Operator (mathematics)

In mathematics, an operator is generally a mapping that acts on the elements of a space to produce other elements of the same space. The most common operators are linear maps, which act on vector spaces. However, when using "linear operator" instead of "linear map", mathematicians mean often actions on vector spaces of functions, which preserve also other 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).

The most common kind of operator encountered are linear operators. Let U and V be vector spaces over a field K. Operator A: U → V is called linear if

for all x, y in U and for all α, β in K.

The importance of linear operators is partially because they are morphisms between vector spaces.

In finite-dimensional case linear operators can be represented by matrices in the following way. Let ${\displaystyle K}$ be a field, and ${\displaystyle U}$ and ${\displaystyle V}$ be finite-dimensional vector spaces over ${\displaystyle K}$. Let us select a basis ${\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{n}}$ in ${\displaystyle U}$ and ${\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{m}}$ in ${\displaystyle V}$. Then let ${\displaystyle \mathbf {x} =x^{i}\mathbf {u} _{i}}$ be an arbitrary vector in ${\displaystyle U}$ (assuming Einstein convention), and ${\displaystyle A:U\to V}$ be a linear operator. Then

...
Wikipedia