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Math Help - To show the function is well defined

  1. #1
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    To show the function is well defined

    Suppose T:V-->V is a nilpotent linear map.Then there exists subspaces W1,W2,...Wk of V such that
    1)T(Wi) is contained in Wi ,for i=1,...,k
    2)Each Wi is T-cyclic subspace of V
    3)V=direct sum of Wi's ,for i=1,...k.
    In the proof of this question ,as T(Wi) is contained in Wi,we get a induced linear map,T':V/Wi-->V/Wi defined by T'([v])=[Tv],for all v in V.
    how do i show that the map is well defined..
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  2. #2
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    Quote Originally Posted by math.dj View Post
    Suppose T:V-->V is a nilpotent linear map.Then there exists subspaces W1,W2,...Wk of V such that
    1)T(Wi) is contained in Wi ,for i=1,...,k
    2)Each Wi is T-cyclic subspace of V
    3)V=direct sum of Wi's ,for i=1,...k.
    In the proof of this question ,as T(Wi) is contained in Wi,we get a induced linear map,T':V/Wi-->V/Wi defined by T'([v])=[Tv],for all v in V.
    how do i show that the map is well defined..


    In this case, to show it is well-defined you must show that [v]:=v+W_i= v'+W_i=:[v']\,\Longrightarrow T([v])=T(v+W_i)=T(v'+W_i)=T([v'])

    Tonio
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