Definition of positively oriented basis

Question

Let $(U_\mathbb{R},A)$ be an oriented $m$-dimensional Euclidean space $(U_\mathbb{R},A)$, where $A$ is an alternating unimodular $m$-th order tensor, and subset $B=\{\vec{u}_1,\cdots,\vec{u}_m\}$ a basis of $U_\mathbb{R}$. I know that when $B$ is orthonormal then it is called a positively oriented basis if $A(\vec{u}_1,\cdots,\vec{u}_m)=1$, but when $B$ is not orthonormal, may I say that it is a positively oriented basis if $A(\vec{u}_1,\cdots,\vec{u}_m)>0$ ?

Answer 1

Yes, that's fine. $A$ is different from zero on all bases. Orientation is choosing one of the two classes of bases according to the sign of the evaluation of $A$.