Linear map - Citizendia

In mathematics, a linear map (also called a linear transformation or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication The term "linear transformation" is in particularly common use, especially for linear maps from a vector space to itself (endomorphisms). In Mathematics, an endomorphism is a Morphism (or Homomorphism) from a mathematical object to itself

In the language of abstract algebra, a linear map is a homomorphism of vector spaces, or a morphism in the category of vector spaces over a given field. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division

1 Definition and first consequences
2 Examples
3 Matrices
4 Examples of linear transformation matrices
5 Forming new linear maps from given ones
6 Endomorphisms and automorphisms
7 Kernel, image and the rank-nullity theorem
8 Algebraic classifications of linear transformations
9 Continuity
10 Applications
11 See also
12 References

Definition and first consequences

Let V and W be vector spaces over the same field K. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division 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 a in K, the following two conditions are satisfied:

$f(x+y)=f(x)+f(y) \,$	additivity
$f(ax)=af(x) \,$	homogeneity

This is equivalent to requiring that for any vectors x₁, . . . , x_m and scalars a₁, . . . , a_m, the equality

$f(a_1 x_1+\cdots+a_m x_m)=a_1 f(x_1)+\cdots+a_m f(x_m)$

holds.

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 conjugation of complex numbers is an R-linear map C → C, but it is not C-linear. In Mathematics, the complex conjugate of a Complex number is given by changing the sign of the Imaginary part. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

A linear map from V to K (with K viewed as a vector space over itself) is called a linear functional. This article deals with Linear maps from a Vector space to its field of scalars These maps may be functionals in the traditional

It immediately follows from the definition that f(0) = 0. Hence linear maps are sometimes called homogeneous linear maps (see linear function). In Mathematics, the term linear function can refer to either of two different but related concepts

Examples

The identity map and zero map are linear. In Mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that

For real numbers, the map $x\mapsto x^2$ is not linear.

For real numbers, the map $x\mapsto x+1$ is not linear.

If A is an m × n matrix, then A defines a linear map from Rⁿ to R^m by sending the column vector x ∈ Rⁿ to the column vector Ax ∈ R^m. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Linear algebra, a column vector or column matrix is an m × 1 matrix, i Conversely, any linear map between finite-dimensional vector spaces can be represented in this manner; see the following section. In Mathematics, the dimension of a Vector space V is the cardinality (i

The integral yields a linear map from the space of all real-valued integrable functions on some interval to R

Differentiation is a linear map from the space of all differentiable functions to the space of all functions. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change

If V and W are finite-dimensional vector spaces over a field F, then functions that send linear maps f : V → W to dim_F(W)-by-dim_F(V) matrices in the way described in the sequel are themselves linear maps.

Matrices

If V and W are finite-dimensional, and one has chosen bases in those spaces, then every linear map from V to W can be represented as a matrix; this is useful because it allows concrete calculations. In Mathematics, the dimension of a Vector space V is the cardinality (i Basis vector redirects here For basis vector in the context of crystals see Crystal structure. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally Conversely, matrices yield examples of linear maps: if A is a real m-by-n matrix, then the rule f(x) = Ax describes a linear map Rⁿ → R^m (see Euclidean space).

Let $\{v_1, \cdots, v_n\}$ be a basis for V. Then every vector v in V is uniquely determined by the coefficients $c_1, \cdots, c_n$ in

$c_1 v_1+\cdots+c_n v_n.$

If f : V → W is a linear map,

$f(c_1 v_1+\cdots+c_n v_n)=c_1 f(v_1)+\cdots+c_n f(v_n),$

which implies that the function f is entirely determined by the values of $f(v_1),\cdots,f(v_n).$

Now let $\{w_1, \dots, w_m\}$ be a basis for W. Then we can represent the values of each $f (v j)$ as

$f(v_j)=a_{1j} w_1 + \cdots + a_{mj} w_m.$

Thus, the function f is entirely determined by the values of $a i, j .$

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 $c_1, \cdots, c_n$ in an n-by-1 matrix C, we have MC = f(v).

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.

