Another Carnival
The Secret Blogging Seminar is hosting the latest installment of the Carnival of Mathematics. They seem to have done a fair job of collecting links, with more (and more varied) entries than I remember seeing before. Enjoy.
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John, I have appreciated your help in the past and I would like to get your insight into yet another question I have concerning linear algebra.
What is the relationship between the set of all linear transformations on a vector space over a field F of dimension n and the set of all nxn matrices with entries from the same field F. Clearly, it is not a one-to-one relation. There may be many, possibly an infinite number of matrices that can represent the same linear transformation with respect to different basis of the space. What exactly is the relation?
Thank you for your time,
Jonathan Greenberg
When you’re dealing with
-linear functions between free 
-modules (all vector spaces — modules over a field — are free), you can pick a basis for each of the domain and range as I discuss here. Once you do, then any such linear transformation can be written down uniquely as a matrix with respect to those bases, as I discuss in that link. Also, given a choice of bases any matrix corresponds to exactly one linear transformation.
But as you notice, there’s that choice of basis. Changing the basis of a free module is the same as applying an invertible linear transformation from that module to itself. That is,
is the space of linear transformations from a rank 
module to a rank 
module. Then there is an action of the group 
(invertible 
matrices with entries in 
) on the one side, and an action of 
on the other. What you’re really asking about is what the set of transformations looks like once we quotient out by these group actions.
And that’s where the story gets complicated. Over an algebraically closed field (like the complex numbers), the answer is contained in the Jordan normal form, which is a standard part of any good senior-level linear algebra course. It gives a canonical representative of each orbit of the group action. Beyond that it’s a huge mess. Sorry I can’t be more accurate.
John, thank you for your insight into my question. I will have to think about your answer as I study linear algebra in more depth.
JG