Understanding Isomorphisms for Linear Transformations

In summary, a linear transformation is an isomorphism if its associated matrix is invertible. This means that if the determinant of the matrix is 0, then the transformation is not invertible and therefore not an isomorphism. It appears that the statement and conclusion are correct, with the exception of a typo in which "invertable" should be spelled as "invertible".
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Homework Statement


I have a question about isomorphisms -- I'm not sure if this is the right forum to post this in though.

A linear transformation is an isomorphism if the matrix associated to the transformation is invertable. This means that if the determinant of a transformation matrix = 0, then the transformation is not invertable and thus not an isomorph.

Just wondering if this statement / conclusion is correct? Thanks :)

Homework Equations

The Attempt at a Solution

 
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  • #2
says said:

Homework Statement


I have a question about isomorphisms -- I'm not sure if this is the right forum to post this in though.

A linear transformation is an isomorphism if the matrix associated to the transformation is invertable. This means that if the determinant of a transformation matrix = 0, then the transformation is not invertable and thus not an isomorph.

Just wondering if this statement / conclusion is correct? Thanks :)

Homework Equations

The Attempt at a Solution

Looks correct (as you talk about matrices, the implicit assumption is that we are talking about a finite dimensional vector space).

EDIT: there is a typo: it must be invertible.
 
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FAQ: Understanding Isomorphisms for Linear Transformations

1. What is the definition of isomorphism?

Isomorphism is a concept in mathematics and computer science that refers to a mapping or transformation between two objects or systems that preserves their structure and properties. In other words, the two objects are essentially identical in terms of their underlying structure.

2. How is isomorphism different from homomorphism?

While both isomorphism and homomorphism involve mappings between objects, isomorphism specifically refers to mappings that preserve structure and properties, while homomorphism refers to mappings that preserve operations and relationships between objects.

3. What are some examples of isomorphism?

One example of isomorphism is in group theory, where two groups may be isomorphic if they have the same structure and operations, but their elements may have different names or labels. Another example is in graph theory, where two graphs may be isomorphic if they have the same number of vertices and edges, and their connections or edges are the same.

4. How is isomorphism used in science and technology?

Isomorphism has many applications in various fields of science and technology. In mathematics, it is used to identify and classify different structures and systems. In computer science, it is used in the design of algorithms and data structures. It is also used in fields such as chemistry, biology, and physics to study and understand the underlying structure and properties of different systems.

5. Can two objects be isomorphic if they are not exactly the same?

Yes, two objects can be isomorphic even if they are not exactly the same. Isomorphism is based on the underlying structure and properties of the objects, not their physical appearance or specific elements. As long as the two objects have the same structure and properties, they can be considered isomorphic.

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