Showing null space and range are invariant


by adottree
Tags: invariant, null, range, showing, space
adottree
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#1
Apr20-08, 02:37 PM
P: 4
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?
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Pere Callahan
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#2
Apr20-08, 04:44 PM
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Quote Quote by adottree View Post
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
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#3
Apr23-08, 06:53 AM
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Quote Quote by adottree View Post
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 ?


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