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 Exercise [11.09] 
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Joined: 07 May 2009, 16:45
Posts: 62
Post Exercise [11.09]
Grassmann algebra. Show
\boldsymbol{a \wedge b} = \boldsymbol{- b \wedge a}
where
\boldsymbol{a} = \sum_{i=1}^n a_i\boldsymbol{\eta}_i
\boldsymbol{b} = \sum_{j=1}^n b_j\boldsymbol{\eta}_j

\boldsymbol{a \wedge b} = \sum_{i=1}^n a_i\boldsymbol{\eta}_i \wedge \sum_{j=1}^n b_j\boldsymbol{\eta}_j
= \sum_{i=1}^n \sum_{j=1}^n (a_i\boldsymbol{\eta}_i \wedge b_j\boldsymbol{\eta}_j) (distributive law that Penrose says we can use)
= \sum_{i=1}^n \sum_{j=1}^n ((a_i b_j)(\boldsymbol{\eta}_i \wedge  \boldsymbol{\eta}_j)) (more distributive law)
= \sum_{j=1}^n \sum_{i=1}^n (-(b_ja_i)(\boldsymbol{\eta}_j \wedge  \boldsymbol\eta_i)) (anti-commute each wedge product, commute scalars and summations)
= - \sum_{j=1}^n b_j\boldsymbol\eta_j \wedge \sum_{i=1}^n a_i\boldsymbol\eta_i (now do it all backwards)
=-\boldsymbol{b} \wedge \boldsymbol{a}


18 Jul 2009, 21:57
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