Linear map - WikiHQ

In mathematics, and more specifically in linear algebra, a linear map (or linear mapping) is a particular kind of function between vector spaces, which respects the basic operations of vector addition and scalar multiplication. A standard example of a linear map is an <math>m\times n</math> matrix, which takes vectors in <math>n</math>-dimensions into vectors in <math>m</math>-dimensions in a way that is compatible with addition of vectors, and multiplication of vectors by scalars.

A linear map is a homomorphism of vector spaces. Thus, a linear map <math>T:V\to W</math> satisfies , where <math>a</math> and <math>b</math> are scalars, and <math>x</math> and <math>y</math> are vectors (elements of the vector space ). A linear mapping always maps the origin of <math>V</math> to the origin of , and linear subspaces of <math>V</math> onto linear subspaces in <math>W</math> (possibly of a lower dimension); for example, it maps a plane through the origin in <math>V</math> to either a plane through the origin in , a line through the origin in , or just the origin in . Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations.

Definition and first consequences

Let <math>V</math> and <math>W</math> be vector spaces over the same field , such as the real or complex numbers.

A function <math>f: V \to W</math> is said to be a linear map if for any two vectors <math display="inline">\mathbf{u}, \mathbf{v} \in V</math> and any scalar <math>c \in K</math> the following two conditions are satisfied:

Additivity / operation of addition <math display=block>f(\mathbf{u} + \mathbf{v}) = f(\mathbf{u}) + f(\mathbf{v})</math>
Homogeneity of degree 1 / operation of scalar multiplication <math display=block>f(c \mathbf{u}) = c f(\mathbf{u})</math>

Thus, a linear map is said to be operation preserving. In other words, it does not matter whether the linear map is applied before (the right sides of the above examples) or after (the left sides of the examples) the operations of addition and scalar multiplication.

By the associativity of the addition operation denoted as +, for any vectors <math display="inline"> \mathbf{u}_1, \ldots, \mathbf{u}_n \in V</math> and scalars , the following equality holds:

<math display="block">f(c_1 \mathbf{u}_1 + \cdots + c_n \mathbf{u}_n) = c_1 f(\mathbf{u}_1) + \cdots + c_n f(\mathbf{u}_n).</math> Thus a linear map is one which preserves linear combinations.

Denoting the zero elements of the vector spaces <math>V</math> and <math>W</math> by <math display="inline">\mathbf{0}_V</math> and <math display="inline">\mathbf{0}_W</math> respectively, it follows that . Let <math>c = 0</math> and <math display="inline">\mathbf{v} \in V</math> in the equation for homogeneity of degree 1:

<math display="block">f(\mathbf{0}_V) = f(0\mathbf{v}) = 0f(\mathbf{v}) = \mathbf{0}_W.</math>

A linear map <math>V \to K</math> with <math>K</math> viewed as a one-dimensional vector space over itself is called a linear functional.

These statements generalize to any left-module <math display="inline">{}_R M</math> over a ring <math>R</math> without modification, and to any right-module upon reversing of the scalar multiplication.

