Understanding Isomorphisms in Linear Transformations?

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An isomorphism in linear transformations refers to a bijective linear transformation between two vector spaces, indicating that they are structurally identical. To determine if a linear transformation is an isomorphism, one must show that it is both injective and surjective. The discussion highlights that having the kernel of one transformation equal to the range of another can be relevant in establishing isomorphism. It clarifies that linear transformations themselves cannot be isomorphic to one another but can be isomorphisms between vector spaces. Understanding these concepts is crucial for grasping the relationship between different vector spaces.
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Hey I have a problem in which I need to show that a linear transformation is an isomorphism with another linear transformation. However I don't really understand what an isomorphism is, and how you would even determine it??

I already showed that the Kernal of one transformation was the same as the Range of another transformation is that helps any? I am looking for help on the concept and idea, I wouldn't consider this to be a homework problem? I'm new here, don't really know the rules and regulations that well.
 
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Two vector spaces V and W are said to be isomorphic if there is a bijective (invertible) linear transformation T:V->W, in which case T is called an isomorphism from V to W. It doesn't make sense to speak of a linear transformation being an isomorphism with another linear transformation, they would just be isomorphisms on their own. Posting the question would help clear things up more.
 

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