Timeline for Continuous change of basis (and on the definition of determinant)
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Nov 13, 2012 at 6:10 | comment | added | Gabriel Nivasch | quid: You understood my intent. Thanks! | |
| Nov 12, 2012 at 19:52 | comment | added | user9072 | [...] that you might have an easier time to find what you seem to be looking for in multivariate analysis books (as opposed to linear algebra ones). But then also some Lin Alg books contain a discussion of this, Axler's Lineare Algebra Done Right is an example I believe. In still other words, you should be able to find your final 'I think...' worked out in such books in one way or another were the volume is defined via some grid of cubes to stay in your wording. Or I misunderstood your intent entirely. | |
| Nov 12, 2012 at 19:36 | comment | added | user9072 | @Gabriel Nivasch: Did I say that? I don't think so. I stressed that at least the proofs but possibly also a result will show the equality of the two. Typically there will be a definition of some notion of volume via Riemann integral say, as you mention yourself. And there will be a transformation formula with some determinant inside. Apply this formula for the unit cube under a linear map, shows that the volume of the parallepiped (the image of the unit cube under this linear map) and the volume of the unit cube are linked via the determinant. My point is [...] | |
| Nov 12, 2012 at 5:45 | comment | added | Gabriel Nivasch | @quid: I don't have those books in front of me, but I really hope they don't just say that the determinant was shown to equal the volume in Linear Algebra... | |
| Nov 11, 2012 at 22:25 | comment | added | Margaret Friedland | @quid: transformation-of-variable formula in integral calculus says the same thing as the above, just at the particular context of cotangent spaces. | |
| Nov 11, 2012 at 21:21 | comment | added | user9072 | @Gabriel Nivasch: regarding the volume question, if you check (standard) books on multivariate analysis you will find transformation formulas for the higher-dim integrals that involve a determinant. The proofs (or possibly some result) will show that you could define the volume differently (via integrating the shape) and then show it matches a differently defined determiannt. | |
| Nov 11, 2012 at 21:10 | comment | added | Gabriel Nivasch | Of course I'm not going to define a topology in my course, and I'm not going to define volume either. But I want to know whether to tell my students "The determinant can be shown to equal the volume" or "The volume is the determinant by definition". And regarding the "orientation" thing, I can start by arguing that in the plane one cannot continuously transform $(e_1,e_2)$ into $(e_2,e_1)$ without making the vectors parallel or anti-parallel at some point; then go to 3-space and discuss the same for right-handed and left-handed bases; then discuss the general notion of orientation. | |
| Nov 11, 2012 at 19:59 | comment | added | Gabriel Nivasch | Isn't the volume normally defined by dividing space into an infinite grid of tiny cubes, counting how many cubes are inside, and taking the limit? | |
| Nov 11, 2012 at 19:35 | comment | added | Margaret Friedland | Usually there is more than one way to define a mathematical notion, so one can proceed one way or another, depending on the purpose. How do you propose to define the volume? And certainly it is possible to try to define orientation using continuous motions, but this would require defining topology on the space of matrices and to count the pathwise components in it. It would be interesting, but if I was teaching an introductory linear algebra course, I would not try it on the students... | |
| Nov 11, 2012 at 19:01 | comment | added | Gabriel Nivasch | Yes, I know those are the definitions, and that's exactly my question: Must the volume of the parallelepiped be defined in terms of the determinant? Can't they be shown to be equal? And must the orientation be defined in terms of the sign of the determinant? Can't it be defined more naturally in terms of continuous motion, and then show that there are really only 2 equivalence classes which are given by the sign of the determinant? | |
| Nov 11, 2012 at 18:30 | history | answered | Margaret Friedland | CC BY-SA 3.0 |