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In the Preliminaries to Grove and Benson "Finite Reflection Groups' On page 1 (see attachment) we find the following:

"If [itex] \{ x_1 , x_2, ... x_n \} [/itex] is a basis for V, let [itex] V_i [/itex] be the subspace spanned by [itex] \{ x_1, ... , x_{i-1} , x_{i+1}, ... x_n \} [/itex], excluding [itex] x_i [/itex].

If [itex] 0 \ne y_i \in {V_i}^{\perp} [/itex], then [itex] < x_j , y_i > \ = 0 [/itex] for all [itex] j \ne i [/itex], but [itex] < x_i , y_i > \ne 0 [/itex] , for otherwise [itex] y_i \in V^{\perp} = 0 [/itex]"

I do not completely follow the argument as to why [itex] < x_i , y_i > \ \ne 0 [/itex] despite Grove and Bensons attempt to explain it. Can someone please (very explicitly) show why this is true?

Why would [itex] y_i \in V^{\perp} [/itex]be equal to 0 if [itex] < x_i , y_i > \ = \ 0 [/itex]?necessarily

Another issue I have is the folowing:

[itex] y_i [/itex] is defined as a non-zero vector belonging to [itex] {V_i}^{\perp} [/itex].

How do we know that [itex] y_i \in V^{\perp} [/itex] ?

Peter

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# Linear Algebra Preliminaries in Finite Reflection Groups

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