Understand homomorphisms from Z^a --> Z^b

In summary, the conversation discusses understanding homomorphisms ##\alpha: Z^a -> Z^b## and their matrix representations. It is mentioned that the bases for ##Z^a## and ##Z^b## are denoted as ##(e_1,e_2,...,e_a)## and ##(f_1,f_2,...,f_b)## respectively. The speaker also asks for insights and clarifies whether the discussion is about ring, group, or ##\mathbb{Z}-##modules. They also inquire about defining the transformation of elements ##(0,\ldots,1,\ldots)## in terms of ##\mathbb{Z}-##modules. It is noted that
  • #1
PsychonautQQ
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10
I want to understand all possible homomorphisms ##\alpha: Z^a -> Z^b## as well as understand what a matrix representation for an arbitrary one of these homomorphisms would look like. Furthermore, under what conditions does a homomorphism have a matrix representation?

To begin, let ##(e_1,e_2,...,e_a)## be a bases for ##Z^a## and ##(f_1,f_2,...,f_b)## be a basis for ##Z^b##.

And eh, yeah, can someone give me some insights into my inquiries? Thanks!
 
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  • #2
Ring, group or ##\mathbb{Z}-##modules? The elements ##(0,\ldots,1,\ldots)## are the generators, so how about defining what happens to them? As ##\mathbb{Z}-##modules this isn't very different from ordinary vector space transformations, just with integer matrices.
 

Related to Understand homomorphisms from Z^a --> Z^b

1. What is a homomorphism?

A homomorphism is a mathematical function that preserves the structure of a mathematical object. In other words, it maps one mathematical object onto another in a way that preserves the operations and relationships between elements.

2. How is a homomorphism defined in the context of Z^a and Z^b?

In the context of Z^a and Z^b, a homomorphism is a function that maps each element in Z^a to an element in Z^b in a way that preserves the addition and multiplication operations between elements.

3. What is the significance of understanding homomorphisms from Z^a to Z^b?

Understanding homomorphisms from Z^a to Z^b allows us to better understand the relationship between different mathematical objects. It also allows us to prove theorems and solve problems by using homomorphisms as a tool.

4. Can a homomorphism from Z^a to Z^b be bijective?

Yes, a homomorphism from Z^a to Z^b can be bijective. This means that the function is both injective (one-to-one) and surjective (onto), and it preserves the operations and relationships between elements.

5. Are there different types of homomorphisms from Z^a to Z^b?

Yes, there are different types of homomorphisms from Z^a to Z^b, including group homomorphisms, ring homomorphisms, and module homomorphisms. Each type of homomorphism has its own specific properties and applications.

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