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- terminology - What does isomorphic mean in linear algebra . . .
Here an isomorphism just a bijective linear map between linear spaces Two linear spaces are isomorphic if there exists a linear isomorphism between them
- what exactly is an isomorphism? - Mathematics Stack Exchange
An isomorphism is a particular type of map, and we often use the symbol $\cong$ to denote that two objects are isomorphic to one another Two objects are isomorphic there is a $1$ - $1$ map from one object onto the other that preserves all of the structure that we're studying That second part is important, but it's often implied from context
- What does it mean when two Groups are isomorphic?
For sets: isomorphic means same cardinality, so cardinality is the "classifier" For vector spaces: isomorphic means same dimension, so dimension (i e , cardinality of a base) is our classifier I is a bit more complex but still not too difficult (you'll probably encounter it in your book sooner or later) to classify finite abelian groups
- Isomorphic groups beyond the isomorphism: is this also true for . . .
Each isomorphism has an inverse, which is also an isomorphism between the groups So yes: "being isomorphic" goes beyond the isomorphism in that strict sense What we mean when we say two things in mathematics, not just group theory, are isomorphic is that the algebraic structure remains the same up to a relabelling of all the constituent parts Consider, for example, the classical
- abstract algebra - What is exactly the meaning of being isomorphic . . .
11 I know that the concept of being isomorphic depends on the category we are working in So specifically when we are building a theory, like when we define the natural numbers, or the real numbers, or geometry, I often hear that people say that such a structure is complete in the sense that any other set that satisfy their properties is
- What is the difference between homomorphism and isomorphism?
Isomorphisms capture "equality" between objects in the sense of the structure you are considering For example, $2 \mathbb {Z} \ \cong \mathbb {Z}$ as groups, meaning we could re-label the elements in the former and get exactly the latter This is not true for homomorphisms--homomorphisms can lose information about the object, whereas isomorphisms always preserve all of the information For
- Whats an Isomorphism? - Mathematics Stack Exchange
So all that treating isomorphic objects as equal is completely justified in homotopy type theory By itself that wouldn't be that exciting, but homotopy type theory is a (fairly minor in some ways) extension of Martin Löf type theory which has been studied by type theorists and computer scientists and implemented for decades
- Whats the difference between isomorphism and homeomorphism?
I think that they are similar (or same), but I am not sure Can anyone explain the difference between isomorphism and homeomorphism?
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