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Every nxn matrix can be written as a linear combination of matrices in GL(n,F)

  1. Feb 8, 2012 #1
    1. The problem statement, all variables and given/known data
    Prove: Every nxn matrix can be written as a linear combination of matrices in GL(n,F).


    2. Relevant equations
    GL(n,F) = the set of all nxn invertible matrices over the field F together with the operation of matrix multiplication.


    3. The attempt at a solution
    I know all matrices in GL(n,F) are invertible and hence have linearly independent columns and rows. I was thinking perhaps there is something about the joint bases for the n-dimensional column and row spaces, respectively, that could provide a basis for M_{nxn}(F), which has dimension of n^2. But I'm not really sure if that works.
     
  2. jcsd
  3. Feb 10, 2012 #2
    Okay, I figured it out. Nevermind (although once a post gets buried two screens back, it's not likely to be answered anyway, even if it has zero replies...).
     
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