Unique linear transformations

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SUMMARY

The discussion centers on the uniqueness of linear transformations between two vector spaces with finite, linearly independent sets of vectors. It is established that a linear transformation is unique only if both sets serve as bases for their respective vector spaces and the mapping is defined in a specific order. If the sets are not bases or if the mapping is not explicitly defined, multiple linear transformations can exist. The key method to demonstrate uniqueness involves representing the linear transformation with a matrix corresponding to the specified bases.

PREREQUISITES
  • Understanding of vector spaces and linear independence
  • Knowledge of linear transformations and their properties
  • Familiarity with matrix representation of linear transformations
  • Concept of bases in vector spaces
NEXT STEPS
  • Study the properties of vector spaces and linear independence
  • Learn about linear transformations and their uniqueness criteria
  • Explore matrix representations of linear transformations in different bases
  • Investigate examples of linear transformations between finite-dimensional vector spaces
USEFUL FOR

Mathematicians, students of linear algebra, and anyone interested in understanding the properties of linear transformations between vector spaces.

complexhuman
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Unique linear transformations!

Problems agiain :cry: :cry: :cry:

Say I have 2 vector spaces with some finite number of vectors(can assume linear independency)...how can I show that the linear transformation between the two is unique?

Thanks in advance!
 
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Hint: translate the problem using a system of equations.
 
complexhuman said:
Problems agiain :cry: :cry: :cry:

Say I have 2 vector spaces with some finite number of vectors(can assume linear independency)...how can I show that the linear transformation between the two is unique?

Thanks in advance!

Perhaps I am misunderstanding something here. Do you mean that you are given a finite, linearly independent set of vectors in each space and the linear transformation must map each vector in one set into a corresponding vector in the other?

They way you have stated it, you can't prove the linear transformation is unique- it isn't. If the two sets are not bases for their vector spaces, then the linear transformation is not unique. Even if they are bases, unless you are requiring that the linear transformation map a specific vector in one set into a specific vector in the other then different linear transformations can map one set into the other, just rearranging which vector maps into which.

Assuming that you are given a basis for one space, in a specific order, and the linear transformation must map that into a basis for the other space, also in a given order, then you can show that the linear transformation is unique. One way would be to see what matrix represents that linear transformation in those bases.
 

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