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