- #1

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## Homework Statement:

- If (u,v,w) is a positive basis, so (u^v, v^w,w^u) is too.

## Relevant Equations:

- All below

I think we can say that (u,v,u^v) is a positive basis, so as (w^v,v,w) and (u,w^u,w). (1)

So

u^v = βw

v^w = γu

w^u = λv

where λ, β, and γ > 0 (*)

(u^v, v^w,w^u) = (βw,γu,λv)

\begin{vmatrix}

0 & 0 & β \\

γ & 0 & 0 \\

0 & λ & 0 \\

\end{vmatrix}

This determinant is positive by (*)

What you think about?

So

u^v = βw

v^w = γu

w^u = λv

where λ, β, and γ > 0 (*)

(u^v, v^w,w^u) = (βw,γu,λv)

\begin{vmatrix}

0 & 0 & β \\

γ & 0 & 0 \\

0 & λ & 0 \\

\end{vmatrix}

This determinant is positive by (*)

What you think about?