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In [[mathematics]], a '''linear transformation''' (also called '''linear operator''' or '''linear map''') is a [[function (mathematics)|function]] between two [[vector space]]s [[restrictive clause|that]] preserves the operations of vector addition and [[scalar]] multiplication. In other words, it "preserves [[linear combination]]s".
:''This article is about operators in [[mathematics]], for other kinds of operators see [[operator (disambiguation)]].''
In the language of [[abstract algebra]], a linear transformation is a [[homomorphism]] of vector spaces.
In [[matematica]], un '''operatore''' è un qualche tipo di [[funzione (matematica)| funzione]]; nel caso venga specificato un particolare tipo di [[operando]] come [[dominio di funzione]] non è nient'altro che una specifica funzione. L'utilizzo maggiormente frequente è nel caso di applicazioni tra [[spazio vettoriale | spazi vettoriali]]; questo tipo di operatore è contraddistinto
== Definition and first consequences ==
In [[mathematics]], an '''operator''' is some kind of [[function (mathematics)|function]]; if it comes with a specified type of [[operand]] as [[function ___domain]], it is no more than another way of talking of functions of a given type. The most frequently met usage is a mapping between [[vector space]]s; this kind of operator is distinguished by taking one vector and returning another. For example, consider an [[enlargement]], say by a factor of √2; such as is required to take one size of paper to another. It can also be applied geometrically to vectors as operands.
Formally, if ''V'' and ''W'' are vector spaces over the same ground [[field (mathematics)|field]] ''K'', we say that ''f'' : ''V'' → ''W'' is a linear transformation if for any two vectors ''x'' and ''y'' in ''V'' and any [[scalar]] ''a'' in ''K'', we have
In many important cases, operators transform functions into other functions. We also say an operator ''maps'' a function to another. The operator itself is a function, but has an attached ''type'' indicating the correct operand, and the kind of function returned. This extra data can be defined formally, using [[type theory]]; but in everyday usage saying ''operator'' flags its significance. Functions can therefore conversely be considered operators, for which we forget some of the type baggage, leaving just labels for the ___domain and codomain.
:<math>f(x+y)=f(x)+f(y) \,</math> (additivity)
:<math>f(ax)=af(x) \,</math> (homogeneity).
This is equivalent to saying that ''f'' "preserves linear combinations", i.e., for any vectors ''x''<sub>1</sub>, ..., ''x''<sub>''m''</sub> and scalars ''a''<sub>1</sub>, ..., ''a''<sub>''m''</sub>, we have
:<math>f(a_1 x_1+\cdots+a_m x_m)=a_1 f(x_1)+\cdots+a_m f(x_m).</math>
Occasionally, ''V'' and ''W'' can be considered as vector spaces over different ground fields, and it is then important to specify which field was used for the definition of "linear". If ''V'' and ''W'' are considered as spaces over the field ''K'' as above, we talk about ''K''-linear maps. For example, the conjugation of [[complex numbers]] is an '''R'''-linear map '''C''' → '''C''', but it is not '''C'''-linear.
==Operators and levels of abstraction==
== Examples ==
To begin with, the usage of '''''operator''''' in mathematics is subsumed in the usage of ''[[function (mathematics)|function]]'': an operator can be taken to be some special kind of function. The word is generally used to call attention to some aspect of its nature as a function. Since there are several such aspects that are of interest, there is no completely consistent terminology. Common are these:
* If ''A'' is an ''m'' × ''n'' [[matrix (mathematics)|matrix]], then ''A'' defines a linear transformation from '''R'''<sup>''n''</sup> to '''R'''<sup>''m''</sup> by sending the [[column vector]] ''x'' ∈ '''R'''<sup>''n''</sup> to the column vector ''Ax'' ∈ '''R'''<sup>''m''</sup>. Every linear transformation between [[finite-dimensional]] vector spaces arises in this fashion; see the following section.
