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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 further help in order to fully understand the proof of Theorem 12.20 on page 275 ... ...

The relevant text reads as follows:

In the above proof by Browder we read the following:

" ... ... \(\displaystyle ^{\tau }{ ( A \alpha ) } = ^{\tau }{ ( } \sum_{ \sigma \text{ in } S_r } \varepsilon ( \sigma ) ^{ \sigma }{ \alpha } ) = \sum_{ \sigma \text{ in } S_r } \varepsilon ( \tau ) \varepsilon ( \tau \sigma ) ^{ \tau \sigma }{ \alpha } \) ... ... ... "

My question is as follows:

Can someone please explain and demonstrate why/how we have that

\(\displaystyle ^{\tau }{ ( } \sum_{ \sigma \text{ in } S_r } \varepsilon ( \sigma ) ^{ \sigma }{ \alpha } ) = \sum_{ \sigma \text{ in } S_r } \varepsilon ( \tau ) \varepsilon ( \tau \sigma ) ^{ \tau \sigma }{ \alpha }\)

Help will be much appreciated ... ...

Peter

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So that MHB readers can see and understand Browder's definitions and notation in the text given above I am providing the text of Browder's section on alternating tensors ... as follows:

Hope that helps ...

Peter

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

I need some further help in order to fully understand the proof of Theorem 12.20 on page 275 ... ...

The relevant text reads as follows:

In the above proof by Browder we read the following:

" ... ... \(\displaystyle ^{\tau }{ ( A \alpha ) } = ^{\tau }{ ( } \sum_{ \sigma \text{ in } S_r } \varepsilon ( \sigma ) ^{ \sigma }{ \alpha } ) = \sum_{ \sigma \text{ in } S_r } \varepsilon ( \tau ) \varepsilon ( \tau \sigma ) ^{ \tau \sigma }{ \alpha } \) ... ... ... "

My question is as follows:

Can someone please explain and demonstrate why/how we have that

\(\displaystyle ^{\tau }{ ( } \sum_{ \sigma \text{ in } S_r } \varepsilon ( \sigma ) ^{ \sigma }{ \alpha } ) = \sum_{ \sigma \text{ in } S_r } \varepsilon ( \tau ) \varepsilon ( \tau \sigma ) ^{ \tau \sigma }{ \alpha }\)

Help will be much appreciated ... ...

Peter

========================================================================================

So that MHB readers can see and understand Browder's definitions and notation in the text given above I am providing the text of Browder's section on alternating tensors ... as follows:

Hope that helps ...

Peter

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