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Shift Operator Properties

  1. Jan 31, 2012 #1
    1. The problem statement, all variables and given/known data
    http://desmond.imageshack.us/Himg810/scaled.php?server=810&filename=screenshot20120131at923.png&res=medium [Broken]

    3. The attempt at a solution
    So in particular I want to look at the last part of this problem. That is, "Show that [itex]S^n = 0[/itex]"

    I know that [itex]dim(KerS^k) = k[/itex] and therefore, [itex]dim(ImS^k)= n-k[/itex]. If k=n, [itex]dim(ImS^k)= n- n = 0[/itex] which implies that [itex]ImS^k = \left\{ 0 \right\} [/itex].

    I'm having trouble drawing from this the conclusion that [itex]S^n = 0[/itex]. Is S an isomorphism? If so, does that mean that the only thing that can map to 0 in the codomain is 0 in the domain?
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Jan 31, 2012 #2

    Dick

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    You mean [itex]ImS^n = \left\{ 0 \right\}[/itex]. That means for all vectors x, [itex]S^n(x)=0[/itex]. Doesn't that mean [itex]S^n=0[/itex]? Of course S isn't an isomorphism! It has a nonzero kernel. Not sure what's bothering you here.
     
    Last edited: Jan 31, 2012
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