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| In [[mathematics]], a '''linear map''' (also called a '''linear mapping''', '''linear [[Transformation (function)|transformation]]''' or, in some contexts, '''[[linear function]]''') is a [[function (mathematics)|mapping]] {{math|''V'' ↦ ''W''}} between two [[Module (mathematics)|module]]s (including [[vector space]]s) that preserves (in the sense defined below) the operations of addition and [[scalar (mathematics)|scalar]] multiplication. An important special case is when {{math|''V'' {{=}} ''W''}}, in which case the map is called a '''linear operator''', or an [[endomorphisms|endomorphism]] of {{math|''V''}}. Sometimes the definition of a [[linear function]] coincides with that of a linear map, while in [[analytic geometry]] it does not.
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| A linear map always [[map (mathematics)|maps]] linear subspaces to linear subspaces (possibly of a lower dimension); for instance it maps a plane through the origin to a plane, straight line or point.
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| In the language of [[abstract algebra]], a linear map is a [[homomorphism]] of modules. In the language of [[category theory]] it is a [[morphism]] in the [[category of modules]] over a given [[Ring (mathematics)|ring]].
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| == Definition and first consequences ==
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| Let ''V'' and ''W'' be vector spaces over the same [[field (mathematics)|field]] ''K''. A function ''f'': ''V'' → ''W'' is said to be a ''linear map'' if for any two vectors '''x''' and '''y''' in ''V'' and any scalar α in ''K'', the following two conditions are satisfied:
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| {|
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| | style="padding:0 20pt"|<math>f(\mathbf{x}+\mathbf{y}) = f(\mathbf{x})+f(\mathbf{y}) \!</math>
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| | [[Additive function|additivity]]
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| | style="padding:0 20pt"|<math>f(\alpha \mathbf{x}) = \alpha f(\mathbf{x}) \!</math>
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| | [[Homogeneous function|homogeneity]] of degree 1
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| |}
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| This is equivalent to requiring the same for any linear combination of vectors, i.e. that for any vectors ''x''<sub>1</sub>, ..., ''x<sub>m</sub>'' ∈ ''V'' and scalars ''a''<sub>1</sub>, ..., ''a<sub>m</sub>'' ∈ ''K'', the following equality holds:
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| : <math>f(a_1 \mathbf{x}_1+\cdots+a_m \mathbf{x}_m) = a_1 f(\mathbf{x}_1)+\cdots+a_m f(\mathbf{x}_m). \!</math>
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| Denoting the zero elements of the vector spaces ''V'' and ''W'' by '''0'''<sub>''V''</sub> and '''0'''<sub>''W''</sub> respectively, it follows that ''f''('''0'''<sub>''V''</sub>) = '''0'''<sub>''W''</sub> because letting α = 0 in the equation for homogeneity of degree 1,
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| :<math>f(\mathbf{0}_{V}) = f(0 \cdot \mathbf{0}_{V}) = 0 \cdot f(\mathbf{0}_{V}) = \mathbf{0}_{W} .</math>
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| Occasionally, ''V'' and ''W'' can be considered to be vector spaces over different fields. It is then necessary to specify which of these ground fields is being used in 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 [[complex conjugate|conjugation]] of [[complex numbers]] is an '''R'''-linear map '''C''' → '''C''', but it is not '''C'''-linear.
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| A linear map from ''V'' to ''K'' (with ''K'' viewed as a vector space over itself) is called a [[linear functional]].
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| These statements generalize to any left-module <sub>''R''</sub>''M'' over a ring ''R'' without modification.
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| == Examples ==
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| * The zero map is always linear
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| * The [[Identity function|identity map]] of any vector space is a linear operator
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| * Any [[homothecy]] centered in the origin of a vector space, <math>v\mapsto cv</math> where ''c'' is a scalar, is a linear operator.
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| * For real numbers, the map <math>x\mapsto x^2</math> is not linear.
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| * For real numbers, the map <math>x\mapsto x+1</math> is not linear (but is an [[affine transformation]]; <math>y=x+1</math> is a [[linear equation]], as used in [[analytic geometry]].)
