Greub, Linear Algebra, introduces the concept of Oriented vector spaces. Basically, given a skew symmetric $n$-linear form on a $n$ dimensional space (a determinant form) two determinant forms $\Delta$ and $\Delta'$ are equivalent iff $$ \Delta=\alpha \Delta' \quad\to\quad \alpha > 0. $$ A basis is positive (for a given orientation) iff $\Delta(v_1,\cdots,v_n)>0$.
Q1 Is the dual basis positive iff the basis is positive?
I think yes. The dual determinant form is defined by $$ \Delta^*(v^1,\cdots, v^n)\Delta(v_1,\cdots, v_n)= \det\big((v^iv_j)\big) $$ which for dual basis becomes (Am I correct?) $$ \Delta^*(v^1,\cdots, v^n)\Delta(v_1,\cdots, v_n)= 1, $$ thus one is positive iff the other also is.
Q2 In the following screen shot I do not understand the sentence and not on the choice of the representing determinant functions[=determinant form]
If I change $\Delta_E$ to $\Delta'_E:=-\Delta_E$, then $\lambda$ has to change sign to preserve the equality? Probably the sentence is correct, but I miss something conceptual.