Linear algebra is a branch of mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions concerned with the study of vectors In elementary mathematics, physics, and engineering, a Euclidean vector is a geometric object that has both a magnitude (or length) and direction. A Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by, with families of vectors called vector spaces A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex numbers, rational numbers, or even more general fields or linear spaces, and with functions that input one vector and output another, according to certain rules. These functions are called linear maps In mathematics, a linear map is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. The expression "linear operator" is in especially common use, for linear maps from a vector space to itself (endomorphisms). In advanced mathematics, the definition of linear function coincides or linear transformations and are often represented by matrices An item in a matrix is called an entry or an element. The example has entries 1, 9, 13, 20, 55, and 4. Entries are often denoted by a variable with two subscripts, as shown on the right. Matrices of the same size can be added and subtracted entrywise and matrices of compatible sizes can be multiplied. These operations have many of the properties. Linear algebra is central to modern mathematics and its applications. An elementary application of linear algebra is to the solution of a systems of linear equations In mathematics, a system of linear equations is a collection of linear equations involving the same set of variables. For example, in several unknowns. More advanced applications are ubiquitous, in areas as diverse as abstract algebra Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulae and functional analysis Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well as in the study of differential and integral. Linear algebra has a concrete representation in analytic geometry Analytic geometry, also known as coordinate geometry, analytical geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis. This contrasts with the synthetic approach of Euclidean geometry, which treats certain geometric notions as primitive, and uses deductive reasoning based on and is generalized in operator theory. It has extensive applications in engineering Engineering is the discipline, art and profession of acquiring and applying technical, scientific, and mathematical knowledge to design and implement materials, structures, machines, devices, systems, and processes that safely realize a desired objective or invention, physics Physics is a natural science that involves the study of matter and its motion through space-time, as well as all applicable concepts, such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves, natural sciences In science, the term natural science refers to a naturalistic approach to the study of the universe, which is understood as obeying rules or laws of natural origin and the social sciences The social sciences are the fields of academic scholarship that explore aspects of human society. "Social science" is commonly used as an umbrella term to refer to a plurality of fields outside of the natural sciences. These include: anthropology, archaeology, economics, geography, history, linguistics, political science, international. Nonlinear mathematical models A mathematical model uses mathematical language to describe a system. The process of developing a mathematical model is termed mathematical modelling . Mathematical models are used not only in the natural sciences (such as physics, biology, earth science, meteorology) and engineering disciplines, but also in the social sciences (such as economics, can often be approximated by linear ones.

Contents

History

Many of the basic tools of linear algebra, particularly those concerned with the solution of systems of linear equations, date to antiquity. See, for example, the history of Gaussian elimination In linear algebra, Gaussian elimination is an algorithm for solving systems of linear equations, finding the rank of a matrix, and calculating the inverse of an invertible square matrix. Gaussian elimination is named after German mathematician and scientist Carl Friedrich Gauss. But the abstract study of vectors and vector spaces does not begin until the 1600s. The origin of many of these ideas is discussed in the article on determinants In algebra, the determinant is a special number associated with any square matrix. The fundamental geometric meaning of a determinant is a scale factor for measure when the matrix is regarded as a linear transformation. Thus a 2 × 2 matrix with determinant 2 when applied to a set of points with finite area will transform those points into a set. The method of least squares The method of least squares is used to approximately solve overdetermined systems, i.e. systems of equations in which there are more equations than unknowns. Least squares is often applied in statistical contexts, particularly regression analysis, first used by Gauss Johann Carl Friedrich Gauss (pronounced /ˈɡaʊs/; German: Gauß listen , Latin: Carolus Fridericus Gauss) (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy in the 1790s, is an early and significant application of the ideas of linear algebra.

The subject began to take its modern form in the mid-19th century, which saw many ideas and methods of previous centuries generalized as abstract algebra Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulae. Matrices An item in a matrix is called an entry or an element. The example has entries 1, 9, 13, 20, 55, and 4. Entries are often denoted by a variable with two subscripts, as shown on the right. Matrices of the same size can be added and subtracted entrywise and matrices of compatible sizes can be multiplied. These operations have many of the properties and tensors Tensors are geometric entities introduced into mathematics and physics to extend the notion of scalars, geometric vectors, and matrices to higher orders. Tensors were first conceived by Tullio Levi-Civita and Gregorio Ricci-Curbastro, who continued the earlier work of Bernhard Riemann and Elwin Bruno Christoffel and others, as part of the absolute were introduced and well understood by the turn of the 20th century. The use of these objects in special relativity Special relativity (also known as the special theory of relativity or STR) is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein (after the considerable and independent contributions of Hendrik Lorentz, Henri Poincaré and others) in the paper "On the Electrodynamics of Moving Bodies", statistics Statistics is the formal science of making effective use of numerical data relating to groups of individuals or experiments. It deals with all aspects of this, including not only the collection, analysis and interpretation of such data, but also the planning of the collection of data, in terms of the design of surveys and experiments, and quantum mechanics Quantum mechanics , also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic scales. In advanced topics of QM, some of these did much to spread the subject of linear algebra beyond pure mathematics.

