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The General Linear Group as a basis for all nxn matrices

  1. Feb 9, 2012 #1
    I'm trying to prove that every nxn matrix can be written as a linear combination of matrices in GL(n,F).

    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.

    Is this on the right track?
  2. jcsd
  3. Feb 9, 2012 #2
    The easiest bases for the space of nxn matrices is just the matrices with a one as one entry and zeros everywhere else. If you can show that there is a way to make all of these matrice as linear combinations of invertible nxn matrices (just do it explicitly) you are done.

    Also remember that the general linear group is not a group of matrices but of isomorphisms of vector spaces. depending on the bases one isomorphism can have different matrices.
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