What kind of object is the N-tuple that describes an affine scaling operation in N dimensions, and how does it transform?

Question

I'm building a math package to handle affine transformations and I'm trying to decide what the necessary set of objects and operations are for my purposes. So far I've got spaces, matricies, points, vectors and covectors. I've been calling the n-tuple of factors for building a scaling transform a 'scaling vector' but I'm not sure at all that it really is a vector, nor do I know that it should transform as a vector does if I perform a change of coordinates. Any useful references or terminology would help.