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| * If ''A'' is a real ''m'' × ''n'' [[matrix (mathematics)|matrix]], then ''A'' defines a linear map from '''R'''<sup>''n''</sup> to '''R'''<sup>''m''</sup> by sending the [[column vector]] '''x''' ∈ '''R'''<sup>''n''</sup> to the column vector ''A'''''x''' ∈ '''R'''<sup>''m''</sup>. Conversely, any linear map between [[finite-dimensional]] vector spaces can be represented in this manner; see the following section.
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| * [[Derivative|Differentiation]] defines a linear map from the space of all differentiable functions to the space of all functions. It also defines a linear operator on the space of all [[smooth function]]s.
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| * The (definite) [[integral]] over some [[interval (mathematics)|interval]] ''I'' is a linear map from the space of all real-valued integrable functions on ''I'' to '''R'''
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| * The (indefinite) [[integral]] (or [[antiderivative]]) with a fixed starting point defines a linear map from the space of all real-valued integrable functions on '''R''' to the space of all real-valued functions on '''R'''. Without fixed starting point it does not define a mapping at all, as the presence of a constant of integration in the result means it produces an infinite number of outputs for a single input.
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| * If ''V'' and ''W'' are finite-dimensional vector spaces over a field ''F'', then functions that send linear maps ''f'' : ''V'' → ''W'' to dim<sub>''F''</sub>(''W'') × dim<sub>''F''</sub>(''V'') matrices in the way described in the sequel are themselves linear maps (indeed linear isomorphisms).
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| * The [[expected value]] of a [[Random variable#Definition|random variable]] (which is in fact a function, and as such member of a vector space) is linear, as for random variables ''X'' and ''Y'' we have E[''X'' + ''Y''] = E[''X''] + E[''Y''] and E[''aX''] = ''a''E[''X''], but the [[variance]] of a random variable is not linear.
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| == Matrices ==
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| {{main|Transformation matrix}}
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| If ''V'' and ''W'' are [[finite-dimensional]], and one has chosen [[basis of a vector space|bases]] in those spaces, then every linear map 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 maps: if ''A'' is a real ''m'' × ''n'' matrix, then the rule
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| ''f''('''x''') = ''A'''''x''' describes a linear map '''R'''<sup>''n''</sup> → '''R'''<sup>''m''</sup> (see [[Euclidean space]]).
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| Let {'''v'''<sub>1</sub>, ..., '''v'''<sub>''n''</sub>} be a basis for ''V''. Then every vector '''v''' in ''V'' is uniquely determined by the coefficients ''c''<sub>1</sub>, ..., ''c<sub>n</sub>'' in
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| : <math>c_1 \mathbf{v}_1+\cdots+c_n \mathbf{v}_n.</math>
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| If ''f'': ''V'' → ''W'' is a linear map,
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| : <math>f(c_1 \mathbf{v}_1+\cdots+c_n \mathbf{v}_n)=c_1 f(\mathbf{v}_1)+\cdots+c_n f(\mathbf{v}_n),</math>
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| which implies that the function ''f'' is entirely determined by the values of ''f''('''v'''<sub>1</sub>), ..., ''f''('''v'''<sub>''n''</sub>).
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| Now let {'''w'''<sub>1</sub>, ..., '''w'''<sub>''m''</sub>} be a basis for ''W''. Then we can represent the values of each ''f''('''v'''<sub>''j''</sub>) as
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| : <math>f(\mathbf{v}_j)=a_{1j} \mathbf{w}_1 + \cdots + a_{mj} \mathbf{w}_m.</math>
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| Thus, the function ''f'' is entirely determined by the values of ''a<sub>ij</sub>''.
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| If we put these values into an ''m'' × ''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 ''c''<sub>1</sub>, ..., ''c<sub>n</sub>'' in an ''n'' × 1 matrix ''C'', we have ''MC'' = the ''m'' × 1 matrix whose ''i''th element is the coordinate of ''f''('''v''') which belongs to the base '''w'''<sub>''i''</sub>.
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| A single linear map may be represented by many matrices. This is because the values of the elements of the matrix depend on the bases that are chosen.
