Ring and Linear Transformation

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Homework Help Overview

The problem involves a polynomial \( g(x) \) in the context of linear transformations and rings of operators. It asks to show the invertibility of \( g(T) \) implies that \( g^{-1}(T) \) is also in the ring generated by \( g(T) \).

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are exploring the definitions and implications of the ring \( F[T] \) and its generation by \( g(T) \). There is confusion regarding the phrasing of the problem and what it means for \( g(T) \) to generate \( F[T] \). Some participants suggest clarifying whether \( g(T) \) is meant to generate the entire ring or if it is simply an element of it.

Discussion Status

The discussion is currently focused on clarifying the terms and assumptions of the problem. Participants are questioning the original poster's phrasing and exploring the implications of the definitions involved. No consensus has been reached yet, but there is a productive dialogue about the meaning of the terms used.

Contextual Notes

There is an assumption that \( V \) is a vector space over the field \( F \), which is being discussed but not explicitly confirmed in the original problem statement.

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Let [tex]g(x)\in F[x][/tex], [tex]T\in L(V)[/tex]. Let [tex]F[T][/tex] be a ring generated by [tex]g(T)[/tex]. Show that if [tex]g(T)[/tex] is invertible, then [tex]g^{-1}(T)\in F[T][/tex].

No idea what do do.
 
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The statement of the problem is confusing. I'm going to assume V is a vector space over the field F.

Now, F[T] is not just "a" ring -- it is a very specific ring. It is (isomorphic to) the subring of L(V) that is generated by F and T.

So, I'm confused when you say that you let it be "a ring generated by g(T)".

Did you mean to say something to the effect if:

"Suppose g(T) generates F[T]"

or maybe

"Let F[g(T)] be the ring generated by g(T)"

or even just

"So g(T) is an element of F[T]"

?


(I suspect you meant the first one -- work out what that really means)
 
It's either the first or third. Let's assume the first, since I must show that the inverse of g(T) is a positive powered polynomial.
 
Okay, so what does it mean that g(T) generates F[T]? Any interesting particular cases?
 

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