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happyg1
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Homework Statement
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].
Homework Equations
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.
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