Examples

The unique map of the form <math>T:\{\vec{0}\} \to \{\vec{0}\}</math> is linear.
A prototypical example that gives linear maps their name is a function , of which the graph is a line through the origin.thumb|Examples of linear transformations used in computer graphics
More generally, any homothety <math display="inline">\mathbf{v} \mapsto c\mathbf{v}</math> centered in the origin of a vector space is a linear map (here is a scalar).
The zero map <math display="inline">\mathbf x \mapsto \mathbf 0</math> between two vector spaces (over the same field) is linear.
The identity map on any module is a linear operator.
For real numbers, the map <math display="inline">x \mapsto x^2</math> is not linear.
For real numbers, the map <math display="inline">x \mapsto x + 1</math> is not linear (but is an affine transformation).
If <math>A</math> is a <math>m \times n</math> real matrix, then <math>A</math> defines a linear map from <math>\R^n</math> to <math>\R^m</math> by sending a column vector <math>\mathbf x \in \R^n</math> to the column vector . Conversely, any linear map between finite-dimensional vector spaces can be represented in this manner; see ', below.
If <math display="inline">f: V \to W</math> is an isometry between real normed spaces such that <math display="inline"> f(0) = 0</math> then <math>f</math> is a linear map. This result is not necessarily true for complex normed space.
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 functions (a linear operator is a linear endomorphism, that is, a linear map with the same domain and codomain). Indeed, <math display="block">\frac{d}{dx} \left( a f(x) + b g(x) \right) = a \frac{d f(x)}{dx} + b \frac{d g( x)}{dx}.</math>
A definite integral over some interval is a linear map from the space of all real-valued integrable functions on to . Indeed, <math display="block">\int_u^v \left(af(x) + bg(x)\right) dx = a\int_u^v f(x) dx + b\int_u^v g(x) dx . </math>
An indefinite integral (or antiderivative) with a fixed integration starting point defines a linear map from the space of all real-valued integrable functions on <math>\R</math> to the space of all real-valued, differentiable functions on . Without a fixed starting point, the antiderivative maps to the quotient space of the differentiable functions by the linear space of constant functions.
If <math>V</math> and <math>W</math> are finite-dimensional vector spaces over a field , of respective dimensions and , then the function that maps linear maps <math display="inline">f: V \to W</math> to matrices in the way described in ' (below) is a linear map, and even a linear isomorphism.
The expected value of a random variable is a linear function of the random variable: for random variables <math>X</math> and <math>Y</math> we have <math>E[X + Y] = E[X] + E[Y]</math> and . The conditional expectation is as well. But the variance of a random variable is not linear, because for instance .

File:Streckung eines Vektors.gif|The function <math display="inline">f:\R^2 \to \R^2</math> with <math display="inline">f(x, y) = (2x, y)</math> is a linear map. This function scales the <math display="inline">x</math> component of a vector by the factor .

File:Streckung der Summe zweier Vektoren.gif|The function <math display="inline">f(x, y) = (2x, y)</math> is additive: It does not matter whether vectors are first added and then mapped or whether they are mapped and finally added: <math display="inline">f(\mathbf a + \mathbf b) = f(\mathbf a) + f(\mathbf b)</math>

File:Streckung homogenitaet Version 3.gif|The function <math display="inline">f(x, y) = (2x, y)</math> is homogeneous: It does not matter whether a vector is first scaled and then mapped or first mapped and then scaled: <math display="inline">f(\lambda \mathbf a) = \lambda f(\mathbf a)</math>

</gallery>

Linear endomorphisms and isomorphisms

If a linear map is a bijection then it is called a . In the case where , a linear map is called a linear endomorphism. Sometimes the term refers to this case, but the term "linear operator" can have different meanings for different conventions.

Linear extensions

Often, a linear map is constructed by defining it on a subset of a vector space and then to the linear span of the domain.

Suppose <math>X</math> and <math>Y</math> are vector spaces and <math>f : S \to Y</math> is a function defined on some subset .

Then a of <math>f</math> to <math>X,</math> if it exists, is a linear map <math>F : X \to Y</math> defined on <math>X</math> that extends <math>f</math> (meaning that <math>F(s) = f(s)</math> for all ) and takes its values from the codomain of .

When the subset <math>S</math> is a vector subspace of <math>X</math> then a (-valued) linear extension of <math>f</math> to all of <math>X</math> is guaranteed to exist if (and only if) <math>f : S \to Y</math> is a linear map. In particular, if <math>f</math> has a linear extension to <math>\operatorname{span} S,</math> then it has a linear extension to all of .

The map <math>f : S \to Y</math> can be extended to a linear map <math>F : \operatorname{span} S \to Y</math> if and only if whenever <math>n > 0</math> is an integer, <math>c_1, \ldots, c_n</math> are scalars, and <math>s_1, \ldots, s_n \in S</math> are vectors such that , then necessarily .

If a linear extension of <math>f : S \to Y</math> exists then the linear extension <math>F : \operatorname{span} S \to Y</math> is unique and

<math display=block>F\left(c_1 s_1 + \cdots c_n s_n\right) = c_1 f\left(s_1\right) + \cdots + c_n f\left(s_n\right)</math>

holds for all , and <math>s_1, \ldots, s_n</math> as above.

