(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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

2. Relevant equations

3. The attempt at a solution

This is what I have so far:

let [tex] f \in Hom(W,F)[/tex]

Define [tex] f(T^*):V \rightarrow F[/tex]

and [tex](v)((f)T^*)=((v)T)f[/tex]

Ok so then I need to show that [tex]T^*[/tex] is acutually a homomorphism.

So I tried this:

Let [tex]\lambda \in T^*[/tex]

Then

[tex]\lambda(f+g)T^*=\lambda(f(T^*)+g(T^*))[/tex]

[tex]=\lambda((f(T^*)) + \lambda(g(T^*))[/tex]

[tex]=(\lambda f)(T^*) + (\lambda g)(T^*)[/tex]

[tex]=(\lambda f +\lambda g) T^*[/tex]

[tex]=(\lambda f)T^* = (\lambda g)T^*[/tex]

So [tex]T^*[/tex] 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 [tex]f(T^*) \in Hom(V,F)[/tex] that doesn't seem intuituvely difficult, but my problem is WRITING IT DOWN.

Any input will be greatly appreciated.

CC

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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