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Thread: linear maps, rank

  1. #1
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    linear maps, rank

    Let $\displaystyle V=V(m,\mathbb{R})$ and $\displaystyle W=W(n,\mathbb{R})$ and let f be a matrix corresponding to a linear map from V to W. Verify that $\displaystyle rankf=rankf^{t}=rank(Mf^{t}N)$. Gdzie $\displaystyle M\in GL(m,\mathbb{R})$ oraz $\displaystyle N\in GL(n,\mathbb{R})$
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    One way:

    $\displaystyle M(f^t)=[M(f)]^t$

    where:

    $\displaystyle M(f)$ : matrix of $\displaystyle f$ with respect to the basis $\displaystyle B_V$ and $\displaystyle B_W$

    $\displaystyle M(f^t)$ : matrix of $\displaystyle f^t$ with respect to the dual basis $\displaystyle B_W^*$ and $\displaystyle B_V^*$


    Fernando Revilla
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