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Encyclopedia > Linear transformation

In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. In other words, it "preserves linear combinations". Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... Partial plot of a function f. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, scalars are components of vector spaces (and modules), usually real numbers, which can be multiplied into vectors by scalar multiplication. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ...


In the language of abstract algebra, a linear transformation is a homomorphism of vector spaces. Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In abstract algebra, a homomorphism is a structure-preserving map. ...

Contents


Definition and first consequences

If V and W are vector spaces over the same ground field K, one says that f : VW is a linear transformation if for any two vectors x and y in V and any scalar a in K, one has The fundamental concept in linear algebra is that of a vector space or linear space. ... This article presents the essential definitions. ... In mathematics, scalars are components of vector spaces (and modules), usually real numbers, which can be multiplied into vectors by scalar multiplication. ...

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

This is equivalent to stating that f   "preserves linear combinations", that is, for any vectors x1, ..., xm and scalars a1, ..., am, 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 ground fields. It is then normal to specify which of these ground fields 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 CC, but it is not C-linear. The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...


A linear transformation from V to K (with K viewed as a vector space over itself) is called a linear functional. In linear algebra, a branch of mathematics, a linear functional or linear form is a linear function from a vector space to its field of scalars. ...


It follows at once from the definition that f(0) = 0, hence linear transformations are sometimes called homogeneous linear transformations.


Examples

  • If A is an m × n matrix, then A defines a linear transformation from Rn to Rm by sending the column vector xRn to the column vector AxRm. Every linear transformation between finite-dimensional vector spaces arises in this fashion; see the following section.
  • The integral yields a linear map from the space of all real-valued integrable functions on some interval to R
  • Differentiation is a linear transformation from the space of all differentiable functions to the space of all functions.
  • If V and W are finite-dimensional vector spaces over a field F, then functions that map linear transformations f : VW to dimF(W)-by-dimF(V) matrices in the way described in the sequel are themselves linear transformations.

In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In linear algebra, a column vector is an m × 1 matrix, i. ... In mathematics, the dimension of a vector space V is the cardinality (i. ... In calculus, the integral of a function is a generalization of area, mass, volume and total. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... In mathematics, the derivative is defined as the instantaneous rate of change of a function. ...

Matrices

If V and W are finite-dimensional, and one has chosen bases in those spaces, then every linear transformation from V to W can be represented as a 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 RnRm (see Euclidean space). In mathematics, the dimension of a vector space V is the cardinality (i. ... In mathematics, a subset B of a vector space V is said to be a basis of V if it satisfies one of the four equivalent conditions: B is both a set of linearly independent vectors and a generating set of V. B is a minimal generating set of V... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ...


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 : VW is a linear transformation,

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(vj) as

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

Thus, the function f is entirely determined by the values of ai,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 transformation 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: 2-dimensional renderings (ie. ...

  • rotations: no real eigenvectors (complex eigenvalue/eigenvector pairs exist). Example (rotation by 90 degrees counterclockwise):
    A=begin{bmatrix}0 & -1 1 & 0end{bmatrix}.
  • reflection: eigenvectors are perpendicular and parallel to the line of symmetry. The eigenvalues are -1 and 1, respectively. Example (symmetry against the x axis):
    A=begin{bmatrix}1 & 0 0 & -1end{bmatrix}.
  • scaling:
    • uniform scaling: all vectors are eigenvectors, and the eigenvalue is the scale factor. Example (scale by 2 in all directions):
    A=begin{bmatrix}2 & 0 0 & 2end{bmatrix}.
    • directional scaling: eigenvalues are the scale factor and 1
    • directionally differential scaling: eigenvalues are the scale factors
  • squeezing with reciprocal eigenvalues k and r = 1/k:
    A=begin{bmatrix}k & 0 0 & rend{bmatrix}.
  • projection onto a line: vectors on the line are eigenvectors with eigenvalue 1 and vectors in the direction of projection (which may or may not be perpendicular) are eigenvectors with eigenvalue 0. Example:
    A=begin{bmatrix}0 & 0 0 & 1end{bmatrix}.

Rotation of a plane, seen as the rotation of the terrain relative to the plane (exposure time 1. ... In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... IT IS KNOWN AS MARK a lunitice insain int gw brain ... Perpendicular is a geometric term that may be used as a noun or adjective. ... Square with symmetry group D4 Symmetry is a characteristic of geometrical shapes, equations, and other objects; we say that such an object is symmetric with respect to a given operation if this operation, when applied to the object, does not appear to change it. ... In Euclidean geometry, uniform scaling is a linear transformation that enlarges or diminishes objects; the scale factor is the same in all directions; it is also called a homothety. ... A scale factor is a number which scales some quantity. ... In mathematics, a squeeze mapping in linear algebra is a type of linear transformation that preserves Euclidean area of regions in the cartesian plane, but is not a Euclidean motion. ... In linear algebra, a projection is a linear transformation P such that P2 = P, i. ...

