Timeline for Continuous change of basis (and on the definition of determinant)
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Nov 12, 2012 at 13:35 | comment | added | paul garrett | Calculus only defines volumes given a coordinate system... At the very least, we'd presume that changing coordinates in the most trivial possible, but real-life, fashion, namely, "rigid rotations" (and translations) oughtn't change "volume". | |
| Nov 12, 2012 at 5:52 | comment | added | Gabriel Nivasch | Calculus already has a definition of volume, which is by Riemann sums, which boils down to dividing space into a grid of tiny cubes and counting how many cubes are inside your shape. So why should Linear Algebra come and introduce a different notion of volume?? At the very least one needs to prove that the two definitions are equivalent! | |
| Nov 12, 2012 at 3:22 | comment | added | paul garrett | ... so, although many students do need "checks" on reckless and thoughtless facile conclusions, my experience also indicates that many have too-well "learned" to distrust even qualitative conclusions based on prior experience. My specific methodological point is that, although, of course, one must always be cautious, there is nothing else upon which to base speculation than... the past, and prior experience. So, to distrust that, or, worse, to be taught to distrust prior experience too strongly, is simply defeatist. | |
| Nov 12, 2012 at 3:20 | comment | added | paul garrett | I might disagree in a very mild way to @AndreasBlass' comment about, in effect, legitimizing ways of working with new situations. Of course I understand the methodological point, that, on one hand, one should mistrust an untutored intuition. However, there is not a unique way to extend intuition/sensibilities from the familiar to the novel. I truly do think that it is defensible to claim that, whatever else may be going on, rigid rotations preserve volume, for example, even if we admit that we are beyond 3 dimensions, etc. ... (cont'd) | |
| Nov 12, 2012 at 2:04 | comment | added | Andreas Blass | In a first course in linear algebra, I admit that we have a prior notion of volume (and a prior notion of orientation) in dimensions up to 3. I do not claim to have, nor do I suggest that they should have, such prior notions in higher dimensions. I develop the basic properties of determinants in low dimensions by treating them as oriented volumes (pointing out that including the orientation makes the algebra cleaner). Then I extend the algebra to higher dimensions and tell them that this extension can be used to define volume and orientation in spaces where we lack intuition. | |
| Nov 11, 2012 at 22:01 | history | answered | paul garrett | CC BY-SA 3.0 |