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Math Help - Linear map problem

  1. #1
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    Linear map problem

    Let V be a vector space over the field F. and T \in L(V, V) be a linear map.

    Show that the following are equivalent:
    a) Im T \cap Ker T = {0}
    b) If T^2(v) = 0 -> T(v) = 0, v \in V


    Using p -> (q -> r) <-> (p \wedgeq) ->r
    I suppose Im T \cap Ker T = {0} and T ^{2}(v) = 0.
    then I know that T(v) \in Ker T and T(v) \in Im T
    so T(v) = 0.
    I need help on how to prove the other direction.
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  2. #2
    MHF Contributor
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    Re: Linear map problem

    Hey jdm900712.

    Can you make use of the rank-nullity theorem for your map?
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  3. #3
    MHF Contributor

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    Re: Linear map problem

    what you've done is shown a) implies b).

    now suppose we have b). so T2(v) = 0 implies v = 0.

    suppose that w is in im(T) and ker(T). since w is in ker(T), T(w) = 0. since w is in im(T), w = T(v), for some vector v.

    hence 0 = T(w) = T(T(v)) = T2(v).

    by b), this means that v must be 0. hence w = T(v) = T(0) = 0, so every element of the intersection of ker(T) and im(T) is a 0-vector, that is: ker(T)∩im(T) = {0}, which is precisely a).
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