If <math>S</math> is linearly independent then every function <math>f : S \to Y</math> into any vector space has a linear extension to a (linear) map <math>\operatorname{span} S \to Y</math> (the converse is also true).

For example, if <math>X = \R^2</math> and <math>Y = \R</math> then the assignment <math>(1, 0) \to -1</math> and <math>(0, 1) \to 2</math> can be linearly extended from the linearly independent set of vectors <math>S := \{(1,0), (0, 1)\}</math> to a linear map on . The unique linear extension <math>F : \R^2 \to \R</math> is the map that sends <math>(x, y) = x (1, 0) + y (0, 1) \in \R^2</math> to

Every (scalar-valued) linear functional <math>f</math> defined on a vector subspace of a real or complex vector space <math>X</math> has a linear extension to all of .

Indeed, the Hahn–Banach dominated extension theorem even guarantees that when this linear functional <math>f</math> is dominated by some given seminorm <math>p : X \to \R</math> (meaning that <math>|f(m)| \leq p(m)</math> holds for all <math>m</math> in the domain of ) then there exists a linear extension to <math>X</math> that is also dominated by .

Matrices

If <math>V</math> and <math>W</math> are finite-dimensional vector spaces and a basis is defined for each vector space, then every linear map from <math>V</math> to <math>W</math> can be represented by a matrix. This is useful because it allows concrete calculations. Matrices yield examples of linear maps: if <math>A</math> is a real <math>m \times n</math> matrix, then <math>f(\mathbf x) = A \mathbf x</math> describes a linear map <math>\R^n \to \R^m</math> (see Euclidean space).

Let <math>\{ \mathbf {v}_1, \ldots , \mathbf {v}_n \}</math> be a basis for . Then every vector <math>\mathbf {v} \in V</math> is uniquely determined by the coefficients <math>c_1, \ldots , c_n</math> in the field :

<math display="block">\mathbf{v} = c_1 \mathbf{v}_1 + \cdots + c_n \mathbf {v}_n.</math>

If <math display="inline">f: V \to W</math> is a linear map,

<math display="block">f(\mathbf{v}) = f(c_1 \mathbf{v}_1 + \cdots + c_n \mathbf{v}_n) = c_1 f(\mathbf{v}_1) + \cdots + c_n f\left(\mathbf{v}_n\right),</math>

which implies that the function f is entirely determined by the vectors . Now let <math>\{ \mathbf {w}_1, \ldots , \mathbf {w}_m \}</math> be a basis for . Then we can represent each vector <math>f(\mathbf {v}_j)</math> as

<math display="block">f\left(\mathbf{v}_j\right) = a_{1j} \mathbf{w}_1 + \cdots + a_{mj} \mathbf{w}_m.</math>

Thus, the function <math>f</math> is entirely determined by the values of . If we put these values into an <math>m \times n</math> matrix , then we can conveniently use it to compute the vector output of <math>f</math> for any vector in . To get , every column <math>j</math> of <math>M</math> is a vector

<math display="block">\begin{pmatrix} a_{1j} \\ \vdots \\ a_{mj} \end{pmatrix}</math>

corresponding to <math>f(\mathbf {v}_j)</math> as defined above. To define it more clearly, for some column <math>j</math> that corresponds to the mapping ,

<math display="block">\mathbf{M} = \begin{pmatrix}

\ \cdots & a_{1j} & \cdots\ \\

& \vdots & \\

& a_{mj} &

\end{pmatrix}</math>

where <math>M</math> is the matrix of . In other words, every column <math>j = 1, \ldots, n</math> has a corresponding vector <math>f(\mathbf {v}_j)</math> whose coordinates <math>a_{1j}, \cdots, a_{mj}</math> are the elements of column . A single linear map may be represented by many matrices. This is because the values of the elements of a matrix depend on the bases chosen.

