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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
Hope the access to the above text helps ...
Peter
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
Hope the access to the above text helps ...
Peter
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