*To draw attention to the [[function ___domain]], which may itself consist of [[vector (spatial)|vector]]s or [[function (mathematics)|function]]s, rather than just numbers. The [[expectation]] operator in [[probability theory]], for example, has [[random variable]]s as ___domain (and is also a [[functional (mathematics)|functional]]).
*To draw attention to the fact that the ___domain consists of pairs or [[tuple]]s of some sort, in which case ''operator'' is synonymous with the usual mathematical sense of [[operation]].
*To draw attention to the function codomain; for example a ''vector-valued function'' might be called an operator.
* The [[integration|integral]] yields a linear map from the space of all real-valued integrable functions on some [[interval (mathematics)|interval]] to '''R'''
A single operator might conceivably qualify under all three of these. Other important ideas are:
* [[derivative|Differentiation]] is a linear transformation from the space of all differentiable functions to the space of all functions.
*[[Overloading]], in which for example [[addition]], say, is thought of as a single ''operator'' able to act on numbers, vectors, matrices ... .
*Operators are often in practice just [[partial function]]s, a common phenomenon in the theory of [[differential equation]]s since there is no guarantee that the [[derivative]] of a function exists.
*Use of higher operations on operators, meaning that operators are themselves combined.
* If ''V'' and ''W'' are finite-dimensional vector spaces over the field ''F'', then functions that map linear transformations ''f'' : ''V'' → ''W'' to dim<sub>''F''</sub>(''W'')-by-dim<sub>''F''</sub>(''V'') matrices in the way described in the sequel are themselves linear transformations.
These are abstract ideas from mathematics, and [[computer science]]. They may however also be encountered in [[quantum mechanics]]. There [[Dirac]] drew a clear distinction between [[q-number]] or operator quantities, and [[c-number]]s which are conventional [[complex number]]s. The manipulation of q-numbers from that point on became basic to theoretical physics.
== Describing operatorsMatrices ==
If ''V'' and ''W'' are [[finite-dimensional]] and [[basis of a vector space|bases]] have been chosen, then every linear transformation from ''V'' to ''W'' can be represented as a [[matrix (mathematics)|matrix]]; this is useful because it allows concrete calculations. Conversely, matrices yield examples of linear transformations: if ''A'' is a real ''m''-by-''n'' matrix, then the rule
Operators are described usually by the number of operands:
''f''(''x'') = ''Ax'' describes a linear transformation '''R'''<sup>''n''</sup> → '''R'''<sup>''m''</sup> (see [[Euclidean space]]).
Let <math>\{v_1, \cdots, v_n\}</math> be a basis for ''V''. Then every vector ''v'' in ''V'' is uniquely determined by the coefficients <math>c_1, \cdots, c_n</math> in
* [[monadic]] or [[unary operator|unary]] operators take one argument.
:<math>c_1 v_1+\cdots+c_n v_n.</math>
* [[dyadic]] or [[binary operator|binary]] operators take two arguments.
If ''f'' : ''V'' → ''W'' is a linear transformation,
* [[triadic]] or [[ternary operator|ternary/trinary/tertiary]] operators take three arguments.
:<math>f(c_1 v_1+\cdots+c_n v_n)=c_1 f(v_1)+\cdots+c_n f(v_n),</math>
which implies that the function f is entirely determined by the values of <math>f(v_1),\cdots,f(v_n).</math>
Now let <math>\{w_1, \cdots, w_m\}</math> be a basis for ''W''. Then we can represent the values of each <math>f(v_j)</math> as
The number of operands is also called the '''''arity''''' of the operator. If an operator has an arity given as ''n''-ary (or ''n''-adic), then it takes ''n'' arguments. In programming, outside than [[functional programming]], the -ary terms are more often used than the other variants. See [[arity]] for an extensive list of the -ary endings.
:<math>f(v_j)=a_{1j} w_1 + \cdots + a_{mj} w_m.</math>
So the function f is entirely determined by the values of <math>a_{i,j}</math>.
If we put these values into an m-by-n matrix M, then we can conveniently use it to compute the value of f for any vector in ''V''. For if we place the values of <math>c_1, \cdots, c_n</math> in an n-by-1 matrix ''C'', we have ''MC'' = f(''v'').
