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Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, a system of linear equations (or linear system) is a collection of Linear equations involving the same set of Variables For example Vector spaces are a central theme in modern mathematics; thus, linear algebra is widely used in both abstract algebra and functional analysis. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules For functional analysis as used in psychology see the Functional analysis (psychology article Linear algebra also has a concrete representation in analytic geometry and it is generalized in operator theory. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry In Mathematics, operator theory is the branch of Functional analysis which deals with Bounded linear operators and their properties It has extensive applications in the natural sciences and the social sciences, since nonlinear models can often be approximated by linear ones. In Science, the term natural science refers to a naturalistic approach to the study of the Universe, which is understood as obeying rules or law of The social sciences comprise academic disciplines concerned with the study of the social life of human groups and individuals including Anthropology, Communication studies

Contents

History

The history of modern linear algebra dates back to the early 1840s. In 1843, William Rowan Hamilton introduced quaternions, which describe mechanics in three-dimensional space. Sir William Rowan Hamilton (4 August 1805 &ndash 2 September 1865 was an Irish Mathematician, Physicist, and Astronomer who Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician In 1844, Hermann Grassmann published his book Die lineale Ausdehnungslehre (see References). Hermann Günther Grassmann ( April 15, 1809, Stettin ( Szczecin) &ndash September 26, 1877, Stettin) was a Arthur Cayley introduced matrices, one of the most fundamental linear algebraic ideas, in 1857. Arthur Cayley ( August 16 1821 - January 26 1895) was a British Mathematician. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally Despite these early developments, linear algebra has been developed primarily in the twentieth century. It was the focus of one of the first international mathematical societies, the Quaternion Society (1899 - 1913), which aimed to study allied systems of mathematics. A Scientific society, the Quaternion Society was an “International Association for Promoting the Study of Quaternions and Allied Systems of Mathematics”

Matrices were poorly-defined before the development of ring theory within abstract algebra. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules With the coming of special relativity, many practitioners gained appreciation of the subtleties of linear algebra. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial Furthermore, the routine application of Cramer's rule to solve partial differential equations led to the inclusion of linear algebra in standard coursework at universities. Cramer's rule is a Theorem in Linear algebra, which gives the solution of a System of linear equations in terms of Determinants It is named after In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i E. T. Copson wrote, for instance,

“ When I went to Edinburgh as a young lecturer in 1922, I was surprised to find how different the curriculum was from that at Oxford. It included topics such as Lebesgue integration, matrix theory, numerical analysis, Riemannian geometry, of which I knew nothing. In Mathematics, the Integral of a non-negative function can be regarded in the simplest case as the Area between the graph of Matrix theory is a branch of Mathematics which focuses on the study of matrices. Numerical analysis is the study of Algorithms for the problems of continuous mathematics (as distinguished from Discrete mathematics) Elliptic geometry is also sometimes called Riemannian geometry. . . ”

—E. T. Copson, Preface to Partial Differential Equations, 1973

Francis Galton initiated the use of correlation coefficients in 1888. Sir Francis Galton FRS ( 16 February 1822 &ndash 17 January 1911) half-cousin of Charles Darwin, was an In Probability theory and Statistics, correlation, (often measured as a correlation coefficient) indicates the strength and direction of a linear Often more than one random variable is in play and may be cross-correlated. A random variable is a rigorously defined mathematical entity used mainly to describe Chance and Probability in a mathematical way In Signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them In statistical analysis of multivariate random variables the correlation matrix is a natural tool. Statistics is a mathematical science pertaining to the collection analysis interpretation or explanation and presentation of Data. A multivariate random variable or random vector is a vector X = ( X 1. In Probability theory and Statistics, correlation, (often measured as a correlation coefficient) indicates the strength and direction of a linear Thus, statistical study of such random vectors helped establish matrix usage.

More recent developments followed the formulation of the vector space concept into an algebraic structure, and the growth of functional analysis. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, For functional analysis as used in psychology see the Functional analysis (psychology article One can see a diverse set of applications in the list of matrices. This page lists some important classes of matrices used in Mathematics, Science and Engineering:

Elementary introduction

Linear algebra had its beginnings in the study of vectors in Cartesian 2-space and 3-space. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane A vector, here, is a directed line segment, characterized by both its magnitude, represented by length, and its direction. In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points Vectors can be used to represent physical entities such as forces, and they can be added to each other and multiplied with scalars, thus forming the first example of a real vector space. In Physics, a force is whatever can cause an object with Mass to Accelerate. In Linear algebra, Real numbers are called Scalars and relate to vectors in a Vector space through the operation of Scalar multiplication In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added

