Talk:Linear map

Major fixing needed

1. This page is lifted from Wikipedia, but for some reason, parts of the Wikipedia page were deleted.
2. The definition is terrible. First, every linear algebra book I have seen defines the term as "linear transformation", using map as an informal substitute afterward. But googling "linear map" and "linear transformation" shows "linear map" is much more prevalent. Perhaps because I am wrong, but more likely because people are too lazy to type "transformation"? In any case, I think the page name and main term should be "transformation", not "map", just based on my experience. Does anyone have a much different experience?
3. Transformation does not usually refer to a map from a space to itself. Transformation is the most general term, and "operator" has a tendency to refer to a map from a set to itself. This is based on my own experience, but to back me up, here is the American Heritage Dictionary[1]

I thought I was going crazy when I looked up the definition of "operator" given on Wikipedia, saying inputs and outputs must be functions, but even there, the inputs and outputs are drawn from the same set. Can anyone back me up? Or is my experience rather unusual, and other places/cultures/books use transformation as written here and operator for more general maps?Barry R. Smith 17:02, 11 December 2008 (UTC)

Barry, I see your point, but I don't think that "linear map" is "wrong" per se. This seems like a matter of preference or semantics or even what people are used to. Some people like "linear transformation" while others prefer "linear map" and certainly a lot of people (as verified by Google) do use the latter. Perhaps this could indeed be due to the fact that a lot of people are too lazy to write "transformation". BTW, thanks for all your contributions to CZ Mathematics (and also to Richard Pinch), it really needs more maths people getting involved and get the article counts up. Best, Hendra I. Nurdin 01:56, 10 December 2008 (UTC)

I decided I was being lazy myself, and hence hypocritical, so I did a little research to see if my experience is unusual by grabbing a few books off of my shelf. Here is what I find, from the following small sample of books: Halmos, "Finite Dimensional Vector Spaces"; Steven J. Leon, "Linear Algebra"; Herstein, "Topics in Algebra"; M. Artin, "Algebra"; Lang, "Algebra"; Hungerford, "Algebra"; "John B. Fraleigh, "A first course in Abstract Algebra"; Roger Penrose, "The Road to Reality" -- 8 books total.

• The definition of "linear transformation" appears in Ha, Le, He, Ar, Hu, Fr, Pe
• The definition of "linear map" appears in La
• The definition of "linear operator" appears in Ha, Le, Ar

with the following remarks:

1. Lang talks only about modules over general rings, and linear transformation would be inappropriate in this more general context -- all others define the main term as "linear transformation"
2. Ha and Pe deal only with transformations from a space to itself
3. Ha, He, and Pe are the only ones which indicate that "transformation" typically means a map from a space to itself, and by #2, Ha and Pe do this out of necessity.
4. Ha, Le, and Ar all indicate that operator is reserved for maps from a space to itself.

My conclusions:

1. in formal writing at least, "transformation" is the preferred term whenever possible. As our articles are supposed to be encyclopedic, should we not also abide by this?
2. "transformation" has a very very slight tendency to indicate a map from a space to itself, while "operator" in the above sample always indicates such a map.

I think this presents a reasonable case for making the page name and main definition "transformation". I do retract my statement that transformation never indicates that the map from a space to itself, as Herstein is a respected source. However, if this very slight (perhaps this was more common in the past?) tendency is to be mentioned, the same tendency for "operator" must surely be mentioned along with it.Barry R. Smith 17:02, 11 December 2008 (UTC)

Re transformation. From the top of my head, without consulting any books, I would say that physicists say "transformation" when they mean a 1-1 map (a basis transformation, a symmetry transformation, etc.) They use (linear) "operator" for linear maps. The more mathematically inclined physicist will use "linear map", especially when the vector goes to (is mapped into) a different vector space. PS. Please undersign your contributions.--Paul Wormer 12:42, 11 December 2008 (UTC)

For whatever it is worth, Google gives over 3 million hits for "linear transformation", just over 2 million for the term I'd use, "linear mapping", and about a million for "linear map". I do not think it matters much which term we use as an article title, but the other two should exist as redirects. 15:04, 11 December 2008 (UTC)