User:IssaRice/Linear algebra/Determinant as signed volume of transformation: Difference between revisions
(Created page with " Let <math>f : \mathbf R^n \to \mathbf R^n</math> be some function (not necessarily linear) and let <math>\Omega \subseteq \mathbf R^n</math> be some region in space. we will...") |
No edit summary |
||
| Line 1: | Line 1: | ||
Let <math>f : \mathbf R^n \to \mathbf R^n</math> be some function (not necessarily linear) and let <math>\Omega \subseteq \mathbf R^n</math> be some region in space. we will assume we can assign some "volume" to <math>\Omega</math>, e.g. by cutting it up into little cubes and adding up the volumes of the cubes (the volume of a cube is just the product of its edge lengths). | Let <math>f : \mathbf R^n \to \mathbf R^n</math> be some function (not necessarily linear) and let <math>\Omega \subseteq \mathbf R^n</math> be some region in space. we will assume we can assign some "volume" to <math>\Omega</math>, e.g. by cutting it up into little cubes and adding up the volumes of the cubes (the volume of a cube is just the product of its edge lengths). | ||
since f takes this space to itself, the image of <math>\Omega</math> under f, denoted <math>f(\Omega)</math>, is another region in space. let's assume f is nice enough that we can assign a volume to <math>f(\Omega)</math>. we can now ask, what is the volume of <math>f(\Omega)</math>? is it related to the volume of <math>\Omega</math> somehow? does the volume change if we translate <math>\Omega</math>, stretch it, rotate it, etc.? | since f takes this space to itself, the image of <math>\Omega</math> under f, denoted <math>f(\Omega)</math>, is another region in space. let's assume f is nice enough that we can assign a volume to <math>f(\Omega)</math>. we can now ask, what is the volume of <math>f(\Omega)</math>? is it related to the volume of <math>\Omega</math> somehow? does the volume change if we translate <math>\Omega</math>, stretch it, rotate it, etc.? | ||
in general, i don't think we can say anything too interesting here. (consider a quadratic function, where distance to the origin changes how much the volume changes.) but if we restrict attention to the following functions: for all <math>x \in \mathbf R^n</math> and all <math>\Omega \subseteq \mathbf R^n</math>, <math>\operatorname{vol} f(x+\Omega) = \operatorname{vol} f(\Omega)</math>. in other words, f alters volume "globally" in the sense that no matter where you place <math>\Omega</math> in space, the deformed volume is the same. | |||
==References== | ==References== | ||
Revision as of 02:43, 28 December 2019
Let be some function (not necessarily linear) and let be some region in space. we will assume we can assign some "volume" to , e.g. by cutting it up into little cubes and adding up the volumes of the cubes (the volume of a cube is just the product of its edge lengths).
since f takes this space to itself, the image of under f, denoted , is another region in space. let's assume f is nice enough that we can assign a volume to . we can now ask, what is the volume of ? is it related to the volume of somehow? does the volume change if we translate , stretch it, rotate it, etc.?
in general, i don't think we can say anything too interesting here. (consider a quadratic function, where distance to the origin changes how much the volume changes.) but if we restrict attention to the following functions: for all and all , . in other words, f alters volume "globally" in the sense that no matter where you place in space, the deformed volume is the same.
References
- https://www.youtube.com/watch?v=xX7qBVa9cQU -- this is probably the best explanation of the determinant i have ever seen
- sergei treil's linear algebra done wrong has a pretty good explanation. in particular, i like how he first defines determinant for a list of vectors.