Definition of positively oriented basis

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 $$A(\vec{u}_1,\cdots,\vec{u}_m)>0$$ ?

• 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 $A$. – logarithm May 21 at 17:42
• @logarithm That should be an answer instead of a comment. – amd May 21 at 19:53
• Is there anything I could do? – Roberto Dias Algarte May 21 at 20:28

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 $$A$$.