It should be noted that there can be multiple matrices that represent a single linear transformation. This is because the values of the elements of the matrix depend on the bases that are chosen. Similarly, if we are given a matrix, we also need to know the bases that it uses in order to determine what linear transformation it represents.
=== Notations ===
== Forming new linear transformations from given ones ==
There are three major ''systematic'' ways of writing operators and their arguments. These are
* '''prefix''': where the operator name comes ''first'' and the arguments follow, for example:
::Q(''x''<sub>1</sub>, ''x''<sub>2</sub>,...,''x''<sub>n</sub>).
: In [[prefix notation]], the brackets are sometimes omitted if it is known that Q is an ''n''-ary operator.
* '''postfix''': where the operator name comes ''last'' and the arguments precede, for example:
::(''x''<sub>1</sub>, ''x''<sub>2</sub>,...,''x''<sub>n</sub>) Q
: In [[postfix notation]], the brackets are sometimes omitted if it is known that Q is an ''n''-ary operator.
* '''[[infix]]''': where the operator name comes ''between'' the arguments. This is awkward and not commonly used for operators other than binary operators. Infix style is written, for example, as
:: ''x''<sub>1</sub> Q ''x''<sub>2</sub>.
The composition of linear transformations is linear: if ''f'' : ''V'' → ''W'' and ''g'' : ''W'' → ''Z'' are linear, then so is ''g'' o ''f'' : ''V'' → ''Z''.
For operators on a single argument, prefix notation such as −7 is most common, but postfix such as 5! ([[factorial]]) or ''x''* is also usual.
If ''f''<sub>1</sub> : ''V'' → ''W'' and ''f''<sub>2</sub> : ''V'' → ''W'' are linear, then so is their sum ''f''<sub>1</sub> + ''f''<sub>2</sub> (which is defined by (''f''<sub>1</sub> + ''f''<sub>2</sub>)(''x'') = ''f''<sub>1</sub>(''x'') + ''f''<sub>2</sub>(''x'')).
There are other notations commonly met. Writing [[exponent]]s such as 2<sup>8</sup> is really a law unto itself, since it is postfix only as a unary operator applied to 2, but on a slant as binary operator. In some literature, a circumflex is written over the operator name. In certain circumstances, they are written ''unlike'' functions, when an operator has a single [[parameter|argument]] or ''operand''. For example, if the operator name is ''Q ''and the operand a function ''f'', we write ''Qf'' and not usually ''Q''(''f''); this latter notation may however be used for clarity if there is a product — for instance, ''Q''(''fg''). Later on we will use ''Q'' to denote a general operator, and ''x''<sub>i</sub> to denote the ''i''-th argument.
Notations for operators include the following. If ''f''( : ''xV'') is a function of→ ''xW'' is linear and ''Qa'' is thean generalelement operatorof wethe canground writefield ''QK'', actingthen onthe map ''faf'', asdefined by (''af''(Qf)(''x'') = ''a'' (''f''(''x'')), is also linear.
In the finite dimensional case and if bases have been chosen, then the composition of linear maps corresponds to the multiplication of [[matrix (math)|matrices]], the addition of linear maps corresponds to the addition of matrices, and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars.
Operators are often written in [[calligraphy]] to differentiate them from standard functions. For instance, the [[Fourier transform]] (an operator on functions) of f(t) (a function of t), which produces another function F(ω) (a function of ω), would be represented as <math>\mathcal{F}(f(t)) = F(\omega).</math>
== Endomorphisms and automorphisms ==
== Examples of mathematical operators==
A linear transformation ''f'' : ''V'' → ''V'' is an [[endomorphism]] of ''V''; the set of all such endomorphisms End(''V'') together with addition, composition and scalar multiplication as defined above forms an [[associative algebra]] with identity element over the field ''K'' (and in particular a [[ring (algebra)|ring]]). The identity element of this algebra is the [[identity map]] id : ''V'' → ''V''.