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| == Examples of linear transformation matrices ==
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| In two-[[dimension]]al space '''R'''<sup>2</sup> linear maps are described by [[2 × 2 real matrices]]. These are some examples:
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| * [[Rotation (mathematics)|rotation]] by 90 degrees counterclockwise:
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| *: <math>\mathbf{A}=\begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix}</math>
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| * [[Rotation (mathematics)|rotation]] by angle ''θ'' counterclockwise:
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| *: <math>\mathbf{A}=\begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}</math>
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| * [[Reflection (mathematics)|reflection]] against the ''x'' axis:
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| *: <math>\mathbf{A}=\begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix}</math>
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| * [[Reflection (mathematics)|reflection]] against the ''y'' axis:
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| *: <math>\mathbf{A}=\begin{pmatrix}-1 & 0\\ 0 & 1\end{pmatrix}</math>
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| * [[Scaling (geometry)|scaling]] by 2 in all directions:
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| *: <math>\mathbf{A}=\begin{pmatrix}2 & 0\\ 0 & 2\end{pmatrix}</math>
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| * [[shear mapping|horizontal shear mapping]]:
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| *: <math>\mathbf{A}=\begin{pmatrix}1 & m\\ 0 & 1\end{pmatrix}</math>
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| * [[squeeze mapping]]:
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| *: <math>\mathbf{A}=\begin{pmatrix}k & 0\\ 0 & 1/k\end{pmatrix}</math>
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| * [[Projection (linear algebra)|projection]] onto the ''y'' axis:
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| *: <math>\mathbf{A}=\begin{pmatrix}0 & 0\\ 0 & 1\end{pmatrix}.</math>
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| == Forming new linear maps from given ones ==
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| The composition of linear maps is linear: if {{nowrap|''f'' : ''V'' → ''W''}} and {{nowrap|''g'' : ''W'' → ''Z''}} are linear, then so is their [[Relation composition|composition]] {{nowrap|''g'' ∘ ''f'' : ''V'' → ''Z''}}. It follows from this that the [[class (set theory)|class]] of all vector spaces over a given field ''K'', together with ''K''-linear maps as [[morphism]]s, forms a [[category (mathematics)|category]].
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| The [[inverse function|inverse]] of a linear map, when defined, is again a linear map.
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| If {{nowrap|''f''<sub>1</sub> : ''V'' → ''W''}} and {{nowrap|''f''<sub>2</sub> : ''V'' → ''W''}} are linear, then so is their sum {{nowrap|''f''<sub>1</sub> + ''f''<sub>2</sub>}} (which is defined by {{nowrap|1=(''f''<sub>1</sub> + ''f''<sub>2</sub>)('''x''') = ''f''<sub>1</sub>('''x''') + ''f''<sub>2</sub>('''x'''))}}.
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| If {{nowrap|''f'' : ''V'' → ''W''}} is linear and ''a'' is an element of the ground field ''K'', then the map ''af'', defined by {{nowrap|1=(''af'')('''x''') = ''a''(''f''('''x'''))}}, is also linear.
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| Thus the set {{nowrap|''L''(''V'', ''W'')}} of linear maps from ''V'' to ''W'' itself forms a vector space over ''K'', sometimes denoted {{nowrap|Hom(''V'', ''W'')}}. Furthermore, in the case that {{nowrap|1=''V'' = ''W''}}, this vector space (denoted End(''V'')) is an [[associative algebra]] under [[composition of maps]], since the composition of two linear maps is again a linear map, and the composition of maps is always associative. This case is discussed in more detail below.
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| Given again the finite-dimensional case, if bases have been chosen, then the composition of linear maps corresponds to the [[matrix multiplication]], the addition of linear maps corresponds to the [[matrix addition]], and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars.
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| == Endomorphisms and automorphisms ==
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| 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 multiplicative identity element of this algebra is the [[identity function|identity map]] id: ''V'' → ''V''.
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| An endomorphism of ''V'' that is also an [[isomorphism]] 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''). Since the automorphisms are precisely those [[endomorphisms]] which possess inverses under composition, Aut(''V'') is the group of [[Unit (ring theory)|units]] in the ring End(''V'').
