I am trying to understand the meanings of "covariant transformation" and "contravariant transformation" and how they are related. I have read the related Wikipedia article and still feel I cannot state, with mathematical precision, the definition of these terms.
The Wikipedia article states that a covariant transformation, in the context of a vector space, is one that "describes new basis vectors in terms of old basis vectors". This is not a satisfactory definition unless, of course, no other transformations can be described as "covariant". I have seen however the word "covariant" being used to describe other sorts of transformations as "covariant". Namely, the "physicists" definition of co/contravariant transformations where components transform as such-and-such (which makes absolutely zero mathematical sense to me). This leads one to believe that co/contravariant transformations are always defined in terms of derivatives of coordinate changes and I don't believe this is the case.
I understand what co/contra-variant tensors are, at least from a mathematical perspective, so this is not a question about the meanings of "contravariant tensor" or "covariant tensor"; indeed, These concepts have been well-explained here.
My question then, in summary, What are lucid, self-contained and mathematically precise definitions of "covariant transformation" and "contravariant transformation"? A reference to such definitions would also work wonderfully.