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,882
Hobart, Tasmania
I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 12: Multilinear Algebra ... ...

I need some help in order to fully understand the proof of Proposition 12.22 on page 276 ... ...


The relevant text reads as follows:



Browder - 1 - Proposition 12.22  ... ... PART 1 ... .png
Browder - 2 - Proposition 12.22  ... ... PART 2 ... ... .png






In the above proof by Browder we read the following:

" ... ... the Kronecker delta, and hence, using Proposition 12.20, that for any \(\displaystyle r\)-tuples \(\displaystyle I\) and \(\displaystyle J\), not necessarily increasing

\(\displaystyle \tilde{u}^I ( u_{ j_1 }, \cdot \cdot \cdot u_{ j_r } ) = \varepsilon^I_J\)

where \(\displaystyle \varepsilon^I_J\), the "Kronecker epsilon" ... ... "


My question is as follows:

How is Proposition 12.20 used to show that \(\displaystyle \tilde{u}^I ( u_{ j_1 }, \cdot \cdot \cdot u_{ j_r } ) = \varepsilon^I_J\) ... ... could someone please demonstrate the use of Proposition 12.20 to derive this result ...





The above proof refers to Proposition 12.20 so I am providing the text of this proposition as follows:



Browder ... Proposition 12.20 ... .png





The above proof also refers (indirectly) to the basis \(\displaystyle ( \tilde{u}^1, \cdot \cdot \cdot \tilde{u}^n )\) for \(\displaystyle V^*\) ... ... this is mentioned at the start of Section 12.1 ... so I am providing the relevant text as follows:




Browder - 1 - Start of Chapter 12 ... ... PART 1.png
Browder - 2 - Start of Chapter 12 ... ... PART 2     .png
Browder - 3 - Start of Chapter 12 ... ... PART 3     .png




It may also be useful in order to understand the above post for MHB members to have access to Section 12.2 on Alternating Tensors ... so I am providing the same as follows:



Browder - 1 - Start of Section 12.2 ... ... PART 1 .... .png
Browder - 4 - Start of Section 12.2 ... ... PART 4  ....  .png
Browder - 3 - Start of Section 12.2 ... ... PART 3 .... .png
Browder - 4 - Start of Section 12.2 ... ... PART 4  ....  .png



Hope access to the above text is helpful ...

Peter















Hope the access to the above text helps ...

Peter
 

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Last edited:

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,882
Hobart, Tasmania
I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 12: Multilinear Algebra ... ...

I need some help in order to fully understand the proof of Proposition 12.22 on page 276 ... ...


The relevant text reads as follows:











In the above proof by Browder we read the following:

" ... ... the Kronecker delta, and hence, using Proposition 12.20, that for any \(\displaystyle r\)-tuples \(\displaystyle I\) and \(\displaystyle J\), not necessarily increasing

\(\displaystyle \tilde{u}^I ( u_{ j_1 }, \cdot \cdot \cdot u_{ j_r } ) = \varepsilon^I_J\)

where \(\displaystyle \varepsilon^I_J\), the "Kronecker epsilon" ... ... "


My question is as follows:

How is Proposition 12.20 used to show that \(\displaystyle \tilde{u}^I ( u_{ j_1 }, \cdot \cdot \cdot u_{ j_r } ) = \varepsilon^I_J\) ... ... could someone please demonstrate the use of Proposition 12.20 to derive this result ...





The above proof refers to Proposition 12.20 so I am providing the text of this proposition as follows:









The above proof also refers (indirectly) to the basis \(\displaystyle ( \tilde{u}^1, \cdot \cdot \cdot \tilde{u}^n )\) for \(\displaystyle V^*\) ... ... this is mentioned at the start of Section 12.1 ... so I am providing the relevant text as follows:











It may also be useful in order to understand the above post for MHB members to have access to Section 12.2 on Alternating Tensors ... so I am providing the same as follows:










Hope access to the above text is helpful ...

Peter







Almost all of the scanned text in my post is to enable readers to check definitions and notation (only if they need to ...) ... it is not necessary for a reader who knows the topic to read all the text ... maybe just to check the meaning of notation every now and then ..


The essence of my problem is one of the early steps in the proof of the following proposition in Browder's book:


12.22 Proposition The set


##\{ \tilde{u}^I \ : \ \mid I \mid = r, \ I \text{ increasing } \} ##


forms a basis for ##\bigwedge^r ( V^* )##




Now the proof starts by noting that ...


