Thread: Abelian group

Hi, need a bit of help here.

The set of invertible affine transformations of R is a group under composition. Is this group Abelian? Prove it if it is, and if it is not, provide a counter example.

Not sure what the answer is.

Thank You for any help.

Let and over the field . Then are invertible affine transformations. But . So the group is not abelian.

EDIT: Oops, I see the transformation has to be over . Example has been changed to accommodate this.

Originally Posted by shelford

Hi, need a bit of help here.

The set of invertible affine transformations of R is a group under composition. Is this group Abelian? Prove it if it is, and if it is not, provide a counter example.

Not sure what the answer is.

Thank You for any help.

Well an affine transformation is of the form

so create another one

Now just compute

and

If it is Abelian what must be true?

I would like to point out a tangential consideration. This question has an interesting application in the field of data acquisition. I was in a situation once where I was taking in data on a computer, applying one slope and offset (affine transformation), and then applying another slope and offset. The purpose of applying these slopes and offsets is to convert the raw voltage data that you nearly always get off of your data acquisition hardware, into engineering units like psig. I needed to know whether I would get the same result if I switched the order of application. The answer to the question in this thread was also the answer to my query.

Thanks to all for the great help. I understand it now.