Linear Operator Rules for Ker T and Ker S: Solving for T and S in R^3

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Homework Help Overview

The discussion revolves around finding the rule for a linear operator S in relation to another linear operator T defined on R^3. The original poster expresses uncertainty about how to approach the problem and questions whether S can be considered the inverse of T.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants discuss the necessity of finding the kernel of T and question the relationship between the operators T and S, particularly regarding the assumption of S being the inverse of T. Some suggest that if the kernels are non-trivial, T cannot have an inverse.

Discussion Status

There is a mix of attempts to clarify the problem and explore geometric interpretations. While some participants have made progress in finding the kernel of T, uncertainty remains about deriving the rule for S. No consensus has been reached regarding the assumptions about the relationship between T and S.

Contextual Notes

Participants are working under the constraints of the problem's definitions and the properties of linear operators, particularly focusing on the implications of kernel relationships.

Cyannaca
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Let T: R^3 -> R^3 be a linear operator and
T(x;y;z)= (x -y + 3z;2x-y+8z;3x -5y+5z).

Find the rule, for the linear operator S: R^3 -> R^3 such that ker S= I am T and I am S=Ker T.

I'm not really sure how I should start this problem. Also I would like to know if I can assume S is the inverse operator of T.
 
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Seems obvious to me that the first thing to do is to find ker T and I am T. *shrug*

Also I would like to know if I can assume S is the inverse operator of T.

What do you think?
 
" Also I would like to know if I can assume S is the inverse operator of T."

If the two kernals are not trivial, then T does not HAVE an inverse!
 
I found ker T and I am T and I'm still stuck. I know it's not the inverse operator but I still don't know how to find the rule for the linear operator S.
 
Maybe if you described them geometrically, it would spark an idea?
 

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