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| If ''V'' has finite dimension ''n'', then End(''V'') is [[isomorphism|isomorphic]] to the [[associative algebra]] of all ''n'' × ''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'' × ''n'' invertible matrices with entries in ''K''.
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| == Kernel, image and the rank–nullity theorem ==
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| If ''f'' : ''V'' → ''W'' is linear, we define the [[kernel (linear operator)|kernel]] and the [[image (mathematics)|image]] or [[range (mathematics)|range]] of ''f'' by
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| : <math>\operatorname{\ker}(f)=\{\,x\in V:f(x)=0\,\}</math>
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| : <math>\operatorname{im}(f)=\{\,w\in W:w=f(x),x\in V\,\}</math>
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| ker(''f'') is a [[Linear subspace|subspace]] of ''V'' and im(''f'') is a subspace of ''W''. The following [[dimension]] formula is known as the [[rank–nullity theorem]]:
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| : <math> \dim(\ker( f ))+ \dim(\operatorname{im}( f ))= \dim( V ).</math>
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| The number dim(im(''f'')) is also called the ''rank of f'' and written as rank(''f''), or sometimes, ρ(''f''); the number dim(ker(''f'')) is called the ''nullity of f'' and written as null(''f'') or ν(''f''). If ''V'' and ''W'' are finite-dimensional, bases have been chosen and ''f'' is represented by the matrix ''A'', then the rank and nullity of ''f'' are equal to the [[rank of a matrix|rank]] and [[Kernel (matrix)#Subspace properties|nullity]] of the matrix ''A'', respectively.
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| == Cokernel ==
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| {{main|Cokernel}}
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| A subtler invariant of a linear transformation is the [[cokernel|''co''kernel]], which is defined as
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| : <math>\mathrm{coker}\,f := W/f(V) = W/\mathrm{im}(f).</math>
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| This is the ''dual'' notion to the kernel: just as the kernel is a ''sub''space of the ''domain,'' the co-kernel is a [[quotient space (linear algebra)|''quotient'' space]] of the ''target.''
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| Formally, one has the [[exact sequence]]
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| : <math>0 \to \ker f \to V \to W \to \mathrm{coker}\,f \to 0.</math>
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| These can be interpreted thus: given a linear equation ''f''('''v''') = '''w''' to solve,
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| * the kernel is the space of ''solutions'' to the ''homogeneous'' equation ''f''('''v''') = 0, and its dimension is the number of ''degrees of freedom'' in a solution, if it exists;
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| * the co-kernel is the space of ''constraints'' that must be satisfied if the equation is to have a solution, and its dimension is the number of constraints that must be satisfied for the equation to have a solution.
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| The dimension of the co-kernel and the dimension of the image (the rank) add up to the dimension of the target space. For finite dimensions, this means that the dimension of the quotient space ''W''/''f''(''V'') is the dimension of the target space minus the dimension of the image.
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| As a simple example, consider the map ''f'': '''R'''<sup>2</sup> → '''R'''<sup>2</sup>, given by ''f''(''x'', ''y'') = (0, ''y''). Then for an equation ''f''(''x'', ''y'') = (''a'', ''b'') to have a solution, we must have ''a'' = 0 (one constraint), and in that case the solution space is (''x'', ''b'') or equivalently stated, (0, ''b'') + (''x'', 0), (one degree of freedom). The kernel may be expressed as the subspace (''x'', 0) < ''V'': the value of ''x'' is the freedom in a solution – while the cokernel may be expressed via the map ''W'' → '''R''', <math> (a,b) \mapsto (a):</math> given a vector (''a'', ''b''), the value of ''a'' is the ''obstruction'' to there being a solution.
