Understanding Isomorphisms in Linear Transformations?

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In summary, the conversation is about understanding the concept of isomorphism in linear transformations. The individual needs help understanding how to show that a linear transformation is an isomorphism with another linear transformation. They have already shown that the Kernal of one transformation is the same as the Range of another transformation. Isomorphism refers to two vector spaces being bijective and invertible through a linear transformation. It is not possible for a linear transformation to be an isomorphism with another linear transformation, as they would be isomorphisms on their own. Posting the question would provide more clarity on the issue.
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Dgray101
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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.
 

Related to Understanding Isomorphisms in Linear Transformations?

What is an isomorphism?

An isomorphism is a mathematical concept that describes a one-to-one correspondence between two mathematical objects, where the structure and relationships between the objects are preserved.

How do you determine if two mathematical objects are isomorphic?

To determine if two mathematical objects are isomorphic, you must compare their structures and relationships. This involves examining the elements, operations, and properties of each object and checking if they are equivalent. If they are, then the objects are isomorphic.

What is the importance of determining isomorphisms?

Determining isomorphisms allows us to identify and understand similarities between different mathematical objects. This can help us solve problems more efficiently and make connections between seemingly unrelated concepts.

Can any two mathematical objects be isomorphic?

No, not all mathematical objects can be isomorphic. The objects must have similar structures and relationships in order for an isomorphism to exist.

How can isomorphisms be used in real-world applications?

Isomorphisms have applications in various fields such as computer science, chemistry, and biology. They can be used to model and analyze complex systems, identify patterns, and make predictions.

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