# linear transformation

## Primary tabs

Defines:
linear operator
Synonym:
linear map, vector space homomorphism, linear mapping
Type of Math Object:
Definition
Major Section:
Reference

## Mathematics Subject Classification

### Terminology

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.

### Re: Terminology

"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.

### Re: Terminology

"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).

### Re: Terminology

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.