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Image and kernel of iterated linear transformation intersect trivially

  1. Jul 15, 2012 #1
    1. The problem statement, all variables and given/known data
    Given a linear transformation f:V -> V on a finite-dimensional vector space V, show that there is a postive integer m such that im(f^m) and ker(f^m) intersect trivially.


    2. Relevant equations



    3. The attempt at a solution
    Observe that the image and kernel of a linear transformation f:V -> V are each subspaces of V.
    The kernels of the successive iterations of the transformation form an increasing chain of subspaces.
    The images of the successive iterations of the transformation form a decreasing chain of subspaces.
    Since V is finite-dimensional, the chains of the kernels and images must eventually stabilize, say at m1 and m2 respectively. Let m be max{m1,m2}. So the kernel and image of the transformation are stable after m iterations.
    Now let v be some element contained in both the image and the kernel of f^m.
    This means that some element w in V with f^m(w) = v, and that f^m(v) = 0.

    I need to obtain a contradiction, but I have been unsuccessful so far.

    Thanks for your help.
     
  2. jcsd
  3. Jul 15, 2012 #2

    Dick

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    If the image is stabilized then f^m is a 1-1 map of Im(f^m)->Im(f^m). f^m(f^m(w))=f^m(v)=0. Seems like there is a contradiction there to me if you assume v is nonzero. What is it?
     
    Last edited: Jul 16, 2012
  4. Jul 18, 2012 #3
    Thank you, Dick. I am kind of dumb, so it took me a while to see what you meant.
     
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