1) it is pretty easy to check the ring conditions, i mean it clearly is a group under addition it is just basically component wise if you think of it as in
. The multiplication too is easy to check just multiply by another matrix in that form, make sure it is closed and then multiply on the other side by that matrix and see if you get the same thing. Something tells me if you are capable of TeXing up the matrices you are capable of multiplying out two 2x2 matrices. I just did it really quick and it looks like its a commutative ring.
2) pointwise addition is pretty easy to check is a group, the question is about composition. As long as the functions are defined over all of
you should be fine. Certainly the composition of two functions from
will still be a function
.
I am just a little concerned about like if you had your first function as
and then composed it with
like this function would not be defined anywhere, but I assume when they defined the family of functions it has to be defined on all of
. To see that is is definitely not commutative try
and
. Hope this is of some help.
Ah I see NCA already posted, well instead of deleting it ill let you read mine too maybe it might help as well. And I think NCA might check the commutativity of 1) it looks commutative to me.