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Math Help - T-invariant

  1. #1
    Member Last_Singularity's Avatar
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    T-invariant

    Question: Let W be a subspace of vector space V and that T: V \rightarrow V is linear. Prove that subspaces \{0\},V,R(T),N(T) are all T-invariant.

    I feel that the trick is really simple but I just cannot grasp it. Any pointers would be appreciated - thanks.
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  2. #2
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    Quote Originally Posted by Last_Singularity View Post
    Question: Let W be a subspace of vector space V and that T: V \rightarrow V is linear. Prove that subspaces \{0\},V,R(T),N(T) are all T-invariant.

    I feel that the trick is really simple but I just cannot grasp it. Any pointers would be appreciated - thanks.
    \{ 0 \} and V are obvious since T:V \longrightarrow V is linear. Since T(v) \in V for all v \in V we have T(T(v)) \in R(T) and so T^n(v) \in R(T) by definition, if v \in N(T) then T(v)=0 \in N(T) and T^n(v) \in N(T)
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