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Concept
Operator (mathematics)
Summary
In mathematics, an operator is generally a mapping or function that acts on elements of a space to produce elements of another space (possibly and sometimes required to be the same space). There is no general definition of an operator, but the term is often used in place of function when the domain is a set of functions or other structured objects. Also, the domain of an operator is often difficult to characterize explicitly (for example in the case of an integral operator), and may be extended so as to act on related objects (an operator that acts on functions may act also on differential equations whose solutions are functions that satisfy the equation). See Operator (physics) for other examples.
The most basic operators are linear maps, which act on vector spaces. Linear operators refer to linear maps whose domain and range are the same space, for example to .
Such operators often preserve properties, such as continuity. For example, differentiation and indefinite integration are linear operators; operators that are built from them are called differential operators, integral operators or integro-differential operators.
Operator is also used for denoting the symbol of a mathematical operation. This is related with the meaning of "operator" in computer programming, see operator (computer programming).
Linear operator
The most common kind of operator encountered are linear operators. Let U and V be vector spaces over a field K. A mapping A: U → V is linear iffor all x, y in U and for all α, β in K.
This means that a linear operator preserves vector space operations, in the sense that it does not matter whether you apply the linear operator before or after the operations of addition and scalar multiplication. In more technical words, linear operators are morphisms between vector spaces.
In the finite-dimensional case linear operators can be represented by matrices in the following way. Let be a field, and and be finite-dimensional vector spaces over . Let us select a basis in and in .
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