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Thread: linear algebra question

  1. #1
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    linear algebra question

    linear algebra question-screen-shot-2013-02-26-12.06.31-pm.png

    in the question why can you change the order of AB and BA in

    5(AB^T)^T = 5(B^T)^T A^T ?

    where ^T means the transpose of a matrix
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  2. #2
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    Re: linear algebra question

    This is a standard property of transposition ($\displaystyle (AB)^T=B^TA^T$). Let $\displaystyle A$ be an $\displaystyle n\times m$ matrix and let $\displaystyle B$ be an $\displaystyle m\times n$ matrix. Now consider their transposes $\displaystyle A^T$ and $\displaystyle B^T$. Row $\displaystyle i$ of $\displaystyle A$ is exactly the same as column $\displaystyle i$ of $\displaystyle A^T$, similarly column $\displaystyle i$ of $\displaystyle B$ is exactly the same as row $\displaystyle i$ of $\displaystyle B^T$. So you should be able to see that

    $\displaystyle row_i(A)\cdot column_j(B)=row_j(B^T)\cdot column_i(A^T),$

    so it follows that $\displaystyle (AB)^T=B^TA^T$.
    Last edited by hairymclairy; Feb 27th 2013 at 01:04 AM.
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