... ... if ##I = ( i_1, \cdot \cdot \cdot , i_r )## and ##J = ( j_1, \cdot \cdot \cdot , j_r )## are increasing sequences then




##\tilde{u}^I (u_{ j_1}, \cdot \cdot \cdot , u_{ j_r} ) = A ( \tilde{u}^{i_1} \otimes \cdot \cdot \cdot \otimes \tilde{u}^{i_r} ) ( u_{j_1} \ ..... \ u_{j_r} )##




##= \sum_{ \sigma \text{ in } S_r } \varepsilon ( \sigma ) \tilde{u}^{i_1} \otimes \cdot \cdot \cdot \otimes \tilde{u}^{i_r} ( u_{j_{\sigma (1) } } \ ..... \ u_{j_{\sigma (r) } } )##




##= \delta^I_J##


The proof then notes that it follows that for any ##r##-tuples ##I## and ##J##, not necessarily increasing,


##\tilde{u}^I ( u_{ j_1 }, \cdot \cdot \cdot u_{ j_r } ) = \varepsilon^I_J##


where ##\varepsilon^I_J##, the "Kronecker epsilon" is defined to be ##0## unless the sequence ##I## is a rearrangement of the sequence ##J##, and to be ##\varepsilon ( \sigma )##, if the permutation ##\sigma## transforms ##I## to ##J##. ... ...




My question is as follows:


How/why does it follow that for any ##r##-tuples ##I## and ##J##, not necessarily increasing,


##\tilde{u}^I ( u_{ j_1 }, \cdot \cdot \cdot u_{ j_r } ) = \varepsilon^I_J## ... ... ?


Indeed ... can someone please demonstrate that it follows that ##\tilde{u}^I ( u_{ j_1 }, \cdot \cdot \cdot u_{ j_r } ) = \varepsilon^I_J## ...




(NOTE: Browder says using Proposition 12.20 it follows that ,,, ,,, etc etc ... )




Help will be much appreciated ...


Peter














.



Almost all of the scanned text in my post is to enable readers to check definitions and notation (only if they need to ...) ... it is not necessary for a reader who knows the topic to read all the text ... maybe just to check the meaning of notation every now and then ..


The essence of my problem is one of the early steps in the proof of the following proposition in Browder's book:


12.22 Proposition The set


##\{ \tilde{u}^I \ : \ \mid I \mid = r, \ I \text{ increasing } \} ##


forms a basis for ##\bigwedge^r ( V^* )##




Now the proof starts by noting that ...


... ... if ##I = ( i_1, \cdot \cdot \cdot , i_r )## and ##J = ( j_1, \cdot \cdot \cdot , j_r )## are increasing sequences then




##\tilde{u}^I (u_{ j_1}, \cdot \cdot \cdot , u_{ j_r} ) = A ( \tilde{u}^{i_1} \otimes \cdot \cdot \cdot \otimes \tilde{u}^{i_r} ) ( u_{j_1} \ ..... \ u_{j_r} )##




##= \sum_{ \sigma \text{ in } S_r } \varepsilon ( \sigma ) \tilde{u}^{i_1} \otimes \cdot \cdot \cdot \otimes \tilde{u}^{i_r} ( u_{j_{\sigma (1) } } \ ..... \ u_{j_{\sigma (r) } } )##




##= \delta^I_J##


The proof then notes that it follows that for any ##r##-tuples ##I## and ##J##, not necessarily increasing,


##\tilde{u}^I ( u_{ j_1 }, \cdot \cdot \cdot u_{ j_r } ) = \varepsilon^I_J##


where ##\varepsilon^I_J##, the "Kronecker epsilon" is defined to be ##0## unless the sequence ##I## is a rearrangement of the sequence ##J##, and to be ##\varepsilon ( \sigma )##, if the permutation ##\sigma## transforms ##I## to ##J##. ... ...




My question is as follows:


How/why does it follow that for any ##r##-tuples ##I## and ##J##, not necessarily increasing,


##\tilde{u}^I ( u_{ j_1 }, \cdot \cdot \cdot u_{ j_r } ) = \varepsilon^I_J## ... ... ?


Indeed ... can someone please demonstrate that it follows that ##\tilde{u}^I ( u_{ j_1 }, \cdot \cdot \cdot u_{ j_r } ) = \varepsilon^I_J## ...