The matrices of a linear transformation can be represented visually:

Matrix for <math display="inline">T</math> relative to : <math display="inline">A</math>
Matrix for <math display="inline">T</math> relative to : <math display="inline">A'</math>
Transition matrix from <math display="inline">B'</math> to : <math display="inline">P</math>
Transition matrix from <math display="inline">B</math> to : <math display="inline">P^{-1}</math>

frame|The relationship between matrices in a linear transformation|none

Such that starting in the bottom left corner <math display="inline">\left[\mathbf{v}\right]_{B'}</math> and looking for the bottom right corner , one would left-multiply—that is, . The equivalent method would be the "longer" method going clockwise from the same point such that <math display="inline">\left[\mathbf{v}\right]_{B'}</math> is left-multiplied with , or .

Examples in two dimensions

In two-dimensional space R<sup>2</sup> linear maps are described by 2 × 2 matrices. These are some examples:

rotation
by 90 degrees counterclockwise: <math display="block">\mathbf{A} = \begin{pmatrix} 0 & -1\\ 1 & 0\end{pmatrix}</math>
by an angle θ counterclockwise: <math display="block">\mathbf{A} = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}</math>
reflection
through the x axis: <math display="block">\mathbf{A} = \begin{pmatrix} 1 & 0\\ 0 & -1\end{pmatrix}</math>
through the y axis: <math display="block">\mathbf{A} = \begin{pmatrix}-1 & 0\\ 0 & 1\end{pmatrix}</math>
through a line making an angle θ with the origin: <math display="block">\mathbf{A} = \begin{pmatrix}\cos2\theta & \sin2\theta \\ \sin2\theta & -\cos2\theta \end{pmatrix}</math>
scaling by 2 in all directions: <math display="block">\mathbf{A} = \begin{pmatrix} 2 & 0\\ 0 & 2\end{pmatrix} = 2\mathbf{I}</math>
horizontal shear mapping: <math display="block">\mathbf{A} = \begin{pmatrix} 1 & m\\ 0 & 1\end{pmatrix}</math>
skew of the y axis by an angle θ: <math display="block">\mathbf{A} = \begin{pmatrix} 1 & -\sin\theta\\ 0 & \cos\theta\end{pmatrix}</math>
squeeze mapping: <math display="block">\mathbf{A} = \begin{pmatrix} k & 0\\ 0 & \frac{1}{k}\end{pmatrix}</math>
projection onto the y axis: <math display="block">\mathbf{A} = \begin{pmatrix} 0 & 0\\ 0 & 1\end{pmatrix}.</math>

If a linear map is only composed of rotation, reflection, and/or uniform scaling, then the linear map is a conformal linear transformation.

Vector space of linear maps

The composition of linear maps is linear: if <math>f: V \to W</math> and <math display="inline">g: W \to Z</math> are linear, then so is their composition . It follows from this that the class of all vector spaces over a given field K, together with K-linear maps as morphisms, forms a category.

The inverse of a linear map, when defined, is again a linear map.

If <math display="inline">f_1: V \to W</math> and <math display="inline">f_2: V \to W</math> are linear, then so is their pointwise sum , which is defined by .

If <math display="inline">f: V \to W</math> is linear and <math display="inline">\alpha</math> is an element of the ground field , then the map , defined by , is also linear.

Thus the set <math display="inline">\mathcal{L}(V, W)</math> of linear maps from <math display="inline">V</math> to <math display="inline">W</math> itself forms a vector space over , sometimes denoted . Furthermore, in the case that , this vector space, denoted , 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.

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.

Endomorphisms and automorphisms

A linear transformation <math display="inline">f : V \to V</math> is an endomorphism of <math display="inline">V</math>; the set of all such endomorphisms <math display="inline">\operatorname{End}(V)</math> together with addition, composition and scalar multiplication as defined above forms an associative algebra with identity element over the field <math display="inline">K</math> (and in particular a ring). The multiplicative identity element of this algebra is the identity map .

An endomorphism of <math display="inline">V</math> that is also an isomorphism is called an automorphism of . The composition of two automorphisms is again an automorphism, and the set of all automorphisms of <math display="inline">V</math> forms a group, the automorphism group of <math display="inline">V</math> which is denoted by <math display="inline">\operatorname{Aut}(V)</math> or . Since the automorphisms are precisely those endomorphisms which possess inverses under composition, <math display="inline">\operatorname{Aut}(V)</math> is the group of units in the ring .