Examples of linear transformation matrices

Some special cases of linear transformations of two-dimensional space R² are illuminating:

rotation by 90 degrees counterclockwise:

$A=\begin{bmatrix}0 & -1\\ 1 & 0\end{bmatrix}$
rotation by θ degrees counterclockwise:

$A=\begin{bmatrix}\cos(\theta) & -\sin(\theta)\\ \sin(\theta) & \cos(\theta)\end{bmatrix}$
reflection against the x axis:

$A=\begin{bmatrix}1 & 0\\ 0 & -1\end{bmatrix}$
scaling by 2 in all directions:

$A=\begin{bmatrix}2 & 0\\ 0 & 2\end{bmatrix}$
vertical shear mapping:

$A=\begin{bmatrix}1 & m\\ 0 & 1\end{bmatrix}$
squeezing:

$A=\begin{bmatrix}k & 0\\ 0 & 1/k\end{bmatrix}$
projection onto the y axis:

$A=\begin{bmatrix}0 & 0\\ 0 & 1\end{bmatrix}$

Forming new linear maps from given ones

The composition of linear maps is linear: if f : V → W and g : W → Z are linear, then so is g o f : V → Z. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it In Geometry and Linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a Rigid body around a fixed In Geometry and Linear algebra, a rotation is a transformation in a plane or in space that describes the motion of a Rigid body around a fixed In Mathematics, a reflection (also spelled reflexion) is a map that transforms an object into its Mirror image. In Euclidean geometry, uniform scaling or Isotropic scaling is a Linear transformation that enlarges or diminishes objects the Scale factor In Mathematics, a shear or transvection is a particular kind of Linear mapping. In Linear algebra, a squeeze mapping is a type of Linear map that preserves Euclidean Area of regions in the Cartesian plane, but is not a

The inverse of a linear map, when defined, is again a linear map. In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B

If f₁ : V → W and f₂ : V → W are linear, then so is their sum f₁ + f₂ (which is defined by (f₁ + f₂)(x) = f₁(x) + f₂(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.

Thus the set L(V,W) of linear maps from V to W itself forms a vector space over K, sometimes denoted Hom(V,W). Furthermore, in the case that 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. In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive In Mathematics, a composite function represents the application of one function to the results of another 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. In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix In Mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together

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). In Mathematics, an endomorphism is a Morphism (or Homomorphism) from a mathematical object to itself In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real The multiplicative identity element of this algebra is the identity map id : V → V. This article is about the Identity Map software design pattern

An endomorphism of V that is also an isomorphism is called an automorphism of V. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself The composition of two automorphisms is again an automorphism, and the set of all automorphisms of V forms a group, the automorphism group of V which is denoted by Aut(V) or GL(V). In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself Since the automorphisms are precisely those endomorphisms which possess inverses under composition, Aut(V) is the group of units in the ring End(V). In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i

If V has finite dimension n, then End(V) is isomorphic to the associative algebra of all n by n matrices with entries in K. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive The automorphism group of V is isomorphic to the general linear group GL(n, K) of all n by n invertible matrices with entries in K. In Abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation

Kernel, image and the rank-nullity theorem

If f : V → W is linear, we define the kernel and the image or range of f by

$\operatorname{\ker}(f)=\{\,x\in V:f(x)=0\,\}$

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

ker(f) is a subspace of V and im(f) is a subspace of W. In the various branches of Mathematics that fall under the heading of Abstract algebra, the kernel of a Homomorphism measures the degree to which the homomorphism In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage In Mathematics, the range of a function is the set of all "output" values produced by that function The concept of a linear subspace (or vector subspace) is important in Linear algebra and related fields of Mathematics. The following dimension formula, known as the rank-nullity theorem, is often useful:

$\dim(\ker( f )) + \dim(\operatorname{im}( f )) = \dim( V ) \,$

The number dim(im(f)) is also called the rank of f and written as rk(f), or sometimes, ρ(f); the number dim(ker(f)) is called the nullity of f and written as ν(f). In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it 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 and nullity of the matrix A, respectively. The column rank of a matrix A is the maximal number of Linearly independent columns of A.