Forming new linear transformations from given ones

The composition of linear transformations is linear: if f : VW and g : WZ are linear, then so is g o f : VZ.


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


If f : VW 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 forms a vector space over K itself. Furthermore, in the case that V=W, this vector space 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. In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...


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. This article gives an overview of the various ways to multiply matrices. ... The operations on matrices differ from similar operations of scalar algebra in several respects. ...


Endomorphisms and automorphisms

A linear transformation f : VV 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). The identity element of this algebra is the identity map id : VV. 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 and associative manner. ... In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ...


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, the automorphism group of V which is denoted by Aut(V) or GL(V). In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...


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. 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 mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ... 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 a way that respects the given group operations. ... In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ...


Kernel and image

If f : VW is linear, we define the kernel and the image or range of f by 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 fails to be injective. ... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ... In mathematics, the range of a function is the set of all output values produced by that function. ...

ker(f)={,xin V:f(x)=0,}
operatorname{im}(f)={,f(x):xin V,}

ker(f) is a subspace of V and im(f) is a subspace of W. The following dimension formula is often useful: The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ... 2-dimensional renderings (ie. ...

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). 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. In linear algebra, the column rank (row rank respectively) of a matrix A with entries in some field is defined to be the maximal number of columns (rows respectively) of A which are linearly independent. ... It has been suggested that this article or section be merged into kernel (mathematics). ...


A linear transformation f is an injection if and only if ker(f) = {0}. Injection has multiple meanings: In mathematics, the term injection refers to an injective function. ...


Continuity

A linear operator between topological vector spaces, for example normed spaces, may also be continuous and therefore be a continuous linear operator. On a normed space, a linear operator is continuous if and only if it is bounded, for example, when the domain is finite-dimensional. If the domain is infinite-dimensional, then there may be discontinuous linear operators. An example of an unbounded, hence not continuous, linear transformation is differentiation, with the maximum norm (a function with small values can have a derivative with large values). In mathematics, a topological vector space X is a real or complex vector space which is endowed with a Hausdorff topology such that vector addition X × X → X and scalar multiplication K × X → X are continuous (where the product topologies are used and the base field K carries its standard... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can be easily extended to any real vector space Rn. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... In functional analysis and related areas of mathematics, a continuous linear operator or continuous linear mapping is a continuous linear transformation between topological vector spaces. ... In mathematics, the operator norm is a norm defined on the space of bounded operators between two Banach spaces. ... In mathematics, linear maps form an important class of simple functions which preserve the algebraic structure of linear spaces and are often used as approximations to more general functions (see linear approximation). ...


Applications

A specific application of linear transformations 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 (CG) is the field of visual computing, where one utilizes computers both to generate visual images synthetically and to integrate or alter visual and spatial information sampled from the real world. ... 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 techniques are optimization techniques that have been programmed into a compiler. ...


See also

In mathematics, a mapping f : V → W from a complex vector space to another is said to be antilinear (or conjugate-linear or semilinear) if for all a, b in C and all x, y in V. The composition of two antilinear maps is complex-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 transformation between topological vector spaces. ... Simplified view of an artificial neural network A neural network is an interconnected group of biological neurons. ... Computer graphics (CG) is the field of visual computing, where one utilizes computers both to generate visual images synthetically and to integrate or alter visual and spatial information sampled from the real world. ...

References

  • Halmos, Paul R., Finite-Dimensional Vector Spaces, Springer-Verlag, (1993). ISBN 0387900934.

  Results from FactBites:
 
Linear transformation - definition of Linear transformation in Encyclopedia (952 words)
In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it "preserves linear combinations".
Differentiation is a linear transformation from the space of all differentiable functions to the space of all functions.
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.
Affine transformation - Wikipedia, the free encyclopedia (1071 words)
The invertible affine transformations form the affine group, which has the general linear group of degree n as subgroup and is itself a subgroup of the general linear group of degree n+1.
A linear transformation is a function that preserves all linear combinations; an affine transformation is a function that preserves all affine combinations.
A linear subspace of a vector space is a subset that is closed under linear combinations; an affine subspace is one that is closed under affine combinations.
  More results at FactBites »

 
 

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