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

[Cyannaca]

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.

 

[Hurkyl]

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?

 

[HallsofIvy]

" 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!

 

[Cyannaca]

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.

 

[Hurkyl]

Maybe if you described them geometrically, it would spark an idea?