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| An example illustrating the infinite-dimensional case is afforded by the map ''f'': '''R'''<sup>∞</sup> → '''R'''<sup>∞</sup>, <math>\{a_n\} \mapsto \{b_n\}</math> with ''b''<sub>1</sub> = 0 and ''b''<sub>''n'' + 1</sub> = ''a<sub>n</sub>'' for ''n'' > 0. Its image consists of all sequences with first element 0, and thus its cokernel consists of the classes of sequences with identical first element. Thus, whereas its kernel has dimension 0 (it maps only the zero sequence to the zero sequence), its co-kernel has dimension 1. Since the domain and the target space are the same, the rank and the dimension of the kernel add up to the same [[cardinal number#Cardinal addition|sum]] as the rank and the dimension of the co-kernel ( <math>\aleph_0 + 0 = \aleph_0 + 1</math> ), but in the infinite-dimensional case it cannot be inferred that the kernel and the co-kernel of an [[endomorphism]] have the same dimension (0 ≠ 1). The reverse situation obtains for the map ''h'': '''R'''<sup>∞</sup> → '''R'''<sup>∞</sup>, <math>\{a_n\} \mapsto \{c_n\}</math> with ''c<sub>n</sub>'' = ''a''<sub>''n'' + 1</sub>. Its image is the entire target space, and hence its co-kernel has dimension 0, but since it maps all sequences in which only the first element is non-zero to the zero sequence, its kernel has dimension 1.
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| === Index ===
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| For a linear operator with finite-dimensional kernel and co-kernel, one may define ''index'' as:
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| : <math>\mathrm{ind}\,f := \dim \ker f - \dim \mathrm{coker}\,f,</math>
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| namely the degrees of freedom minus the number of constraints.
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| For a transformation between finite-dimensional vector spaces, this is just the difference dim(''V'') − dim(''W''), by rank–nullity. This gives an indication of how many solutions or how many constraints one has: if mapping from a larger space to a smaller one, the map may be onto, and thus will have degrees of freedom even without constraints. Conversely, if mapping from a smaller space to a larger one, the map cannot be onto, and thus one will have constraints even without degrees of freedom.
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| The index comes of its own in infinite dimensions: it is how [[homology (mathematics)|homology]]{{Citation needed|date=October 2013}} is defined, which is a central theory in algebra and [[algebraic topology]]; the index of an operator is precisely the [[Euler characteristic]] of the 2-term complex 0 → ''V'' → ''W'' → 0. In [[operator theory]], the index of [[Fredholm]] operators is an object of study, with a major result being the [[Atiyah–Singer index theorem]]{{Citation needed|date=October 2013}}.
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| == Algebraic classifications of linear transformations ==
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| No classification of linear maps could hope to be exhaustive. The following incomplete list enumerates some important classifications that do not require any additional structure on the vector space.
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| Let ''V'' and ''W'' denote vector spaces over a field, ''F''. Let ''T'': ''V'' → ''W'' be a linear map.
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| * ''T'' is said to be ''[[injective]]'' or a ''[[monomorphism]]'' if any of the following equivalent conditions are true:
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| ** ''T'' is [[injective|one-to-one]] as a map of [[Set (mathematics)|sets]].
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| ** ker''T'' = {0<sub>''V''</sub>}
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| ** ''T'' is [[monic morphism|monic]] or left-cancellable, which is to say, for any vector space ''U'' and any pair of linear maps ''R'': ''U'' → ''V'' and ''S'': ''U'' → ''V'', the equation ''TR'' = ''TS'' implies ''R'' = ''S''.
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| ** ''T'' is [[inverse (ring theory)|left-invertible]], which is to say there exists a linear map ''S'': ''W'' → ''V'' such that ''ST'' is the [[Identity function|identity map]] on ''V''.
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| * ''T'' is said to be ''[[surjective]]'' or an ''[[epimorphism]]'' if any of the following equivalent conditions are true:
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| ** ''T'' is [[surjective|onto]] as a map of [[Set (mathematics)|sets]].
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| ** [[cokernel|coker]] ''T'' = {0<sub>''W''</sub>}
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| ** ''T'' is [[epimorphism|epic]] or right-cancellable, which is to say, for any vector space ''U'' and any pair of linear maps ''R'': ''W'' → ''U'' and ''S'': ''W'' → ''U'', the equation ''RT'' = ''ST'' implies ''R'' = ''S''.
