Finding the map between two images

[king vitamin]

Homework Statement

Let D* be the parallelogram with vertices at (-1,3), (0,0), (2,-1), and (1,2), and let D be the rectangle D = [0,1] X [0,1]. Find a T such that D is the image set of D8 under T.

Homework Equations

Not much to say other than T(D*) = D

The Attempt at a Solution

This section of the book is on mapping, and the book does not give any clues on how to find maps given images, so I'm pretty stumped. This is a vector calculus text, by the way, in case you're wondering about context.

Should I try to find a map that connects the vertices? Or one that transforms the sides? (I've tried both of these with no luck) Or is there a general method for solving this?

 

[mighty2000]

Thank you for your question. This problem can be solved by finding a linear transformation that maps the vertices of D* to the vertices of D. Here is a step-by-step solution:

1. Write out the coordinates of the vertices of D* and D:

D*: (-1,3), (0,0), (2,-1), (1,2)
D: (0,0), (1,0), (1,1), (0,1)

2. Notice that the vertices of D* are not in the same order as the vertices of D. We need to rearrange the vertices of D* so that they match the order of the vertices of D. This can be done by rotating the parallelogram 90 degrees clockwise. The new vertices of D* will be: (3,1), (0,0), (-1,2), (2,3).

3. Now, we need to find a linear transformation T that maps the vertices of D* to the vertices of D. This can be done by solving a system of equations. Let T represent the linear transformation and (x,y) represent a point in D*. Then, we have the following equations:

T(-1,3) = (0,0)
T(3,1) = (1,0)
T(-1,2) = (1,1)
T(2,3) = (0,1)

Solving this system of equations, we get the following transformation:
T(x,y) = (2x-y+1, -x+y+1)

4. Finally, we can check if this transformation actually maps D* to D. Let's take the point (-1,3) from D* and apply T to it:
T(-1,3) = (2(-1)-3+1, -(-1)+3+1) = (0,0)

This is the vertex (0,0) of D, which is exactly what we wanted. Similarly, if we apply T to the other vertices of D*, we will get the remaining vertices of D.

Therefore, the transformation T(x,y) = (2x-y+1, -x+y+1) maps D* to D.

I hope this helps and let me know if you have any further questions. Good luck with your studies!