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ajayguhan
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What is the exact technical difference between a linear operator and linear function?
Jorriss said: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.
That's how linear operators were distinguished from linear transformations when I learned about them. I.e., a linear operator maps a space to itself, hence the matrix for the operator is necessarily square.chiro said:I think linear operators must be square matrices will linear maps can be any sort of configuration (i.e. non-square).
Mark44 said:That's how linear operators were distinguished from linear transformations when I learned about them. I.e., a linear operator maps a space to itself, hence the matrix for the operator is necessarily square.
pwsnafu said:I've never see "linear transform" used with infinite dimensional spaces. It always seems to be "operator theory" or "bounded linear operator" or "closed operator".
Linear transform seems to be more of a term you see in linear algebra rather than linear analysis.
ajayguhan said: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?
Mark44 said: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.
R136a1 said:Well, you do have Fourier transforms, Gelfand transforms, Laplace transforms, Hilbert transforms,... All of these are on space which are usually pretty infinite-dimensional.
A linear operator is a mathematical object that maps a vector space to itself, while a linear function maps a vector space to a scalar. In other words, a linear operator operates on vectors, while a linear function operates on scalars.
Linear operators can be seen as generalizations of linear functions. In fact, a linear function can be represented as a linear operator acting on a one-dimensional vector space. Additionally, the properties of linearity hold for both linear operators and linear functions.
The derivative operator, denoted as D, is an example of a linear operator. It maps a function to its derivative, and satisfies the properties of linearity: D(f+g) = D(f) + D(g) and D(af) = aD(f), where f and g are functions and a is a scalar.
The function f(x) = 2x+3 is an example of a linear function. It maps a scalar (x) to another scalar (2x+3), and satisfies the properties of linearity: f(x+y) = f(x) + f(y) and f(ax) = af(x), where x and y are scalars and a is a constant.
Linear operators and linear functions have many practical applications in various fields such as physics, engineering, and economics. For example, in physics, linear operators are used to describe the evolution of quantum systems, while linear functions are used to model physical processes. In economics, linear functions are used to represent supply and demand curves, while linear operators are used to analyze market dynamics.