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Orientation of a vector space

  1. Jun 20, 2011 #1
    A non-zero alternating tensor w splits the bases of V into two disjoint groups, those with [itex]\omega(v_1,\cdots,v_n)>0[/itex] and those for which [itex]\omega(v_1,\cdots,v_n)<0[/itex].

    So when we speak of the orientation of a vector space, we need to say the orientation with respect to a certain tensor, correct?
  2. jcsd
  3. Jun 20, 2011 #2
    Hi yifli! :smile:

    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 [itex]\mathbb{R}^3[/itex], 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).
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