(NOTE: Browder says using Proposition 12.20 it follows that ,,, ,,, etc etc ... )




Help will be much appreciated ...


Peter














.










Hope the access to the above text helps ...

Peter



For those who might be put of by all the scanned text in the above post ... a summary of my question appears below ...

[ ***NOTE*** Almost all of the scanned text in my first post above is to enable readers to check definitions and notation (only if they need to ...) ... it is not necessary for a reader who knows the topic to read all the text ... maybe just to check the meaning of notation every now and then ... ]


The essence of my problem is one of the early steps in the proof of the following proposition in Browder's book:


12.22 Proposition The set


\(\displaystyle \{ \tilde{u}^I \ : \ \mid I \mid = r, \ I \text{ increasing } \} \)


forms a basis for \(\displaystyle \bigwedge^r ( V^* ) \)



Now the proof starts by noting that ...


... ... if \(\displaystyle I = ( i_1, \cdot \cdot \cdot , i_r )\) and \(\displaystyle J = ( j_1, \cdot \cdot \cdot , j_r )\) are increasing sequences then




\(\displaystyle \tilde{u}^I (u_{ j_1}, \cdot \cdot \cdot , u_{ j_r} ) = A ( \tilde{u}^{i_1} \otimes \cdot \cdot \cdot \otimes \tilde{u}^{i_r} ) ( u_{j_1} \ ..... \ u_{j_r} )\)




\(\displaystyle = \sum_{ \sigma \text{ in } S_r } \varepsilon ( \sigma ) \tilde{u}^{i_1} \otimes \cdot \cdot \cdot \otimes \tilde{u}^{i_r} ( u_{j_{\sigma (1) } } \ ..... \ u_{j_{\sigma (r) } } )\)




\(\displaystyle = \delta^I_J\)




The proof then notes that it follows that for any \(\displaystyle r\)-tuples \(\displaystyle I\) and \(\displaystyle J\), not necessarily increasing,


\(\displaystyle \tilde{u}^I ( u_{ j_1 }, \cdot \cdot \cdot u_{ j_r } ) = \varepsilon^I_J\)


where \(\displaystyle \varepsilon^I_J\), the "Kronecker epsilon" is defined to be \(\displaystyle 0\) unless the sequence \(\displaystyle I\) is a rearrangement of the sequence \(\displaystyle J\), and to be \(\displaystyle \varepsilon ( \sigma )\), if the permutation \(\displaystyle \sigma\) transforms \(\displaystyle I\) to \(\displaystyle J\). ... ...




My question is as follows:


How/why does it follow that for any \(\displaystyle r\)-tuples \(\displaystyle I\) and \(\displaystyle J\), not necessarily increasing,


\(\displaystyle \tilde{u}^I ( u_{ j_1 }, \cdot \cdot \cdot u_{ j_r } ) = \varepsilon^I_J\) ... ... ?


Indeed ... can someone please demonstrate that it follows that \(\displaystyle \tilde{u}^I ( u_{ j_1 }, \cdot \cdot \cdot u_{ j_r } ) = \varepsilon^I_J\) ...




(NOTE: Browder says using Proposition 12.20 it follows that ,,, ,,, etc etc ... )




Help will be much appreciated ...


Peter



***EDIT 1***

Although the result that \(\displaystyle \tilde{u}^I ( u_{ j_1 }, \cdot \cdot \cdot u_{ j_r } ) = \varepsilon^I_J\) seems reasonable to me ... how/why Proposition 12.20 is involved in establishing this result is a mystery to me ... If anyone can indicate/explain the role Proposition 12.20 plays in establishing the result (and why 12.20 is necessary ...) then that would be most helpful ... ...



***EDIT 2***

A worry I have is that it seems to me that it is quite possible for the expression


\(\displaystyle = \sum_{ \sigma \text{ in } S_r } \varepsilon ( \sigma ) \tilde{u}^{i_1} \otimes \cdot \cdot \cdot \otimes \tilde{u}^{i_r} ( u_{j_{\sigma (1) } } \ ..... \ u_{j_{\sigma (r) } } )\)

to be -1 as surely for some permutations \(\displaystyle \sigma\) we will have \(\displaystyle \varepsilon ( \sigma ) = -1\) ...

... ... but ... ... this surely would mean


\(\displaystyle \tilde{u}^I (u_{ j_1}, \cdot \cdot \cdot , u_{ j_r} ) \neq \delta^I_J\) ... ...

Can someone please clarify this issue ...








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