Editing Operator (mathematics) (section)

== Linear operators ==
{{Main|Linear operator}}

The most common kind of operators encountered are ''linear operators''. Let {{mvar|U}} and {{mvar|V}} be [[vector space]]s over some [[field (mathematics)|field]] {{mvar|K}}. A [[map (mathematics)|''mapping'']] <math>\operatorname{A} :  U \to V </math> is [[linear (mathematics)|''linear'']] if <math display="block">\operatorname{A}\left( \alpha \mathbf{x} + \beta \mathbf{y} \right) = \alpha \operatorname{A} \mathbf{x} + \beta \operatorname{A} \mathbf{y}\ </math> for all {{math|'''x'''}} and {{math|'''y'''}} in {{mvar|U}}, and for all {{math|''α'', ''β''}} in {{mvar|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 [[morphism]]s between vector spaces.
In the finite-dimensional case linear operators can be represented by [[Matrix (mathematics)|matrices]] in the following way. Let {{mvar|K}} be a field, and <math>U</math> and {{mvar|V}} be finite-dimensional vector spaces over {{mvar|K}}. Let us select a basis <math>\ \mathbf{u}_1, \ldots, \mathbf{u}_n </math> in {{mvar|U}} and <math>\mathbf{v}_1, \ldots, \mathbf{v}_m </math> in {{mvar|V}}. Then let <math>\mathbf{x} = x^i \mathbf{u}_i</math> be an arbitrary vector in <math>U</math> (assuming [[Einstein convention]]), and <math>\operatorname{A}: U \to V </math> be a linear operator. Then<math display="block">\ \operatorname{A}\mathbf{x} = x^i \operatorname{A}\mathbf{u}_i = x^i \left( \operatorname{A}\mathbf{u}_i \right)^j \mathbf{v}_j ~.</math> Then <math>a_i^j \equiv \left( \operatorname{A}\mathbf{u}_i \right)^j </math>, with all <math>a_i^j\in K </math>, is the matrix form of the operator <math> \operatorname{A} </math> in the fixed basis <math>\{ \mathbf{u}_i \}_{i=1}^n</math>. The tensor <math>a_i^j </math> does not depend on the choice of <math>x</math>, and <math>\operatorname{A}\mathbf{x} = \mathbf{y} </math> if <math>a_i^j x^i = y^j</math>. Thus in fixed bases {{mvar|n}}-by-{{mvar|m}} matrices are in [[bijective]] correspondence to linear operators from <math>U </math> to <math>V</math>. 

The important concepts directly related to operators between finite-dimensional vector spaces are the ones of [[Matrix rank|rank]], [[determinant]], [[inverse operator]], and [[eigenspace]].

Linear operators also play a great role in the infinite-dimensional case. The concepts of rank and determinant cannot be extended to infinite-dimensional matrices. This is why very different techniques are employed when studying linear operators (and operators in general) in the infinite-dimensional case. The study of linear operators in the infinite-dimensional case is known as [[functional analysis]] (so called because various classes of functions form interesting examples of infinite-dimensional vector spaces).

The space of [[sequence]]s of real numbers, or more generally sequences of vectors in any vector space, themselves form an infinite-dimensional vector space. The most important cases are sequences of real or complex numbers, and these spaces, together with linear subspaces, are known as [[sequence space]]s. Operators on these spaces are known as [[sequence transformation]]s.

Bounded linear operators over a [[Banach space]] form a [[Banach algebra]] in respect to the standard operator norm. The theory of Banach algebras develops a very general concept of [[Spectrum (functional analysis)|spectra]] that elegantly generalizes the theory of eigenspaces.