Algebraic classifications of linear transformations

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.

Let V and W denote vector spaces over a field, F. Let T:V → W be a linear map.

T is said to be injective or a monomorphism if any of the following equivalent conditions are true:
- T is one-to-one as a map of sets. In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism.
- ker T = 0
- T is 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.
- T is left-invertible, which is to say there exists a linear map S:W → V such that ST is the identity map on V. In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to In Mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that

T is said to be surjective or an epimorphism if any of the following equivalent conditions are true:
- T is onto as a map of sets. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every In Category theory an epimorphism (also called an epic morphism or an epi) is a Morphism f: X &rarr Y which In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every
- coker T = 0
- T is 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. In Category theory an epimorphism (also called an epic morphism or an epi) is a Morphism f: X &rarr Y which
- T is right-invertible, which is to say there exists a linear map S:W → V such that TS is the identity map on V. In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to In Mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that

T is said to be an isomorphism if it is both left- and right-invertible. In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective 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. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and

If T: V → V is an endomorphism, then:
- If, for some positive integer n, the n-th iterate of T, Tⁿ, is identically zero, then T is said to be nilpotent. In Mathematics, an element x of a ring R is called nilpotent if there exists some positive Integer n such that
- If T T = T, then T is said to be idempotent
- If T = k I, where k is some scalar, then T is said to be a scaling transformation or scalar multiplication map. Idempotence ˌaɪdɨmˈpoʊtəns describes the property of operations in Mathematics and Computer science which means that multiple applications of the operation

Continuity

Main article: Discontinuous linear map

A linear operator between topological vector spaces, for example normed spaces, may also be continuous and therefore be a continuous linear operator. In Mathematics, Linear maps form an important class of "simple" functions which preserve the algebraic structure of Linear spaces and are often In Mathematics, a topological vector space is one of the basic structures investigated in Functional analysis. In Mathematics, with 2- or 3-dimensional vectors with real -valued entries the idea of the "length" of a vector is intuitive and can easily be extended to In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Functional analysis and related areas of Mathematics, a continuous linear operator or continuous linear mapping is a continuous Linear On a normed space, a linear operator is continuous if and only if it is bounded, for example, when the domain is finite-dimensional. In Functional analysis (a branch of Mathematics) a bounded linear operator is a Linear transformation L between Normed vector spaces If the domain is infinite-dimensional, then there may be discontinuous linear operators. In Mathematics, Linear maps form an important class of "simple" functions which preserve the algebraic structure of Linear spaces and are often An example of an unbounded, hence not continuous, 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).

Applications

A specific application of linear maps is in the field of computational neuroscience. An example of a system being modeled is the innervation of V1 (primary visual cortex) by the retina. This transformation is called the logmap transformation. This kind of transformation is known as a domain coordinate transformation and provides a mathematical model of how neural states can be conferred within the system (CNS and PNS), when a change of state is required, such as from the retina to V1 as previously mentioned.

Another specific application 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. Computer graphics are Graphics created by Computers and more generally the Representation and Manipulation of Pictorial Data In Linear algebra, Linear transformations can be represented by matrices.

Another application of these transformations is in compiler optimizations of nested-loop code, and in parallelizing compiler techniques. Compiler optimization is the process of tuning the output of a Compiler to minimize or maximize some attribute of an Executable computer program

References

Halmos, Paul R., Finite-Dimensional Vector Spaces, Springer-Verlag, (1993). A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable In Mathematics, a mapping f: V → W from a Complex vector space to another is said to be antilinear (or conjugate-linear In Linear algebra, Linear transformations can be represented by matrices. In Functional analysis and related areas of Mathematics, a continuous linear operator or continuous linear mapping is a continuous Linear Traditionally the term neural network had been used to refer to a network or circuit of biological neurons. Computer graphics are Graphics created by Computers and more generally the Representation and Manipulation of Pictorial Data Paul Richard Halmos ( March 3 1916 &mdash October 2 2006) was a Hungarian -born Jewish American Mathematician ISBN 0-387-90093-4.

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