Trying to find an isometry T(X)=MX

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In summary, the speaker is trying to find an isometry that satisfies the equation T(aU+bV)≠aT(U)+bT(V). They have tried different possibilities, including using a matrix M without an inverse, but have not found a suitable solution. They are seeking help and it is suggested that they look into the Mazur-Ulam theorem or consider using a translation function, such as T(x) = x+w, which is an isometry but not linear.
  • #1
viviane363
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Hi, I have a question:
I am trying to find an isometry such that T(aU+bV)≠aT(U)+bT(V).
I have tried so many possibilities. I gave T(X)=MX given that M is a matrix that doesn't have an inverse. But i still can't find a nice matrix that will make the proposition possible.
Help please
 
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  • #2


viviane363 said:
Hi, I have a question:
I am trying to find an isometry such that T(aU+bV)≠aT(U)+bT(V).
I have tried so many possibilities. I gave T(X)=MX given that M is a matrix that doesn't have an inverse. But i still can't find a nice matrix that will make the proposition possible.
Help please
If T is given by a matrix, T(X)=MX, then T(aU+bV)=aT(U)+bT(V) is always true, isometry or not. You might like to search for the Mazur-Ulam theorem.
 
  • #3


Maybe a translation? T(x) = x+w for some fixed vector w. It is an isometry. It is not linear (as you request). But it is affine.
 

1. What is an isometry?

An isometry is a type of transformation that preserves the distance between points in a given space. In other words, when an isometry is applied to a figure or object, the resulting image is congruent to the original figure or object.

2. How is an isometry represented mathematically?

An isometry can be represented by a matrix, denoted as M. This matrix has the property that when it is multiplied by a vector representing a point in the original space, the resulting vector represents the corresponding point in the transformed space.

3. What is the purpose of finding an isometry T(X)=MX?

The purpose of finding an isometry T(X)=MX is to be able to transform a given figure or object into a congruent image. This can be useful in various fields such as computer graphics, architecture, and engineering.

4. How do you determine the correct matrix M for a given isometry?

The correct matrix M for a given isometry can be determined by using the properties of isometries. For example, if the isometry is a translation, the matrix M will have the translation vector as its last column. If the isometry is a reflection, the matrix M will have a combination of 1s and -1s along the main diagonal and 0s elsewhere.

5. Can any figure or object be transformed using an isometry?

Yes, any figure or object can be transformed using an isometry as long as the transformed image is congruent to the original figure or object. However, the specific isometry used may vary depending on the desired transformation.

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