1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    Can you prove {0} ⊂ Im T ∩ Ker T? If so, all you have left to show is Im T ∩ Ker T ⊂ {0}.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook