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linear transformation

linear operator
linear map, vector space homomorphism, linear mapping
Type of Math Object: 
Major Section: 

Mathematics Subject Classification

15A04 no label found


While in principle, the terms "map", "mapping",
"function", "transformation", etc are synonyms,
my impression is that the different terms have acquired
distinct meanings. This is more a matter of connotation than
denotation. However consistent usage that conforms to
prevalent norms should make for clearer communication.

The generic term is "mapping", or "map" - although
mapping seems to be the preferred term.

The word "function" should be reserverd for a "mapping"
whose domain and codomain are sets of "numbers" in
some general sense.

The word "transformation" should be reseverd for a
"mapping" where the domain and codomain coincide.
The basic idea is that one can compose a transformation
with itself.

The word "operator" should be reserved for "transformations"
whose domain/codomain is a set of "functions".

The word "functional" should be reserved for a mapping whose
domain is a set of "functions" and whose codomain is a set of
"numbers", in some general sense.

"Linear operator" is used in my Linear Algebra textbook (Friedberg-Insel-Spence) so frequently (essentially half the theorems begin "Let T be a linear operator...") that I have to ask if their usage really is nonstandard.

"function" tends to have a more precise definition, in that for { (a, b) } in the graph of f, ( a ) are distinct (alternatively, f(a) has up to one value).

After looking around at texts and other references
I have to back away from my position somewhat.

The choice of terminology:

mapping vs function vs transformation vs operator

is completely unstandardized. I come from a differential
geometry background, where "mapping" is the general term
i.e. you can have mappings between manifolds, whereas
function is a "scalar field", i.e. a mapping that plunks
numbers down at points of your manifold.

So probably my original post was just expressing my
own prejudices/prefernces.

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