I've frequently heard that a Fourier transform is "just a change of basis".
However, I'm not sure whether that's correct, in terms of the terminology of "change of basis" versus "transformation" in linear algebra.
Is a Fourier transform of a function merely a change of basis (i.e. the identity transformation from the original vector space onto itself, with merely a change of basis vectors), or is indeed a linear transformation from the original vector space onto another vector space (the "Fourier" vector space, so to speak)?
Also: Does the same answer hold for other similar transforms (e.g. Laplace transforms)? Or is there a different terminology for those?