Modern linear algebra has been extended to consider spaces of arbitrary or infinite dimension. A vector space of dimension n is called an n-space. Most of the useful results from 2- and 3-space can be extended to these higher dimensional spaces. Although people cannot easily visualize vectors in n-space, such vectors or n-tuples are useful in representing data. In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple Since vectors, as n-tuples, are ordered lists of n components, it is possible to summarize and manipulate data efficiently in this framework. For example, in economics, one can create and use, say, 8-dimensional vectors or 8-tuples to represent the Gross National Product of 8 countries. Economics is the social science that studies the production distribution, and consumption of goods and services. One can decide to display the GNP of 8 countries for a particular year, where the countries' order is specified, for example, (United States, United Kingdom, France, Germany, Spain, India, Japan, Australia), by using a vector (v1, v2, v3, v4, v5, v6, v7, v8) where each country's GNP is in its respective position. The United States of America —commonly referred to as the The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom, the UK or Britain,is a Sovereign state located This article is about the country For a topic outline on this subject see List of basic France topics. Germany, officially the Federal Republic of Germany ( ˈbʊndəsʁepuˌbliːk ˈdɔʏtʃlant is a Country in Central Europe. Spain () or the Kingdom of Spain (Reino de España is a country located mostly in southwestern Europe on the Iberian Peninsula. India, officially the Republic of India (भारत गणराज्य inc-Latn Bhārat Gaṇarājya; see also other Indian languages) is a country For a topic outline on this subject see List of basic Japan topics. For a topic outline on this subject see List of basic Australia topics.

A vector space (or linear space), as a purely abstract concept about which theorems are proved, is part of abstract algebra, and is well integrated into this discipline. In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements Some striking examples of this are the group of invertible linear maps or matrices, and the ring of linear maps of a vector space. 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, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real Linear algebra also plays an important part in analysis, notably, in the description of higher order derivatives in vector analysis and the study of tensor products and alternating maps. In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector

In this abstract setting, the scalars with which an element of a vector space can be multiplied need not be numbers. The only requirement is that the scalars form a mathematical structure, called a field. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In applications, this field is usually the field of real numbers or the field of complex numbers. In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted Linear maps take elements from a linear space to another (or to itself), in a manner that is compatible with the addition and scalar multiplication given on the vector space(s). In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that The set of all such transformations is itself a vector space. If a basis for a vector space is fixed, every linear transform can be represented by a table of numbers called a matrix. 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 The detailed study of the properties of and algorithms acting on matrices, including determinants and eigenvectors, is considered to be part of linear algebra. In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes

One can say quite simply that the linear problems of mathematics - those that exhibit linearity in their behavior - are those most likely to be solved. The word linear comes from the Latin word linearis, which means created by lines. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The word linear comes from the Latin word linearis, which means created by lines. For example differential calculus does a great deal with linear approximation to functions. Differential Calculus, a field in Mathematics, is the study of how functions change when their inputs change The difference from nonlinear problems is very important in practice. This article describes the use of the term nonlinearity in mathematics

The general method of finding a linear way to look at a problem, expressing this in terms of linear algebra, and solving it, if need be by matrix calculations, is one of the most generally applicable in mathematics.

Some useful theorems

Generalisations 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 Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In multilinear algebra one considers multivariable linear transformations, that is, mappings which are linear in each of a number of different variables. In Mathematics, multilinear algebra extends the methods of Linear algebra. This line of inquiry naturally leads to the idea of the tensor product. In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector In the spectral theory of operators control of infinite-dimensional matrices is gained, by applying mathematical analysis in a theory that is not purely algebraic. Analysis has its beginnings in the rigorous formulation of Calculus. In all these cases the technical difficulties are much greater.

See also

Note

  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. This is a list of Linear algebra topics, by Wikipedia page See also List of matrices Glossary of matrix theory Numerical linear algebra is the study of Algorithms for performing Linear algebra computations most notably matrix operations on Computers In Mathematics, the dimension theorem for vector spaces states that a Vector space has a definite Well-defined number of dimensions In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory.

References

See also: List of linear algebra references

Textbooks

Free Online books

History

External links

The University of Edinburgh (Oilthigh Dhùn Èideann founded in 1582 is a renowned centre for teaching and research in Edinburgh, Scotland, UK. Lamar University is a four-year university located in Beaumont Texas, and a member of The Texas State University System.

Dictionary

linear algebra

-noun

  1. (mathematics) The branch of mathematics that deals with vectors, vector spaces, linear transformations and systems of linear equations.
  2. (linear algebra) An algebra over a field.
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