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| ** ''T'' is [[inverse (ring theory)|right-invertible]], which is to say there exists a linear map ''S'': ''W'' → ''V'' such that ''TS'' is the [[Identity function|identity map]] on ''W''.
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| * ''T'' is said to be an ''[[isomorphism]]'' if it is both left- and right-invertible. This is equivalent to ''T'' being both one-to-one and onto (a [[bijection]] of sets) or also to ''T'' being both epic and monic, and so being a [[bimorphism]].
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| * If ''T'': ''V'' → ''V'' is an endomorphism, then:
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| ** If, for some positive integer ''n'', the ''n''-th iterate of ''T'', ''T<sup>n</sup>'', is identically zero, then ''T'' is said to be [[nilpotent]].
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| ** If ''T''<sup>2</sup> = ''T'', then ''T'' is said to be [[idempotent]]
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| ** If ''T'' = ''kI'', where ''k'' is some scalar, then ''T'' is said to be a scaling transformation or scalar multiplication map; see [[scalar matrix]].
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| == Change of basis ==
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| Given a linear map whose matrix is ''A'', in the basis ''B'' of the space it transforms vectors coordinates [u] as [v] = ''A''[u]. As vectors change with the inverse of ''B'', its inverse transformation is [v] = ''B''[v'].
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| Substituting this in the first expression
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| :<math>B[v'] = AB[u'] </math>
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| hence
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| :<math>[v'] = B^{-1}AB[u'] = A'[u'].</math>
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| Therefore the matrix in the new basis is ''A′'' = ''B''<sup>−1</sup>''AB'', being ''B'' the matrix of the given basis.
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| Therefore linear maps are said to be 1-co 1-contra [[Covariance and contravariance of vectors|-variant]] objects, or type (1, 1) [[tensor]]s.
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| == Continuity ==
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| {{main|Discontinuous linear map}}
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| A ''linear transformation'' between [[topological vector space]]s, for example [[normed space]]s, may be [[continuous function (topology)|continuous]]. If its domain and codomain are the same, it will then be a [[continuous linear operator]]. A linear operator on a normed linear space is continuous if and only if it is [[bounded operator|bounded]], for example, when the domain is finite-dimensional. An infinite-dimensional domain may have [[discontinuous linear operator]]s.
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| An example of an unbounded, hence discontinuous, linear transformation is differentiation on the space of smooth functions equipped with the supremum norm (a function with small values can have a derivative with large values, while the derivative of 0 is 0). For a specific example, sin(''nx'')/''n'' converges to 0, but its derivative cos(''nx'') does not, so differentiation is not continuous at 0 (and by a variation of this argument, it is not continuous anywhere).
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| == Applications ==
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| A specific application of linear maps is for geometric transformations, such as those performed in [[computer graphics]], where the translation, rotation and scaling of 2D or 3D objects is performed by the use of a [[transformation matrix]]. Linear mappings also are used as a mechanism for describing change: for example in calculus correspond to derivatives; or in relativity, used as a device to keep track of the local transformations of reference frames.
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| Another application of these transformations is in [[compiler optimizations]] of nested-loop code, and in parallelizing compiler techniques.
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| == See also ==
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| {{Wikibooks|Linear Algebra/Linear Transformations}}
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| * [[Affine transformation]]
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| * [[Linear equation]]
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| * [[Bounded operator]]
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| * [[Antilinear map]]
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| * [[Semilinear transformation]]
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| * [[Continuous linear operator]]
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| == References ==
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| {{reflist}}
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| {{More footnotes|date=March 2009}}
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| * {{Citation | last1=Halmos | first1=Paul R. | author1-link=Paul R. Halmos | title=Finite-dimensional vector spaces | publisher=[[Springer-Verlag]] | location=New York | isbn=978-0-387-90093-3 | year=1974}}
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| * {{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Linear algebra | publisher=[[Springer-Verlag]] | location=New York | isbn=978-0-387-96412-6 | year=1987}}
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| {{linear algebra}}
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| [[Category:Abstract algebra]]
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| [[Category:Functions and mappings]]
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| [[Category:Linear algebra]]
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| [[Category:Transformation (function)]]
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