This section concentrates on illustrating the expressive power of the operator concept in mathematics. Please refer to individual topics pages for further details.
A [[bijective]] endomorphism of ''V'' is called an [[automorphism]] of ''V''. The composition of two automorphisms is again an automorphism, and the set of all automorphisms of ''V'' forms a [[group (math)|group]], the [[automorphism group]] of ''V'' which is denoted by Aut(''V'') or GL(''V'').
=== Linear operators ===
If ''V'' has finite dimension ''n'', then End(''V'') is [[isomorphism | isomorphic]] to the [[associative algebra]] of all ''n'' by ''n'' matrices with entries in ''K''. The automorphism group of ''V'' is [[group isomorphism|isomorphic]] to the [[general linear group]] GL(''n'', ''K'') of all ''n'' by ''n'' invertible matrices with entries in ''K''.
''Main article'': [[Linear transformation]]
== Kernel and image ==
The most common kind of operator encountered are ''linear operators''. In talking about linear operators, the operator is signified generally by the letters ''T'' or ''L''. Linear operators are those which satisfy the following conditions; take the general operator ''T'', the function acted on under the operator ''T'', written as f(x), and the constant a:
:<math>T\ (f(x)+g(x)) = T\ (f(x))+T\ (g(x))</math>
:<math>T\ (af(x)) = a\,T\ (f(x))</math>
If ''f'' : ''V'' → ''W'' is linear, we define the '''[[kernel (algebra)|kernel]]''' and the '''[[image (mathematics)|image]]''' of ''f'' by
Many operators are linear. For example, the differential operator and Laplacian operator, which we will see later.
:<math>\ker(f)=\{\,x\in V:f(x)=0\,\}</math>
:<math>\operatorname{im}(f)=\{\,f(x):x\in V\,\}</math>
ker(''f'') is a [[Linear algebra/Subspace|subspace]] of ''V'' and im(''f'') is a subspace of ''W''. The following [[dimension]] formula is often useful (but note that it only applies if V is finite dimensional):
:<math>
\dim(\ker( f ))
+ \dim(\operatorname{im}( f ))
= \dim( V ) \,</math>
The number dim(im(''f'')) is also called the ''rank of f'' and written as rk(''f''). If ''V'' and ''W'' are finite dimensional, bases have been chosen and ''f'' is represented by the matrix ''A'', then the rank of ''f'' is equal to the [[rank of a matrix|rank of the matrix]] ''A''. The dimension of the kernel is also known as the nullity of the matrix.
Linear operators are also known as [[linear transformation]]s or linear mappings. Many other operators one encounters in mathematics are linear, and linear operators are the most easily studied (Compare with [[nonlinearity]]).
== See also ==
Such an example of a linear transformation between vectors in '''R'''<sup>2</sup> is reflection, given a vector '''x'''=(x<sub>1</sub>, x<sub>2</sub>)
* [[Transformation matrix]]
:Q(x<sub>1</sub>, x<sub>2</sub>)=(-x<sub>1</sub>, x<sub>2</sub>)
* [[wikibooks:Algebra:Linear transformations]]
[[Category:Abstract algebra]]
We can also make sense of linear operators between generalisations of finite-[[dimension|dimensional]] vector spaces. For example, there is a large body of work dealing with linear operators on [[Hilbert space#Bounded Operators|Hilbert spaces]] and on [[Banach space#Linear operators|Banach spaces]]. See also [[operator algebra]].
[[Category:Linear algebra]]
[[de:Lineare Abbildung]]
=== Operators in probability theory ===
[[es:Transformación lineal]]
''Main article'': [[Probability theory]]
[[fr:Application linéaire]]
[[he:טרנספורמציה לינארית]]
Operators are also involved in probability theory.
[[nl:Lineaire transformatie]]
Such operators as expectation, variance, covariance, [[factorial]]s, et al.
