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Encyclopedia > Linear algebra

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. Vector spaces are a central theme in modern mathematics; thus, linear algebra is widely used in both abstract algebra and functional analysis. Linear algebra also has a concrete representation in analytic geometry and it is generalized in operator theory. It has extensive applications in the natural sciences and the social sciences, since nonlinear models can often be approximated by linear ones. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In physics and in vector calculus, a spatial vector, or simply vector, is a concept characterized by a magnitude and a direction. ... 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 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. Definition and first consequences Formally, if V and W are... In mathematics and linear algebra, a system of linear equations is a set of linear equations such as 3x1 + 2x2 − x3 = 1 2x1 − 2x2 + 4x3 = −2 −x1 + ½x2 − x3 = 0. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Algebra is a branch of mathematics concerning the study of structure, relation and quantity. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of spaces of functions. ... Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra. ... In mathematics, operator theory is the branch of functional analysis which deals with bounded linear operators and their properties. ... The lunar farside as seen from Apollo 11 Natural science is the rational study of the universe via rules or laws of natural order. ... This article or section does not adequately cite its references or sources. ...

Contents

History

The history of modern linear algebra dates back to the years 1843 and 1844. In 1843, William Rowan Hamilton (from whom the term vector stems) introduced the quaternions. In 1844, Hermann Grassmann published his book Die lineale Ausdehnungslehre (see References). Arthur Cayley introduced matrices, one of the most fundamental linear algebraic ideas, in 1857. These early references belie the fact that linear algebra is mainly a development of the twentieth century: the number-like objects called matrices were hard to place before the development of ring theory in abstract algebra. With the coming of special relativity many practitioners gained appreciation of the subtleties of linear algebra. Furthermore, the routine application of Cramer's rule to solve partial differential equations led to inclusion of linear algebra in standard coursework at universities. For instance, E.T. Copson wrote: Sir William Rowan Hamilton (August 4, 1805 – September 2, 1865) was an Irish mathematician, physicist, and astronomer who made important contributions to the development of optics, dynamics, and algebra. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... Hermann Günther Grassmann (April 15, 1809, Stettin – September 26, 1877, Stettin) was a German polymath, renowned in his day as a linguist and now admired as a mathematician. ... Arthur Cayley (August 16, 1821 - January 26, 1895) was a British mathematician. ... 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. ... 1857 was a common year starting on Thursday (see link for calendar). ... In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... Cramers rule is a theorem in linear algebra, which gives the solution of a system of linear equations in terms of determinants. ... In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ...

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...

—E.T. Copson, Preface to Partial Differential Equations, 1973 In mathematics, the integral of a function of one real variable can be regarded as the area of a plane region bounded by the graph of that function. ... Matrix theory is a branch of mathematics which focuses on the study of matrices. ... Numerical analysis is the study of approximate methods for the problems of continuous mathematics (as distinguished from discrete mathematics). ... In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ...

Francis Galton initiated the use of correlation coefficients in 1888. Often more than one random variable is in play and they may be cross-correlated. In statistical analysis of multivariate random variables the correlation matrix is a natural tool. Thus statistical study of such random vectors helped develop matrix usage. Sir Francis Galton F.R.S. (February 16, 1822 – January 17, 1911), half-cousin of Charles Darwin, was an English Victorian polymath, anthropologist, eugenicist, tropical explorer, geographer, inventor, meteorologist, proto-geneticist, psychometrician, and statistician. ... Positive linear correlations between 1000 pairs of numbers. ... A random variable is a mathematical function that maps outcomes of random experiments to numbers. ... In statistics, the term cross-correlation is sometimes used to refer to the covariance cov(X, Y) between two random vectors X and Y, in order to distinguish that concept from the covariance of a random vector X, which is understood to be the matrix of covariances between the scalar... Statistics is the science and practice of developing knowledge through the use of empirical data expressed in quantitative form. ... A multivariate random variable or random vector is a vector X = (X1, ..., Xn) whose components are scalar-valued random variables on the same probability space (Ω, P). ... Positive linear correlations between 1000 pairs of numbers. ...


