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Thread: 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 $\displaystyle F$ be a field and $\displaystyle V$, $\displaystyle W$ be vector-spaces over $\displaystyle F$.
    Let the dimension of $\displaystyle V/F = \aleph_{\alpha}$.
    Let $\displaystyle T:V\mapsto W$ be a linear transformation then is $\displaystyle \mbox{dim}(\mbox{ker} T) + \mbox{dim}(\mbox{Im} T) = \aleph_{\alpha}$?
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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 $\displaystyle F$ be a field and $\displaystyle V$, $\displaystyle W$ be vector-spaces over $\displaystyle F$.
    Let the dimension of $\displaystyle V/F = \aleph_{\alpha}$.
    Let $\displaystyle T:V\mapsto W$ be a linear transformation then is $\displaystyle \mbox{dim}(\mbox{ker} T) + \mbox{dim}(\mbox{Im} T) = \aleph_{\alpha}$?
    your question basically is whether or not we have $\displaystyle \dim V = \dim K + \dim(V/K),$ for any subspace $\displaystyle K$ of $\displaystyle V,$

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

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

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