What determines the orientation of a vector space?

[yifli]

A non-zero alternating tensor w splits the bases of V into two disjoint groups, those with $\omega(v_1,\cdots,v_n)>0$ and those for which $\omega(v_1,\cdots,v_n)<0$.

So when we speak of the orientation of a vector space, we need to say the orientation with respect to a certain tensor, correct?

 

[micromass]

Hi yifli!

You are correct, specifying the tensor will specify the orientation of the vector space.
However, what we usually do is specifying the positive bases directly. In $\mathbb{R}^3$, for example, these bases are determined by the right-hand rule. Also note that specifying a positive basis, is equivalent to specifying a certain tensor (since there exist a unique tensor that sends this basis to 1).