Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: The General Linear Group as a basis for all nxn matrices
  1. Normal nxn matrices (Replies: 6)

Loading...