Main structures

The main structures of linear algebra are vector spaces A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex numbers, rational numbers, or even more general fields and linear maps In mathematics, a linear map is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. The expression "linear operator" is in especially common use, for linear maps from a vector space to itself (endomorphisms). In advanced mathematics, the definition of linear function coincides between them. A vector space is a set A set is a collection of distinct objects, considered as an object in its own right. Sets are one of the most fundamental concepts in mathematics. Although it was invented at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In whose elements can be added together and multiplied by the scalars, or numbers. In many physical applications, the scalars are real numbers In mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real numbers may be thought of as points on an, R. More generally, the scalars may form any field In abstract algebra, a field is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms. The most commonly used fields are the field of real numbers, the field of complex numbers, and the field of rational numbers, but there are also finite fields, fields of functions, various algebraic F—thus one can consider vector spaces over the field Q of rational numbers In mathematics a rational number is any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number. The set of all rational numbers is usually denoted by a boldface Q , which stands for quotient, the field C of complex numbers A complex number is a number consisting of a real and imaginary part. It can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit with the property i 2 = −1. The complex numbers contain the ordinary real numbers, but extend them by adding in extra numbers and correspondingly expanding the, or a finite field In abstract algebra, a finite field or Galois field is a field that contains only finitely many elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory. The finite fields are classified by size; there is exactly one finite field up to isomorphism of size pk for each prime p and Fq. These two operations must behave similarly to the usual addition and multiplication of numbers: addition is commutative In mathematics, commutativity is the property that changing the order of something does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. The commutativity of simple operations, such as multiplication and addition of numbers, was for many years implicitly assumed and the and associative In mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is, rearranging the parentheses in such, multiplication distributes In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. For example: over addition, and so on. More precisely, the two operations must satisfy a list of axioms chosen to emulate the properties of addition and scalar multiplication of Euclidean vectors In elementary mathematics, physics, and engineering, a Euclidean vector is a geometric object that has both a magnitude (or length) and direction. A Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by in the coordinate n-space Rn. One of the axioms stipulates the existence of zero vector, which behaves analogously to the number zero with respect to addition. Elements of a general vector space V may be objects of any nature, for example, functions The mathematical concept of a function expresses the intuitive idea that one quantity completely determines another quantity (the value, or the output). A function assigns a unique value to each input of a specified type. The argument and the value may be real numbers, but they can also be elements from any given sets: the domain and the codomain or polynomials In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of splitting fields, and to the understanding of a linear, but when viewed as elements of V, they are frequently called vectors.

Given two vector spaces V and W over a field F, a linear transformation is a map In mathematics and related technical fields, the term map or mapping is a synonym for function. Thus, for example, a partial map is a partial function, and a total map is a total function. Related terms like domain, codomain, injective, continuous, etc. can be applied equally to maps and functions, with the same meaning

that is compatible with addition and scalar multiplication:

for any vectors u,v ∈ V and a scalar r ∈ F.

A fundamental role in linear algebra is played by the notions of linear combination In mathematics, linear combinations is a concept central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article, span, and linear independence In linear algebra, a family of vectors is linearly independent if none of them can be written as a linear combination of finitely many other vectors in the collection. A family of vectors which is not linearly independent is called linearly dependent. For instance, in the three-dimensional real vector space we have the following example of vectors and basis In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space or free module, and such that no element of the set can be represented as a linear combination of the others. In other words, a basis is a linearly independent spanning set, or more simply put a "coordinate system& and the dimension In mathematics, the dimension of a vector space V is the cardinality of a basis of V. It is sometimes called Hamel dimension or algebraic dimension to distinguish it from other types of dimension. All bases of a vector space have equal cardinality (see dimension theorem for vector spaces) and so the dimension of a vector space is uniquely defined of a vector space. Given a vector space V over a field F, an expression of the form

