[College - Algebra] Commuting Linear Operators

[QED Stepfiles]

Hey everybody,

This problem was on a homework that was already due... and I did it, but I did it in an incredibly convoluted way. So I was just wondering if anybody could find a better solution for it, since I'm sure one must exist:

Consider the vector space consisting of linear operators on V, some finite dimensional vector space. Given some member T of this vector space, we define Z(T) as the subspace consisting of all linear operators S that commute with T.

Show that the dimension of Z(T) is greater than or equal to Dim(V).

I had simplified the problem by considering Jordan canonical form, but then it just went downhill from there and got pretty messy. If anybody can find a more elegant way to do it, that'd be great!

EDIT: Whoops, forgot to mention that all vector spaces are over algebraically closed fields.

[sumzup]

If you asked this 6 months ago, I would be able to help you, but sadly, all of this stuff went out the window over the intervening time period. Obviously no one on FFR appears to know Linear Algebra in depth, so it's probably best to ask around elsewhere.