Homomorphisms and isomorphisms

In summary, the problem is asking to prove that T is a homomorphism and determine when it is an isomorphism. For T to be an isomorphism, the elements w_i must form a basis of V. Additionally, if vT=0, then the linear combination of the w_i that v is sent to must also equal zero.
  • #1
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Homework Statement



Let[tex]v_1,v_2,...v_n[/tex] be a basis of V and let [tex]w_1,w_2,...w_n[/tex] be any n elements in V. Define T on V by
[tex](\lambda_1 v_1+\lambda_2 v_1+...+\lambda_n v_n)T=\lambda_1w_1+...\lambda_n w_n.[/tex]
a)Show that R is a homomorphism of V into itself.
b)When is T an isomorphism?

Homework Equations


An isomorphism is a 1-1 mapping.


The Attempt at a Solution


I think I have part a) done, although It makes more sense to me to prove that T is a Hom, not R...not sure where the R came from.

For part b, I am thinking that since the [tex]v_i[/tex] are a basis that the Iso would be true if the w's were a basis also?
IK know that an iso occurs when the Kernel is zero..but I'm still kinda confused here.
Any hints or advice will be appreciated.
Thanks,
CC
 
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  • #2
Obviously, in order for T to be an iso, w_i must be a basis - the image lies in the span of the w_i, and that can only be all of V if they are a basis. That says T an iso implies w_i a basis. What about the converse?

Now, suppose that v is sent to zero by T, i.e. vT=0. Then what does that say about the linear combination of the w_i that v is sent to?
 
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What is a homomorphism?

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

What is an isomorphism?

An isomorphism is a special type of homomorphism that is bijective, meaning it is both one-to-one and onto. This means that it not only preserves the structure of a mathematical object, but also allows for a one-to-one correspondence between elements of the two structures.

What is the difference between a homomorphism and an isomorphism?

The main difference between a homomorphism and an isomorphism is that an isomorphism is bijective, while a homomorphism is not necessarily bijective. This means that an isomorphism has an inverse function, while a homomorphism does not necessarily have an inverse.

How are homomorphisms and isomorphisms used in mathematics?

Homomorphisms and isomorphisms are used in mathematics to study the structure and properties of different mathematical objects. They allow for comparisons and connections to be made between different structures, and can help simplify and solve complex problems.

What are some real-world applications of homomorphisms and isomorphisms?

Homomorphisms and isomorphisms have applications in a variety of fields, including computer science, physics, and chemistry. For example, in computer science, they are used in cryptography to encode and decode messages, while in physics, they are used to describe the symmetries of physical systems.

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