Timeline for Oriented volume and determinants: Circularity [duplicate]
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Jul 20, 2015 at 6:28 | comment | added | Jochen Wengenroth | @DeaneYang Okay, this a way to avoid the circularity. Thanks. | |
| Jul 18, 2015 at 20:26 | comment | added | Deane Yang | Jochen, to quibble with this, in my definition of orientation the number of possible orientations is the number of connected components of $GL(n)$. Also, you can use the polar decomposition of invertible matrices to show that the number of components is 2. It's just that using the determinant is much easier. | |
| Jul 18, 2015 at 4:58 | history | closed |
Igor Rivin Deane Yang CommunityBot |
Duplicate of Continuous change of basis (and on the definition of determinant) [closed] | |
| Jul 18, 2015 at 4:44 | comment | added | Jochen Wengenroth | @DeaneYang Your suggestion thus uses determinants (to show that $GL(n)$ has two path components) in the definitionof orientation. | |
| Jul 17, 2015 at 22:10 | review | Close votes | |||
| Jul 17, 2015 at 22:21 | |||||
| Jul 17, 2015 at 21:38 | comment | added | Michael Hardy | I think I've seen some definition of a "frame" according to which that concept is more general that that of a basis. ${}\qquad{}$ | |
| Jul 17, 2015 at 19:43 | comment | added | Tom Goodwillie | Don't forget that a zero-dimensional vector space has only one frame but still has two orientations. (Sorry.) | |
| Jul 17, 2015 at 19:32 | comment | added | Deane Yang | But it's way easier just to define orientation directly using the determinant of the change-of-basis matrix and observing that it's well-defined. | |
| Jul 17, 2015 at 19:18 | comment | added | Deane Yang | 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. | |
| Jul 17, 2015 at 18:40 | comment | added | Eric Wofsey | Do you literally just want any definition of orientation without using determinants, or do you want one that can be used to motivate determinants to students? The former is considerably easier than the latter. | |
| Jul 17, 2015 at 18:33 | history | edited | Michael Hardy | CC BY-SA 3.0 |
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| Jul 17, 2015 at 18:32 | comment | added | Michael Hardy | @PaulReynolds : Could you explain what a frame is? Or perhaps a real frame? And then make your comments into an answer? ${}\qquad{}$ | |
| Jul 17, 2015 at 18:28 | comment | added | Paul Reynolds | Are you really using determinants? Why not define an orientation as the choice of a connected component of the torsor of real frames? | |
| Jul 17, 2015 at 17:40 | history | asked | Jochen Wengenroth | CC BY-SA 3.0 |