Contradictions in authoritative definition of "transformation": Must it be invertible?

Question

What is the authoritative definition of transformation? While transformations are clearly maps, sources seems to disagree as to:

1. whether the map must be invertible
2. whether the map must be from a set to itself or can be to another set
3. whether the map must be a bijection

Of course these three questions are related.

Surprisingly, while the term is used ubiquitously, I can find no authoritative definition of it. I can't even find a formal definition of it in a well accepted text. E.g. Google shows numerous informal uses; even restricting Google to .edu shows only introductory notes, not higher level formal definitions. And Wikipedia's definitions are apparently inconsistent from one page to the next!

Likewise with the way the term is used on math.SE. Below are a few examples:

What is the authoritative (or canonical) definition of transformation?

Sources that transformation need not be invertible:

• https://en.wikipedia.org/wiki/Transformation_(function) "a transformation is a function f, usually with some geometrical underpinning, that maps a set X to itself"
• https://en.wikipedia.org/wiki/Linear_map " a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping $V \to W$ between two vector spaces that preserves the operations of vector addition and scalar multiplication... if a linear map is a bijection then it is called a linear isomorphism."
• What does a non-invertible affine transformation look like geometrically in terms of rotation/shear/scaling?
• What is the difference between the terms transformation and map?

Sources that state or imply it must be invertible:

• https://en.wikipedia.org/wiki/Affine_transformation "An affine transformation is a bijection..."
• https://en.wikipedia.org/wiki/Geometric_transformation "a geometric transformation is any bijection..." yet this seems to contradict itself because it continues "any bijection of a set to itself (or to another such set)"
• What shape do we get when we shear an ellipse? And more generally, do affine transformations always map conic sections to conic sections? - the questions and answers all assume it is invertible
• https://www.math.cmu.edu/~ctj/Articles/affine-transformations.pdf - the corollary and "important property" implicitly assume transformations are invertible

Answer 1

Since you are already assuming "While transformations are clearly maps", it is reasonable to say that a transformation is indeed just a map, and nothing more.

Of course, if you combine this with other words, like "affine transformation", you obtain a much more restricted meaning. Since the affine group ${\rm Aff}(\Bbb R^n)$ can be realised as a subgroup of $GL_{n+1}(\Bbb R)$, it is not surprising that affine transformations are invertible.

This has nothing to do with "transformation". You can also consider "elements", say of an integral domain. Then of course not all elements are prime in general. But an "irreducible element" is prime. And a "unit element" is not prime. So if your title were "Contradictions in authoritative definition of "element" in $R$: Must it be prime?", then again the answer is no.

So whenever looking for a definition of one word, you cannot take a context with other adjectives added.