If <math display="inline">V</math> has finite dimension , then <math display="inline"> \operatorname{End}(V)</math> is isomorphic to the associative algebra of all <math display="inline">n \times n</math> matrices with entries in . The automorphism group of <math display="inline">V</math> is isomorphic to the general linear group <math display="inline">\operatorname{GL}(n, K)</math> of all <math display="inline">n \times n</math> invertible matrices with entries in .

Kernel, image and the rank–nullity theorem

If <math display="inline">f: V \to W</math> is linear, we define the kernel and the image or range of <math display="inline">f</math> by

<math display="block">\begin{align}

\ker(f) &= \{\,\mathbf x \in V: f(\mathbf x) = \mathbf 0\,\} \\

\operatorname{im}(f) &= \{\,\mathbf w \in W: \mathbf w = f(\mathbf x), \mathbf x \in V\,\}

\end{align}</math>

<math display="inline">\ker(f)</math> is a subspace of <math display="inline">V</math> and <math display="inline">\operatorname{im}(f)</math> is a subspace of . The following dimension formula is known as the rank–nullity theorem:

<math display="block">\dim(\ker( f )) + \dim(\operatorname{im}( f )) = \dim( V ).</math>

The number <math display="inline">\dim(\operatorname{im}(f))</math> is also called the rank of <math display="inline">f</math> and written as , or sometimes, ; the number <math display="inline">\dim(\ker(f))</math> is called the nullity of <math display="inline">f</math> and written as <math display="inline">\operatorname{null}(f)</math> or .

Algebraic classifications of linear transformations

No classification of linear maps could be exhaustive. The following incomplete list enumerates some important classifications that do not require any additional structure on the vector space.

Let and denote vector spaces over a field and let be a linear map.

Monomorphism

is said to be injective or a monomorphism if any of the following equivalent conditions are true:

is one-to-one as a map of sets.
is monic or left-cancellable, which is to say, for any vector space and any pair of linear maps and , the equation implies .
is left-invertible, which is to say there exists a linear map such that is the identity map on .

Epimorphism

is said to be surjective or an epimorphism if any of the following equivalent conditions are true:

is onto as a map of sets.
is epic or right-cancellable, which is to say, for any vector space and any pair of linear maps and , the equation implies .
is right-invertible, which is to say there exists a linear map such that is the identity map on .

Isomorphism <span class="anchor" id="isomorphism"></span>

is said to be an isomorphism if it is both left- and right-invertible. This is equivalent to being both one-to-one and onto (a bijection of sets) or also to being both epic and monic, and so being a bimorphism.

If is an endomorphism, then:

If, for some positive integer , the th iterate of , , is identically zero, then is said to be nilpotent.
If , then is said to be idempotent
If , where is some scalar, then is said to be a scaling transformation or scalar multiplication map; see scalar matrix.

Change of basis

Given a linear map which is an endomorphism whose matrix is , in the basis of the space it transforms vector coordinates as . As vectors change with the inverse of (vectors coordinates are contravariant) its inverse transformation is .

Substituting this in the first expression

<math display="block">B\left[v'\right] = AB\left[u'\right]</math>

hence

<math display="block">\left[v'\right] = B^{-1}AB\left[u'\right] = A'\left[u'\right].</math>

Therefore, the matrix in the new basis is , being the matrix of the given basis.

Therefore, linear maps are said to be 1-co- 1-contra-variant objects, or type (1, 1) tensors.

Continuity

A linear transformation between topological vector spaces, for example normed spaces, may be 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, for example, when the domain is finite-dimensional. An infinite-dimensional domain may have discontinuous linear operators.

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 is ). For a specific example, converges to , but its derivative does not, so differentiation is not continuous at (and by a variation of this argument, it is not continuous anywhere).

Applications

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.

Another application of these transformations is in compiler optimizations of nested-loop code, and in parallelizing compiler techniques.

Notes

Bibliography