I think, that I have understood the folowing argument in Grove & Benson. Can someone please check my argument?
Basically, to repeat the text I am trying to understand: (see attachement for Page 1 of Grove and Benson)In the Preliminaries to Grove and Benson "Finite Reflection Groups"
On page 1 (see attachment) we find the following:
"If is a basis for V, let be the subspace spanned by , excluding .
If , then for all , but , for otherwise "
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Thus given " is a basis for V, let be the subspace spanned by , excluding "
I am trying to show that:"If , then for all , but , for otherwise "
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First show the following:
If = 0 for all j then must equal zero ........ (1)
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Proof of (1)
must belong to V [since it belongs to a subspace of V]
Thus we can express as follows:[
tex] y_i = a_1 , x_1 + a_2 , x_2 + .... + a_n , x_n [/tex]
= .... (2)
= .... (3)
If every inner product in (2) or (3) is zero then clearly [itex] y_i [/itex] = 0
{Problem! Expansion (2) or (3) requires to be orthonormal! But of course it could be made orthonormal!}
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But we know (1) does not hold because we have assumed
But we also know that = 0 for all since the with belong to which is orthogonal to .Thus , for otherwise
Is this reasoning correct?
I would appreciate it very much if someone can confirm the correctness of my reasoning.[Problem: Grove and Benson actually conclude the above argument by saying the following: (see attachement)
But , for otherwise but I think this is a typo as they mean ?? Am I correct? ]
Hope someone can help.
Peter