If $\phi$ is a orientation preserving linear automorphism, do we really need to choose the same orientation for the domain and the codomain?

Question

In the book of Linear Algebra by Werner Greub, at pages $131-132$, it is given that

and

However, when $\phi$ is a linear automorphism, it says that $\Delta_F = \Delta_E$, and derives the following result, however, as far as I see, there is no obligation for such thing, so is this really just an assumption of the author, or do we really need to set $\Delta_F$ as $\Delta_E$ ? Note that the author is saying that "Then we have $\Delta_F = \Delta_F$", which totally confuses me.

I mean I tried with the case where $\Delta_F$ and $\Delta_E$ represents different orientations for $E$, and the result also changed with this choice, so is there anything I'm missing ? and is my results are correct ?

Answer 1

In discussing orientation of linear transformations, one (naturally and customarily) fixes an orientation for each vector space under consideration. Consequently, if $E = F$, then $\Delta_{E} = \Delta_{F}$.