[[ja:線型写像]]
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[[pl:Przekształcenie liniowe]]
[[ru:Линейное отображение]]
=== Operators in calculus ===
[[sv:Linjär operation]]
[[Calculus]] is, essentially, the study of one particular operator, and its behavior embodies and exemplifies the idea of the operator very clearly. The key operator studied is the [[differential operator]]. It is ''linear'', as are many of the operators constructed from it.
==== The differential operator ====
''Main article'': [[Differential operator]]
The [[differential operator]] is an operator which is fundamentally used in Calculus to denote the action of taking a derivative. Common notations are such ''d/dx'', y'(x) to denote the derivative of y(x). However here we will use the notation that is closest to the operator notation we have been using, that is, using D ''f'' to represent the action of taking the derivative of f.
==== Integral operators ====
Given that integration is an operator as well (inverse of differentiation), we have some important operators we can write in terms of integration.
===== Convolution =====
''Main article'': [[Convolution]]
The ''convolution'' of two functions is a mapping from two functions to one other, defined by an integral as follows:
If ''x''<sub>1</sub>=f(t) and ''x''<sub>2</sub>=g(t), define the operator Q such that;
: <math>Q\ x_1\ x_2\ = \int f(t) g(\tau - t) dt</math>
which we write as <math>(f*g)(\tau)</math>.
===== Fourier transform =====
''Main article'': [[Fourier transform]]
The Fourier transform is used in many areas, not only in mathematics, but in physics and in signal processing, to name a few. It is another integral operator; it is useful mainly because it converts a function on one (spatial) ___domain to a function on another (frequency) ___domain, in a way that is effectively [[invertible]]. Nothing significant is lost, because there is an inverse transform operator. In the simple case of [[periodic function]]s, this result is based on the theorem that any continuous periodic function can be represented as the sum of a series of [[sine wave]]s and cosine waves:
:<math> f(t) = {a_0 \over 2} + \sum_{n=1}^{\infty}{ a_n \cos ( \omega n t ) + b_n \sin ( \omega n t ) } </math>
When dealing with general function '''R->C''', the transform takes up an [[integral]] form:
:<math> f(t) = {1 \over \sqrt{2 \pi}} \int_{- \infty}^{+ \infty}{g( \omega ) \cdot \exp {( i \omega t )} \cdot d \omega } </math>
===== Laplacian transform =====
''Main article:'' [[Laplace transform]]
The ''Laplace transform'' is another integral operator and is involved in simplifying the process of solving differential equations.
Given f=f(s), it is defined by:
: <math>F(s) = (\mathcal{L}f)(s) =\int_0^\infty e^{-st} f(t)\,dt.</math>
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=== Fundamental operators on scalar and vector fields===
''Main articles:'' [[vector calculus]], [[scalar field]], [[gradient]], [[divergence]], and [[curl]]
Three main operators are key to [[vector calculus]], the operator ∇, known as [[gradient]], where at a certain point in a scalar field forms a vector which points in the direction of greatest change of that scalar field. In a vector field, the [[divergence]] is an operator that measures a vector field's tendency to originate from or converge upon a given point. [[Curl]], in a vector field, is a vector operator that shows a vector field's tendency to rotate about a point.
==Operators in physics==
''Main article:'' [[Operator (physics)]]
In [[physics]], an operator often takes on a more specialized meaning than in mathematics. Operators as [[observable]]s are a key part of the theory of [[quantum mechanics]]. In that context ''operator'' often means a [[linear transformation]] from a [[Hilbert space]] to another, or (more abstractly) an element of a [[C*-algebra]].
==See also==
* [[Logical operator]]
* [[function (mathematics)]]
* [[unary operation]]
* [[binary operation]]
* [[ternary operation]]
* [[relational operator]]
* [[Common operator notation|common operator notation]].
[[Category:Algebra]]
[[Category:Mathematical notation]]
[[ca:Operador matemàtic]]
[[de:Operator (Mathematik)]]
[[he:אופרטור (מתמטיקה)]]
[[it:Operatore]]
[[ja:演算子]]
[[sv:Operator]]
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