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Math Help - Generalized Rank Nullity Theorem

  1. #1
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    Generalized Rank Nullity Theorem

    I was wondering if there is a generalization of this theorem from finite numbers to cardinal numbers.
    More specifically, let F be a field and V, W be vector-spaces over F.
    Let the dimension of V/F = \aleph_{\alpha}.
    Let T:V\mapsto W be a linear transformation then is \mbox{dim}(\mbox{ker} T) + \mbox{dim}(\mbox{Im} T) = \aleph_{\alpha}?
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  2. #2
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    Quote Originally Posted by ThePerfectHacker View Post
    I was wondering if there is a generalization of this theorem from finite numbers to cardinal numbers.
    More specifically, let F be a field and V, W be vector-spaces over F.
    Let the dimension of V/F = \aleph_{\alpha}.
    Let T:V\mapsto W be a linear transformation then is \mbox{dim}(\mbox{ker} T) + \mbox{dim}(\mbox{Im} T) = \aleph_{\alpha}?
    your question basically is whether or not we have \dim V = \dim K + \dim(V/K), for any subspace K of V,

    where \dim here means the cardinality of a basis of a vector space. the answer is yes! let A=\{v_{i}: \ i \in I \}

    be a basis for K and C=\{w_j + K: \ j \in J \} a basis for V/K. let B=\{w_j: \ j \in J\}. it's easy to see that

    |B|=|C|, \ A \cap B = \emptyset, and A \cup B is a basis for V. Q.E.D.
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