Elementary introduction

Linear algebra had its beginnings in the study of vectors in Cartesian 2-space and 3-space. A vector, here, is a directed line segment, characterized by both its magnitude, represented by length, and its direction. 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. Fig. ... The geometric definition of a line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. ... In physics, force is an influence that may cause a body 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 which a vector can be multiplied by a number to produce another vector. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... 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 many people cannot easily visualize vectors in n-space, such vectors or n-tuples are useful in representing data. 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. 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. In mathematics, a tuple is a finite sequence of objects (a list of a limited number of objects). ... Face-to-face trading interactions on the New York Stock Exchange trading floor. ... Measures of national income and output are used in economics to estimate the value of goods and services produced in an economy. ...


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. Some striking examples of this are the group of invertible linear maps or matrices, and the ring of linear maps of a vector space. 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. Look up theorem in Wiktionary, the free dictionary. ... This picture illustrates how the hours in a clock form a group. ... 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 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. ...


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 applications, this field is usually the field of real numbers or the field of complex numbers. 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). 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. 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 abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... 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. Definition and first consequences Formally, if V and W are... In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ... 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, computing, linguistics, and related disciplines, an algorithm is a finite set of well-defined instructions for accomplishing some task which, given an initial state, will terminate in a defined end-state. ... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... 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. ...


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. For example differential calculus does a great deal with linear approximation to functions. The difference from nonlinear problems is very important in practice. The word linear comes from the Latin word linearis, which means created by lines. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... The word linear comes from the Latin word linearis, which means created by lines. ... Differential calculus is the theory of and computations with differentials; see also derivative and calculus. ... In mathematics, nonlinear systems represent systems whose behavior is not expressible as a sum of the behaviors of its descriptors. ...


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

In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ... In mathematics, the cardinality of a set is a measure of the number of elements of the set. There are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. ... In mathematics, the dimension of a vector space V is the cardinality (i. ... In mathematics, the idea of inverse element generalises both of the concepts of negation, in relation to addition (see additive inverse), and reciprocal, in relation to multiplication. ... It has been suggested that this article or section be merged with Logical biconditional. ... In algebra, a determinant is a function depending on n that associates a scalar, det(A), to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In mathematics, the idea of inverse element generalises both of the concepts of negation, in relation to addition (see additive inverse), and reciprocal, in relation to multiplication. ... 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. Definition and first consequences Formally, if V and W are... 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 linear algebra, an n-by-n (square) matrix is called invertible, non-singular, or regular if there exists an n-by-n matrix such that where denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication. ... In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In linear algebra, a positive-definite matrix is a Hermitian matrix which in many ways is analogous to a positive real number. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. ... In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i. ...

Generalisation 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 deals with the 'several variables' problem of mappings linear in each of a number of different variables, inevitably leading to the tensor concept. 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. In all these cases the technical difficulties are much greater. 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 to lie in a field, the scalars may lie in an arbitrary ring. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, multilinear algebra extends the methods of linear algebra. ... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ...


See also

This is a list of linear algebra topics, by Wikipedia page. ... This is a list of important publications in mathematics, organized by field. ... Numerical linear algebra is often at the heart of many engineering and computational science problems, such as image and signal processing, computational finance, materials science simulations, structural biology, datamining, and bioinformatics just to name a few. ...

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.

In mathematics, a module is a finitely-generated module if it has a finite generating set. ... 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

  • Beezer, Rob, A First Course in Linear Algebra, licensed under GFDL.
  • Fearnley-Sander, Desmond, Hermann Grassmann and the Creation of Linear Algebra, American Mathematical Monthly 86 (1979), pp. 809–817.
  • Grassmann, Hermann, Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik: dargestellt und durch Anwendungen auf die übrigen Zweige der Mathematik, wie auch auf die Statik, Mechanik, die Lehre vom Magnetismus und die Krystallonomie erläutert, O. Wigand, Leipzig, 1844.
  • Jim Hefferon: Linear Algebra (Online textbook)
  • Edwin H. Connell: Elements of Abstract and Linear Algebra (Online textbook)

GNU logo (similar in appearance to a gnu) The GNU Free Documentation License (GNU FDL or simply GFDL) is a copyleft license for free content, designed by the Free Software Foundation (FSF) for the GNU project. ...

External links

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