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Operator and function 
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#1
Oct2313, 11:21 AM

P: 134

What is the exact technical difference between a linear operator and linear function?



#2
Oct2313, 11:33 AM

P: 3,002

Wikipedia gives a good description of the similarity between the two:
http://en.wikipedia.org/wiki/Linear_operator 


#3
Oct2313, 12:42 PM

Newcomer
P: 341

To me, they are the same thing.



#4
Oct2313, 12:54 PM

P: 237

Operator and function
I use linear function, linear operator, linear map, and linear transformation all interchangeably, though you come to notice that different communities of mathematicians have different preferred terminology.



#5
Oct2313, 04:30 PM

P: 1,066

Though I do not believe this is standard if IIRC, Axler defines linear maps (functions) as maps between arbitrary vector spaces but reserves the term operator for maps between the same vector space.
Most use the terms interchangeably in my experience. 


#6
Oct2313, 08:20 PM

P: 4,573

I think linear operators must be square matrices will linear maps can be any sort of configuration (i.e. nonsquare).



#7
Oct2313, 08:30 PM

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#8
Oct2413, 01:08 AM

Sci Advisor
P: 834

Linear transform seems to be more of a term you see in linear algebra rather than linear analysis. 


#9
Oct2413, 08:16 AM

P: 134

Can we say that linear operator and linear function are generally used as synonyms but to be more precise and technically linear operator denotes square matrix since it maps space to itself whereas linear function denotes a rectangular matrix since it maps a vector of one space to different space.
Is that right? 


#10
Oct2413, 10:41 AM

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P: 21,313

That's pretty close. I would say it this way: a linear operator maps a vector space to itself, which implies that the matrix is square. A linear function maps an arbitrary vector space to a possibly different vector space, which implies that the matrix is not square if the spaces are of different dimension.



#11
Oct2413, 12:26 PM

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#12
Oct2413, 06:19 PM

Sci Advisor
P: 834




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