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Linear map problem

  1. Feb 10, 2013 #1
    1. The problem statement, all variables and given/known data
    Let V be a vector space over the field F. and T [itex]\in[/itex] L(V, V) be a linear map.
    Show that the following are equivalent:

    a) Im T [itex]\cap[/itex] Ker T = {0}
    b) If T[itex]^{2}[/itex](v) = 0 -> T(v) = 0, v[itex]\in[/itex] V

    2. Relevant equations



    3. The attempt at a solution
    Using p -> (q -> r) <-> (p[itex]\wedge[/itex]q) ->r
    I suppose Im T [itex]\cap[/itex] Ker T = {0} and T[itex]^{2}[/itex](v) = 0.
    then I know that T(v)[itex]\in[/itex] Ker T and T(v)[itex]\in[/itex] Im T
    so T(v) = 0.

    I need help on how to prove the other direction.
     
  2. jcsd
  3. Feb 10, 2013 #2

    vela

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    Can you prove {0} ⊂ Im T ∩ Ker T? If so, all you have left to show is Im T ∩ Ker T ⊂ {0}.
     
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