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Thread: Basis and Dim

  1. #1
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    Basis and Dim

    Let $\displaystyle \mathbb{F} $ be any field. Let $\displaystyle K_n $ be the set of all $\displaystyle n \times n $ matrices $\displaystyle A $ with entries from $\displaystyle \mathbb{F} $ satisfying $\displaystyle \sum_{i=1}^n a_{ij} = 0 $ for each $\displaystyle j $. How do I find a basis for $\displaystyle K_n $ and how do I find dim$\displaystyle (K_n) $ ?

    I really need help understanding how to find a basis and dim for any question like this.
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  2. #2
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    Quote Originally Posted by funnyinga View Post
    Let $\displaystyle \mathbb{F} $ be any field. Let $\displaystyle K_n $ be the set of all $\displaystyle n \times n $ matrices $\displaystyle A $ with entries from $\displaystyle \mathbb{F} $ satisfying $\displaystyle \sum_{i=1}^n a_{ij} = 0 $ for each $\displaystyle j $. How do I find a basis for $\displaystyle K_n $ and how do I find dim$\displaystyle (K_n) $ ?

    I really need help understanding how to find a basis and dim for any question like this.
    for any $\displaystyle 1 \leq i,j \leq n$ define $\displaystyle e_{ij}$ to be the $\displaystyle n \times n$ matrix with (i,j)-entry 1 and everywhere else 0. now put $\displaystyle a_{nj}=-\sum_{i=1}^{n-1}a_{ij}$ in $\displaystyle A$ to see that $\displaystyle A=\sum_{j=1}^n \sum_{i=1}^{n-1} a_{ij}(e_{ij}-e_{nj}).$

    hence $\displaystyle \mathcal{B}=\{e_{ij}-e_{nj}: \ \ 1 \leq i \leq n-1, \ 1 \leq j \leq n \}$ is a basis for $\displaystyle K_n$ and therefore $\displaystyle \dim K_n=n(n-1).$
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