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One often reads about oriented volumes as a motivation for determinants. In two dimensions you can scetch some nice pictures which may convince students that it is a good idea to have a closer look at alternating multi-linear functionals.

However, my feeling is that you are cheating because, if you come to the point where you really define orientations of vector spaces you use determinants.

Hence the question: Is there a definition of orientation which does not rely one determinants?

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    $\begingroup$ Are you really using determinants? Why not define an orientation as the choice of a connected component of the torsor of real frames? $\endgroup$
    – Paul Reynolds
    Commented Jul 17, 2015 at 18:28
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    $\begingroup$ @PaulReynolds : Could you explain what a frame is? Or perhaps a real frame? And then make your comments into an answer? ${}\qquad{}$ $\endgroup$
    – Michael Hardy
    Commented Jul 17, 2015 at 18:32
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    $\begingroup$ It suffices to distinguish between the two different orientations. An orientation is associated with a basis of the vector space, commonly known as a frame. So start with a frame and declare it to have positive orientation. Any other frame has positive orientation, if there exists a continuous path in the space of all frames joining the first frame to the other and negative orientation otherwise. In other words, orientation identifies the two connected components of $GL(n)$. The fact that there are two separate components and that orientation is well-defined is proved using the determinant. $\endgroup$
    – Deane Yang
    Commented Jul 17, 2015 at 19:18
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    $\begingroup$ Don't forget that a zero-dimensional vector space has only one frame but still has two orientations. (Sorry.) $\endgroup$
    – Tom Goodwillie
    Commented Jul 17, 2015 at 19:43
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    $\begingroup$ @DeaneYang Your suggestion thus uses determinants (to show that $GL(n)$ has two path components) in the definitionof orientation. $\endgroup$
    – Jochen Wengenroth
    Commented Jul 18, 2015 at 4:44

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