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--[[Utente:Unit|Unit]] ([[Discussioni utente:Unit|msg]])In [[analisi funzionale]], un '''operatore unitario''' è un [[operatore lineare]] ''U'' su uno [[spazio di Hilbert]] che soddisfa le seguenti richieste:
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".
:''U*U=UU*=I''
In the language of [[abstract algebra]], a linear transformation is a [[homomorphism]] of vector spaces.
:Il dominio di U coincide con l'intero spazio di Hilbert
La proprietà è euqivalente a una qualunque delle seguenti:
== Definition and first consequences ==
* ''U'' è una [[isometria]] [[suriettiva]]
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
:<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>
* ''U'' è [[suriettiva]] e preserva il [[prodotto interno]] sullo spazio di Hilbert, così che per tutti i [[Vettore (matematica)|vettori]] ''x'' e ''y'' dello spazio di hilbert vale
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.
:<math>\langle Ux, Uy \rangle = \langle x, y \rangle.</math>
* 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.
''Ogni matrice unitaria è un operatore unitario ''
* The [[integration|integral]] yields a linear map from the space of all real-valued integrable functions on some [[interval (mathematics)|interval]] to '''R'''
* [[derivative|Differentiation]] is a linear transformation from the space of all differentiable functions to the space of all functions.
In [[functional analysis]], a '''unitary operator''' is a [[bounded linear operator]] ''U'' on a [[Hilbert space]] satisfying
* 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.
:''U*U=UU*=I''
== Matrices ==
where ''I'' is the [[identity]] operator. This property is equivalent to any of the following:
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
''f''(''x'') = ''Ax'' describes a linear transformation '''R'''<sup>''n''</sup> → '''R'''<sup>''m''</sup> (see [[Euclidean space]]).
* ''U'' is a [[surjective]] [[isometry]]
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
:<math>c_1 v_1+\cdots+c_n v_n.</math>
If ''f'' : ''V'' → ''W'' is a linear transformation,
:<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>
* ''U'' is [[surjective]] and preserves the [[inner product]] on the Hilbert space, so that for all [[vector]]s ''x'' and ''y'' in the Hilbert space,
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
:<math>f(v_j)=a_{1j}\langle w_1Ux, +Uy \cdotsrangle += a_{mj}\langle w_mx, y \rangle. t= p \cdot x'</math>
So the function f is entirely determined by the values of <math>a_{i,j}</math>.
[[Unitary matrix|Unitary matrices]] are precisely the unitary operators on finite-dimensional Hilbert spaces, so the notion of a unitary operator is a generalisation of the notion of a unitary matrix.
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'').
Unitary operators implement [[isomorphism]]s between [[operator algebra]]s.
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.
--[[Utente:Unit|Unit]] ([[Discussioni utente:Unit|msg]]) --[[Utente:Unit|Unit]] ([[Discussioni utente:Unit|msg]]) 22:21, 8 gen 2012 (CET) --22:21, 8 gen 2012 (CET)
== Forming new linear transformations from given ones ==
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''.
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'')).
If ''f'' : ''V'' → ''W'' is linear and ''a'' is an element of the ground field ''K'', then the map ''af'', defined by (''af'')(''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.
== Endomorphisms and automorphisms ==
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''.
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'').
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''.
== Kernel and image ==
If ''f'' : ''V'' → ''W'' is linear, we define the '''[[kernel (algebra)|kernel]]''' and the '''[[image (mathematics)|image]]''' of ''f'' by
:<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.
== See also ==
* [[Transformation matrix]]
* [[wikibooks:Algebra:Linear transformations]]
[[Category:Abstract algebra]]
[[Category:Linear algebra]]
[[de:Lineare Abbildung]]
[[es:Transformación lineal]]
[[fr:Application linéaire]]
[[he:טרנספורמציה לינארית]]
[[nl:Lineaire transformatie]]
[[ja:線型写像]]
[[pl:Przekształcenie liniowe]]
[[ru:Линейное отображение]]
[[sv:Linjär operation]]
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