Is there a difference between transform and transformation?

Question

I was told that there is a difference between a transform and a transformation. Can anyone point out clearly?

For example, is Laplace transform not a transformation?

Answer 1

A transformation is certainly a process. The word "transform" can be used as short for "transfomation", but also for a transformed object. An example for the first usage is

"The Fourier transform is an automorphism of the Schwartz space ${\cal S}\,$",

an example for the second usage is

"The Laplace transform of $t\mapsto e^{\lambda t}$ $\>(t\geq0)$ is $s\mapsto{\displaystyle{1\over s-\lambda}}\>$".

Answer 2

From the accepted question on ELU: Transform or transformation?

In the particular domain you're referencing, both transform and transformation have an established history of usage.

For example, we speak of the Hough transform or the Fourier transform. We also talk about affine transformations or homothetic transformations.

I'm not aware of any specific rules, but in general, I've noticed that "named" objects tend to be referenced as the foo transform, while unnamed or otherwise generic ones use the foo transformation.

Answer 3

This is a solidly common-usage question, which I was asked many decades ago, and has now come up again.

In common usage, when referring to the alteration of elements of a set, as an abstraction, (or as Niall says, a "process"), the word transformation is always applied. The set is transformed, action is a transformation.

When referring to the transformation itself as an object, as a thing that performs the transformation, the word transform is often applied. (This includes the "named" transformations that Ooker refers to.) But clearly, there is a direct correspondence here between the action and the actor --- it could be viewed as a matter of emphasis.

But the terminology isn't cut-and-dried at all.

P.S. The terms operator and transform are usually restricted to mappings whose range and domain are homomorphic. These are used in very much the same way, except, I don't think I've ever seen transform applied to mappings of finite-dimensional spaces.