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Math Help - Rank

  1. #1
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    Rank

    Hey there,

    If anyone could give me a hint that would be awesome.

    Let V be a finite-dimensional vector space and let  S,T \in \mathcal{L}(V).

    prove rank(S+T)<= rankS+rankT
    All I have is:

    Take any x from V.

    Rank (S+T) implies dim((S+T)(x)) = dim(S(x)+T(x))

    But I have no clue what to do from here. I know it's really trivial, but I'm stuck.

    Hints?
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by manygrams View Post
    Hey there,

    If anyone could give me a hint that would be awesome.



    All I have is:

    Take any x from V.

    Rank (S+T) implies dim((S+T)(x)) = dim(S(x)+T(x))

    But I have no clue what to do from here. I know it's really trivial, but I'm stuck.

    Hints?
    Hint:
    Note first that \left(S+T)(V)\subseteq S(V)+T(V). To see this merely note that if (S+T)(v)\in (S+T)(V) then (S+T)(v)=S(v)+T(v)\in S(V)+T(V).

    Solution:

    Spoiler:
    Thus, \dim\left((S+T)(V)\right)\leqslant \dim\left(S(V)+T(V)\right)=\dim S(V)+\dim T(V)-\dim\left(S(V)\cap T(V)\right)\leqslant \dim S(V)+\dim T(V).
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