Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Showing null space and range are invariant

  1. Apr 20, 2008 #1
    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?
  2. jcsd
  3. Apr 20, 2008 #2
    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?
  4. Apr 23, 2008 #3


    User Avatar
    Staff Emeritus
    Science Advisor

    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 ?
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Showing null space and range are invariant
  1. Null space (Replies: 8)

  2. The null space (Replies: 3)