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

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

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  • $\begingroup$ This is a verbatim copy of @logarithm's comment above. $\endgroup$ – Carlos Esparza May 26 at 17:48

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