# Algebriac Structures,hom wrecker

happyg1

## Homework Statement

Let $$T:V \rightarrow W$$ be a homomorphism. Using T, define a homomorphism $$T^*om(W,F) \rightarrow Hom(V,F)$$.

## The Attempt at a Solution

This is what I have so far:
let $$f \in Hom(W,F)$$
Define $$f(T^*):V \rightarrow F$$
and $$(v)((f)T^*)=((v)T)f$$

Ok so then I need to show that $$T^*$$ is acutually a homomorphism.
So I tried this:
Let $$\lambda \in T^*$$
Then
$$\lambda(f+g)T^*=\lambda(f(T^*)+g(T^*))$$
$$=\lambda((f(T^*)) + \lambda(g(T^*))$$
$$=(\lambda f)(T^*) + (\lambda g)(T^*)$$
$$=(\lambda f +\lambda g) T^*$$
$$=(\lambda f)T^* = (\lambda g)T^*$$

So $$T^*$$ is a homomorphism.

I'm not sure if this is the correct approach since I have a slippery grasp on this stuff. My Prof says I need to also show that $$f(T^*) \in Hom(V,F)$$ that doesn't seem intuituvely difficult, but my problem is WRITING IT DOWN.

Any input will be greatly appreciated.

CC

Last edited:

## Answers and Replies

Homework Helper
Precisely WHAT algebraic structures are V and W and what is F? My guess is that V and W are vector spaces over field F but I don't know for sure.

happyg1
Your are correct, V and W are vector spaces over a field F.
Sorry.

Homework Helper
Poor lad, having to write things on the wrong side like that.

I would say to your professor that it is bloody trivial bby definition that fT* is in Hom(V,F): it is a linear map from V to F.

happyg1
yeah, I don't know why we are putting things on the right like that, but we do.
I agree that $$f(T^*) \in Hom(V,F)$$ is trivial, but apparently he wants it.
Am I correct in my attempt at the other part?

Thanks

CC

Homework Helper
You're putting things on the right because you're looking at right modules for rings. There is a good reason for this, but not an interesting one for you.

fT* is in Hom(V,F) *by definition*. I.e. you defined it as a function from V to F, and the rest of you proof shows it is a linear map i.e. an element of Hom(V,F). Thus *my* response to you professor is "we define a function from V to F as follows: if v is in V then vfT*=(vT)f (IF YOU WANT MORE DETAILS, THEN vT IS IN V SINCE V IS A FUNCTION FROM V TO V, AND f IS A FUNCTION FROM V TO F. Now we show that this is actually a linear map. INSERT YOUR PROOF"

the caps aren't 'shouty'. You should think about replacing them with something more appropriate. Oh, and don't copy the rest of this into your work either. A professor can tell precisely what you write from a mathematician's writing. Put it in your language.

happyg1
Hey,
Thanks for the response.
Just so you guys know, My Prof knows me well enough to spot ANYTHING that's not precisely in my own language, so no worries.

Am I correct in my attempt that $$T^*$$ is a Hom above? I think since no one has said I'm completely wrong that I must be on the right track.

Thanks again,
CC