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Math Help - Basis and Dim

  1. #1
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    Basis and Dim

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

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

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