The General Linear Group as a basis for all nxn matrices

  • Thread starter fishshoe
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  • #1
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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?
 

Answers and Replies

  • #2
133
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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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