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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 12.2 on pages 277 - 278 ... ...Proposition 12.2 and its proof read as follows:
In the above proof by Browder (near the end of the proof) we read the following:
" ... ... To see also that ##A( \beta \otimes \alpha ) = 0##, we observe that ## \beta \otimes \alpha = \ ^{ \sigma }( \alpha \otimes \beta )## where ##\sigma## is the permutation which sends ##(1, \cdot \cdot \cdot , r+s )## to ##(r+1, \cdot \cdot \cdot , r+s, 1 \cdot \cdot \cdot , r )## ... ... "My question ... or more accurately problem ... is that given ##\sigma## as defined by Browder I cannot verify that ##\beta \otimes \alpha = \ ^{ \sigma }( \alpha \otimes \beta )## is true ...My working is as follows:
Let ##\alpha \in \bigwedge^r## and let ##\beta \in \bigwedge^s## ... ...
Then we have ...
##\beta \otimes \alpha (v_1, \cdot \cdot \cdot , v_{ r+s } ) = \beta ( v_1, \cdot \cdot \cdot , v_s ) \alpha ( v_{ s+1 }, \cdot \cdot \cdot , v_{ r+s } )##
and
##\alpha \otimes \beta (v_1, \cdot \cdot \cdot , v_{ r+s } ) = \alpha ( v_1 , \cdot \cdot \cdot , v_r ) \beta ( v_{ r+1 }, \cdot \cdot \cdot , v_{ r+s } )##Now consider ##\sigma## where ...
... ##\sigma## sends ##(1, \cdot \cdot \cdot , r+s )## to ##(r+1, \cdot \cdot \cdot , r+s, 1 \cdot \cdot \cdot , r )##We have
##^{ \sigma } ( \alpha \otimes \beta ) (v_1, \cdot \cdot \cdot , v_{ r+s } )##
##= \alpha \otimes \beta ( v_{ r+1 }, \cdot \cdot \cdot , v_{ r+s }, v_1, \cdot \cdot \cdot , v_r )## ... ... hmm ... this does not appear to be correct ...BUT ... ... if we consider ##\sigma_1## where ...
... ##\sigma_1## sends ##(1, \cdot \cdot \cdot , r+s )## to ##(s+1, \cdot \cdot \cdot , r+s, 1 \cdot \cdot \cdot , s )##
then we have
##^{ \sigma_1 } ( \alpha \otimes \beta ) (v_1, \cdot \cdot \cdot , v_{ r+s } )##
##= ( \alpha \otimes \beta ) ( v_{ s+1 }, \cdot \cdot \cdot , v_{ r+s }, v_1, \cdot \cdot \cdot , v_s )##
##= \alpha ( v_{ s+1 }, \cdot \cdot \cdot , v_{ r+s } ) \beta ( v_1, \cdot \cdot \cdot , v_s )##
##= \beta \otimes \alpha## ...
Given that my working differs from Browder ... I suspect I have made an error ...
Can someone please point out the error(s) in my working ...
Help will be much appreciated ...
Peter
I am currently reading Chapter 12: Multilinear Algebra ... ...
I need some help in order to fully understand the proof of Proposition 12.2 on pages 277 - 278 ... ...Proposition 12.2 and its proof read as follows:
In the above proof by Browder (near the end of the proof) we read the following:
" ... ... To see also that ##A( \beta \otimes \alpha ) = 0##, we observe that ## \beta \otimes \alpha = \ ^{ \sigma }( \alpha \otimes \beta )## where ##\sigma## is the permutation which sends ##(1, \cdot \cdot \cdot , r+s )## to ##(r+1, \cdot \cdot \cdot , r+s, 1 \cdot \cdot \cdot , r )## ... ... "My question ... or more accurately problem ... is that given ##\sigma## as defined by Browder I cannot verify that ##\beta \otimes \alpha = \ ^{ \sigma }( \alpha \otimes \beta )## is true ...My working is as follows:
Let ##\alpha \in \bigwedge^r## and let ##\beta \in \bigwedge^s## ... ...
Then we have ...
##\beta \otimes \alpha (v_1, \cdot \cdot \cdot , v_{ r+s } ) = \beta ( v_1, \cdot \cdot \cdot , v_s ) \alpha ( v_{ s+1 }, \cdot \cdot \cdot , v_{ r+s } )##
and
##\alpha \otimes \beta (v_1, \cdot \cdot \cdot , v_{ r+s } ) = \alpha ( v_1 , \cdot \cdot \cdot , v_r ) \beta ( v_{ r+1 }, \cdot \cdot \cdot , v_{ r+s } )##Now consider ##\sigma## where ...
... ##\sigma## sends ##(1, \cdot \cdot \cdot , r+s )## to ##(r+1, \cdot \cdot \cdot , r+s, 1 \cdot \cdot \cdot , r )##We have
##^{ \sigma } ( \alpha \otimes \beta ) (v_1, \cdot \cdot \cdot , v_{ r+s } )##
##= \alpha \otimes \beta ( v_{ r+1 }, \cdot \cdot \cdot , v_{ r+s }, v_1, \cdot \cdot \cdot , v_r )## ... ... hmm ... this does not appear to be correct ...BUT ... ... if we consider ##\sigma_1## where ...
... ##\sigma_1## sends ##(1, \cdot \cdot \cdot , r+s )## to ##(s+1, \cdot \cdot \cdot , r+s, 1 \cdot \cdot \cdot , s )##
then we have
##^{ \sigma_1 } ( \alpha \otimes \beta ) (v_1, \cdot \cdot \cdot , v_{ r+s } )##
##= ( \alpha \otimes \beta ) ( v_{ s+1 }, \cdot \cdot \cdot , v_{ r+s }, v_1, \cdot \cdot \cdot , v_s )##
##= \alpha ( v_{ s+1 }, \cdot \cdot \cdot , v_{ r+s } ) \beta ( v_1, \cdot \cdot \cdot , v_s )##
##= \beta \otimes \alpha## ...
Given that my working differs from Browder ... I suspect I have made an error ...
Can someone please point out the error(s) in my working ...
Help will be much appreciated ...
Peter