Linear Algebra Preliminaries in "Finite Reflection Groups" by Grove and Benson

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