How would I prove that an isometry is one to one?

  • Thread starter wurth_skidder_23
  • Start date
  • Tags
    Isometry
In summary, to prove that an isometry is one to one, one must show that the distance between the images of two points is the same as the distance between the points, and this can be mathematically demonstrated by using the definition of distance in an inner product space and the linearity of the transformation. Additionally, one must verify all the axioms of an inner product for the transformed space.
  • #1
How would I prove that an isometry is one to one?

General definition of isometry, A:

<Ax,Ay> = <x,y>

Where < , > is an inner product (scalar product, dot product, etc.)

How do I prove A has to be one to one for this to work?
 
Physics news on Phys.org
  • #2
Note the distance between the images of two points will be the same as the distance between the points. So if two points map to the same image...
 
  • #3
Is there a way to show that mathematically using the adjoint or something similar?
 
  • #4
To show what, that distances are preserved? What is distance in an inner product space?
 
  • #5
distance between two vectors x, y in an inner product space is ||x-y||
 
  • #6
Yes, and [tex]||x-y||^2[/tex] has an expression in terms of the inner product.
 
  • #7
Okay, maybe I should state the whole problem:

Let X and Y be inner product spaces in R with inner products < , >_X and < , >_Y Suppose that T is in L(X,Y). Show that <x,y> = <Tx,Ty>_Y is an inner product on X if and only if T is one to one.
 
  • #8
The <- direction should be pretty easy, you just need to verify all the axioms of an inner product, and you'll need to use the linearity of T. I'm assuming your original question comes from proving the -> direction. It's been pretty much spelled out for you. Just use the definition of distance to show d(x,y)=d(Tx,Ty).
 

1. How do I prove that an isometry is one to one?

One way to prove that an isometry is one to one is by using the definition of an isometry, which states that it preserves distances between points. Therefore, if two points are mapped to the same point under an isometry, their distance from the mapped point should also be the same. If this is not the case, then the isometry is not one to one.

2. Can I use coordinate geometry to prove that an isometry is one to one?

Yes, coordinate geometry can be used to prove that an isometry is one to one. This can be done by showing that the coordinates of two points before and after the isometry are different. If the coordinates are the same, then the isometry is not one to one.

3. What other method can be used to prove that an isometry is one to one?

Another method that can be used to prove that an isometry is one to one is by using the concept of congruence. If two figures are congruent, then their corresponding sides and angles are equal. Therefore, if an isometry maps two congruent figures to different figures, then it is not one to one.

4. Is it possible for an isometry to be both one to one and onto?

Yes, it is possible for an isometry to be both one to one and onto. This means that it is a bijective function, where each point in the original figure is mapped to a unique point in the image, and every point in the image has a corresponding point in the original figure.

5. Can I use a counterexample to prove that an isometry is not one to one?

Yes, a counterexample can be used to prove that an isometry is not one to one. This means finding two points in the original figure that are mapped to the same point in the image, but their distances from the mapped point are not equal. This would contradict the definition of an isometry, thus proving that it is not one to one.

Back
Top