where v1, v2, …, vk are vectors and r1, r2, …, rk are scalars, is called the linear combination of the vectors v1, v2, …, vk with coefficients r1, r2, …, rk. The set of all linear combinations of vectors v1, v2, …, vk is called their span. A linear combination of any system of vectors with all zero coefficients is zero vector of V. If this is the only way to express zero vector as a linear combination of v1, v2, …, vk then these vectors are linearly independent. A linearly independent set of vectors that spans a vector space V is a basis of V. If a vector space admits a finite basis then any two bases have the same number of elements (called the dimension of V) and V is a finite-dimensional vector space. This theory can be extended to infinite-dimensional spaces.

There is an important distinction between the coordinate n-space Rn and a general finite-dimensional vector space V. While Rn has a standard basis {e1, e2, …, en}, a vector space V typically does not come equipped with a basis and many different bases exist (although they all consist of the same number of elements equal to the dimension of V). Having a particular basis {v1, v2, …, vn} of V allows one to construct a coordinate system in V: the vector with coordinates (r1, r2, …, rn) is the linear combination

The condition that v1, v2, …, vn span V guarantees that each vector v can be assigned coordinates, whereas the linear independence of v1, v2, …, vn further assures that these coordinates are determined in a unique way (i.e. there is only one linear combination of the basis vectors that is equal to v). In this way, once a basis of a vector space V over F has been chosen, V may be identified with the coordinate n-space Fn. Under this identification, addition and scalar multiplication of vectors in V correspond to addition and scalar multiplication of their coordinate vectors in Fn. Furthermore, if V and W are an n-dimensional and m-dimensional vector space over F, and a basis of V and a basis of W have been fixed, then any linear transformation T: V → W may be encoded by an m × n matrix An item in a matrix is called an entry or an element. The example has entries 1, 9, 13, 20, 55, and 4. Entries are often denoted by a variable with two subscripts, as shown on the right. Matrices of the same size can be added and subtracted entrywise and matrices of compatible sizes can be multiplied. These operations have many of the properties A with entries in the field F, called the matrix of T with respect to these bases. Therefore, by and large, the study of linear transformations, which were defined axiomatically, may be replaced by the study of matrices, which are concrete objects. This is a major technique in linear algebra.

Vector spaces over the complex numbers

Remarkably, the 2 × 2 complex A complex number is a number consisting of a real and imaginary part. It can be written in the form a + bi, where a and b are real numbers, and i is the standard imaginary unit with the property i 2 = −1. The complex numbers contain the ordinary real numbers, but extend them by adding in extra numbers and correspondingly expanding the matrices were studied before 2 × 2 real In mathematics, the real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of two; or, a real number can be given by an infinite decimal representation, such as 2.4871773339..., where the digits continue in some way; or, the real numbers may be thought of as points on an matrices. Early topics of interest included biquaternions In abstract algebra, the biquaternions are the numbers where w, x, y, and z are complex numbers and the elements of {1, i, j, k} multiply as in the quaternion group. As there are three types of complex number, so there are three types of biquaternion: and Pauli algebra. Investigation of 2 × 2 real matrices revealed the less common split-complex numbers In abstract algebra, the split-complex numbers are a two-dimensional commutative algebra over the real numbers different from the complex numbers. Every split-complex number has the form and dual numbers In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε2 = 0 . The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form z = a + bε with a and b uniquely determined real numbers. The plane, which are at variance with the Euclidean nature of the ordinary complex number plane.

Some useful theorems

For more information regarding the invertibility of a matrix, consult the invertible matrix article.

Generalizations and related topics

Since linear algebra is a successful theory, its methods have been developed in other parts of mathematics. In module theory one replaces the field of scalars by a ring. In multilinear algebra one considers multivariable linear transformations, that is, mappings that are linear in each of a number of different variables. This line of inquiry naturally leads to the idea of the tensor product. Functional analysis mixes the methods of linear algebra with those of mathematical analysis.

See also

Notes

  1. ^ The existence of a basis is straightforward for finitely generated vector spaces, but in full generality it is logically equivalent to the axiom of choice.
  2. ^ Dimension theorem for vector spaces
  3. ^ Pragma's Playground: Matrices for Dummies, http://www.pragmaware.net/articles/matrices/index.php

References

This article includes a list of references or external links, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations where appropriate. (September 2009)

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