Decide whether each map is an isomorphism

In summary, the conversation discusses determining whether a given map is an isomorphism or not. The map in question is from M2×2 to P^3 and is defined by a polynomial expression. The participant mentions that a map is isomorph if it is both one-to-one and onto, and expresses difficulty in showing that the given map is onto due to confusion with polynomials. They request a hint for this problem.
  • #1
danni7070
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0

Homework Statement



Decide whether each map is an isomorphism (if it is an isomorphism then
prove it and if it isn’t then state a condition that it fails to satisfy).

Homework Equations



f : M2×2 ---- P^3 given by:

a b
c d --- c + (d + c)x + (b + a)x^2 + ax^3



The Attempt at a Solution



Ok, I know that map is isomorph if it is one-to-one and onto.

I know it is one-to-one but I'm having problems showing that it is onto because I get confused using polynomials!

Can somebody give me a hint?
 
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  • #2
What does it mean (for a mapping) to be onto?
 

Related to Decide whether each map is an isomorphism

1. What is an isomorphism?

An isomorphism is a mathematical concept that describes a one-to-one relationship between two sets of objects. In simpler terms, it means that two objects or structures are essentially the same, just with different labels or names.

2. How do you decide if a map is an isomorphism?

To decide if a map is an isomorphism, you must first check if it is a bijective function, meaning that each element in the domain has a unique element in the range, and vice versa. Then, you must check if the map preserves the structure and operations of the objects in both sets. If it does, then the map is an isomorphism.

3. What does it mean for a map to preserve structure and operations?

For a map to preserve structure and operations, it means that the map must maintain the relationships between elements in both sets. This includes preserving the order, size, and shape of the objects, as well as the operations performed on them.

4. Can a map be an isomorphism between two different types of objects?

Yes, a map can be an isomorphism between two different types of objects. As long as the map satisfies the criteria of a bijective function and preserves the structure and operations, it can be considered an isomorphism.

5. How is an isomorphism different from an equivalence relation?

An isomorphism is a type of equivalence relation, but not all equivalence relations are isomorphisms. An equivalence relation only requires that the objects in both sets have a certain type of relationship, while an isomorphism also requires the preservation of structure and operations.

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