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 Play Australia. August 16 2008 11:52. Posts 520 | Profile Blog | |
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 travis United States. August 16 2008 11:54. Posts 11643 | Profile Blog | |
| | Life will soon have passed you by. Don't take for granted the beauty that is all around you in each and every moment. |
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| Play Australia. August 16 2008 11:55. Posts 520 | Profile Blog |
On August 16 2008 11:54 travis wrote:
the blue pill
uh oh |
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 crabapple Afghanistan. August 16 2008 12:05. Posts 273 | Profile Blog |
sry i cna't give u an answer, but im pretty sure stuff like this is stated explicitly in some chapter's instructions. usually they will have a compilation of properties. cause matrices and linear algebra is basically a long chain of, "this means this which also means this which is the same as saying 5 other things"
my guess would be that the transform is reversible. |
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| ShOoTiNg_SpElLs Korea (South). August 16 2008 12:08. Posts 678 | Profile |
| Well since the determinant isn't 0, you know the matrix isn't singular. So then the transformation represented by M is invertible, i.e. it is one to one and onto. Last edit: 2008-08-16 12:15:49 |
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| Play Australia. August 16 2008 12:24. Posts 520 | Profile Blog |
On August 16 2008 12:08 ShOoTiNg_SpElLs wrote:
Well since the determinant isn't 0, you know the matrix isn't singular. So then the transformation represented by M is invertible, i.e. it is one to one and onto.
so basically...its just going to transform points from (x,y) to (x', y' ...? |
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| ShOoTiNg_SpElLs Korea (South). August 16 2008 12:33. Posts 678 | Profile |
| Each point (x, y) is mapped to a unique point (x', y', yes (and there exists a transformation which maps (x', y' back to (x, y)). |
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 BottleAbuser Korea (South). August 16 2008 12:37. Posts 1386 | Profile Blog |
A "one to one" relationship means that for every point BEFORE being transformed only goes to one unique point AFTER being transformed, and also that every point that has already been transformed can only get there by transforming a unique starting point.
Onto means that for every point in the codomain (of points) that the function maps to, there is also a point in the domain (starting point) that will yield that after-transformed point.
-_- forgot all the correct terminology |
| | (Love ∈ Life) → ¬((Best Things in Life are Free) ∧ (You Get What You Pay For) → (LIFE SUCKS)) |
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 Hittegods Stockholm. August 16 2008 13:02. Posts 3224 | Profile | |
| | This neo violence, pure self defiance |
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 overpool United States. August 16 2008 13:40. Posts 150 | Profile |
From WikiDeterminants are used to characterize invertible matrices (i.e., exactly those matrices with non-zero determinants)
So basically, since the determinant != 0, there is an "inverse matrix" that can "undo" any transformation. That's the best I can come up with...Last edit: 2008-08-16 13:56:46 |
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| wesbrown United States. August 16 2008 14:22. Posts 31 | Profile |
Suppose the matrix M represents the linear transformation T:V->V, and suppose S is some object in V with hyper-volume A. Then if you apply the linear transformation T to S by T(S)=S', S' has hyper-volume |det(M)|*A. (A negative determinant means the orientation of S is reversed.)
So, in the 2x2 case, if the determinant of M is 1, the linear transformation preserves the orientation and area of all objects in R^2 (or whatever vector space you're working in). For example, the unit square having coordinates (0,0), (0,1), (1,0), and (1,1) is transformed into a quadrilateral with coordinates (0,0), (2,3), (-1,-2), and (1,1) using the matrix you gave. It's easy to check the the area of the quadrilateral is 1, and the orientation is preserved as the point (1,0) is "pulled" to (-1,-2) while the point (0,1) is pulled to (2,3).
If you don't really get the orientation-reversing part, look at the matrix M'=((1,2),(2,3)), which has determinant -1. Transform the unit square using both M and M' and note the location of the corner points. (You already know them under M. For M', they are (0,0), (2,3), (1,2), and (3,5), in the same order as before.) If you think about the movement of the points under M and M', you should notice that (0,1) and (1,0) occupy different relative positions after the transformations. I apologize if this isn't very clear, but you should get it eventually if you draw the pictures.
Edit: I spent 2 minutes in Paint. Notice how the red and blue dots have switched sides with the det=-1 transformation.
![[image loading]](http://img377.imageshack.us/img377/408/lintransvq0.png) Last edit: 2008-08-16 14:35:02 |
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 Grobyc Canada. August 16 2008 15:05. Posts 5315 | Profile Blog |
| holy shit im glad i dropped math for next year T_T |
| | "When you play, you have to start off with a mindset to turn the game into a rape" - iloveoov |
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![[image loading]](http://img292.imageshack.us/img292/5836/90517926tc4.png)
