Showing null space and range are invariant

[adottree]

If V is any vector space and S and T are linear operators on V such that ST=TS show that the null space and the range of T are invariant under S.

I think I need to begin by taking an element of the range of T and having S act on it and show that it stays in V? Can you help get me started?

 

[Pere Callahan]

If V is any vector space and S and T are linear operators on V such that ST=TS show that the null space and the range of T are invariant under S.

I think I need to begin by taking an element of the range of T and having S act on it and show that it stays in V? Can you help get me started?

Showing that it stays in V is not quite what you want to do. You want to show that every element in the range of T remains in the range of T when acted upon by S.

So say, t is an element in the range of T, what can you say about this element?

 

[HallsofIvy]

If V is any vector space and S and T are linear operators on V such that ST=TS show that the null space and the range of T are invariant under S.

I think I need to begin by taking an element of the range of T and having S act on it and show that it stays in V? Can you help get me started?

If x is in the null space of T, then T(x)= 0. Therefore, ST(x)= ?. And so T(S(x))= ?

If y is in the range of T, then there exist x such that T(x)= y. So S(T(x))= S(y). But that is equal to T(S(x)). So S(y) is in ?