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

Invariance of Transformations

  1. Oct 12, 2009 #1
    Hey guys,

    I was wondering how you would go about proving that the image of a transformation T, im(T), is invariant? And following that, how would you prove T(W1 [tex]\bigcap[/tex] W2) is invariant if T(W1) and T(W2) are both invariant.

    On an unrelated note, another questions asks to show that
    TX = X - (P^-1 * X * P) is a linear operation, but no matter what I do, I always come up with it showing that it's in fact not a linear operation. What do you guys think?

    Thanks a lot for any help ^_^.
     
  2. jcsd
  3. Oct 12, 2009 #2

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    Proving the image of a transformation is invariant is pretty straightforward from the definition. Where are you getting stuck?
     
  4. Oct 12, 2009 #3
    Well we haven't yet covered what an invariant transformation is, but the assignment is due before the next lecture :/. Any advice on where to start?
     
  5. Oct 12, 2009 #4

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    A subspace W of a vector space V is called invariant under T if T(W) is a subset of W (so T can be defined as a function from W to W also). Invariance isn't describing the function, it's describing the subspace.
     
  6. Oct 12, 2009 #5
    With that in mind though, how do you show that T(im(T) = im(T)? That is what we're trying to show, right? And what about the intersection of W1 and W2?
     
  7. Oct 12, 2009 #6

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    You want to show that T(im(T)) is a subset of im(T). So if x is in T(im(T)) then x is in im(T). Can you see why this is true?
     
  8. Oct 13, 2009 #7
    I have a loose grasp on why that may be, but what's the formal proof?
     
  9. Oct 16, 2009 #8
    After another week of classes, he still hasn't addressed invariant transformations. I understand that the image is to the column space as the kernel is to the nullspace for a transformations. That is, the image is what is the output of a given transformation. Yet I still don't see how T(im(T)) = im(T). The same goes for showing that the intersection of two spaces is also T invariant. Any help would be much appreciated!
     
  10. Oct 17, 2009 #9

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    You don't want that T(im(T)) = im(T), you just want that it is a subset! Consider the rewording of the question: Show that any element in im(T), say y, has the property that T(y) is in im(T). Do you see why this is always true?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Invariance of Transformations
  1. Invariant subspaces (Replies: 6)

  2. Invariant Subspace (Replies: 6)

  3. Invariant subspace (Replies: 3)

Loading...