in mathematics and physics a vector space also called a linear space is a set whose elements often called vectors may be added together and multiplied scaled by numbers called scalars


scalars are often real numbers but can be complex numbers or more generally elements of any field


the operations of vector addition and scalar multiplication must satisfy certain requirements called vector axioms


real vector space and complex vector space are kinds of vector spaces based on different kinds of scalars real coordinate space or complex coordinate space


vector spaces generalize euclidean vectors which allow modeling of physical quantities such as forces and velocity that have not only a magnitude but also a direction


the concept of vector spaces is fundamental for linear algebra together with the concept of matrices which allows computing in vector spaces


this provides a concise and synthetic way for manipulating and studying systems of linear equations


vector spaces are characterized by their dimension which roughly speaking specifies the number of independent directions in the space


this means that for two vector spaces over a given field and with the same dimension the properties that depend only on the vectorspace structure are exactly the same technically the vector spaces are isomorphic


a vector space is finitedimensional if its dimension is a natural number


otherwise it is infinitedimensional and its dimension is an infinite cardinal


finitedimensional vector spaces occur naturally in geometry and related areas


infinitedimensional vector spaces occur in many areas of mathematics


for example polynomial rings are countably infinitedimensional vector spaces and many function spaces have the cardinality of the continuum as a dimension


many vector spaces that are considered in mathematics are also endowed with other structures


this is the case of algebras which include field extensions polynomial rings associative algebras and lie algebras


this is also the case of topological vector spaces which include function spaces inner product spaces normed spaces hilbert spaces and banach spaces


in this article vectors are represented in boldface to distinguish them from scalars


a vector space over a field f is a nonempty set v together with a binary operation and a binary function that satisfy the eight axioms listed below


in this context the elements of v are commonly called vectors and the elements of f are called scalars


the binary operation called vector addition or simply addition assigns to any two vectors v and w in v a third vector in v which is commonly written as v  w and called the sum of these two vectors


the binary function called scalar multiplicationassigns to any scalar a in f and any vector v in v another vector in v which is denoted av


to have a vector space the eight following axioms must be satisfied for every u v and w in v and a and b in f


when the scalar field is the real numbers the vector space is called a real vector space and when the scalar field is the complex numbers the vector space is called a complex vector space


these two cases are the most common ones but vector spaces with scalars in an arbitrary field f are also commonly considered


in mathematics a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results eg


a linear combination of x and y would be any expression of the form ax  by where a and b are constants


the concept of linear combinations is 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


in the theory of vector spaces a set of vectors is said to be linearly independent if there exists no nontrivial linear combination of the vectors that equals the zero vector


if such a linear combination exists then the vectors are said to be linearly dependent


these concepts are central to the definition of dimension


a vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors


the definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space


linear algebra is central to almost all areas of mathematics


for instance linear algebra is fundamental in modern presentations of geometry including for defining basic objects such as lines planes and rotations


also functional analysis a branch of mathematical analysis may be viewed as the application of linear algebra to function spaces


linear algebra is also used in most sciences and fields of engineering because it allows modeling many natural phenomena and computing efficiently with such models


for nonlinear systems which cannot be modeled with linear algebra it is often used for dealing with firstorder approximations using the fact that the differential of a multivariate function at a point is the linear map that best approximates the function near that point


in mathematics a matrix pl


matrices is a rectangular array or table of numbers symbols or expressions arranged in rows and columns which is used to represent a mathematical object or a property of such an object


matrices are used to represent linear maps and allow explicit computations in linear algebra


therefore the study of matrices is a large part of linear algebra and most properties and operations of abstract linear algebra can be expressed in terms of matrices


for example matrix multiplication represents the composition of linear maps


not all matrices are related to linear algebra


this is in particular the case in graph theory of incidence matrices and adjacency matrices


this article focuses on matrices related to linear algebra and unless otherwise specified all matrices represent linear maps or may be viewed as such


square matrices matrices with the same number of rows and columns play a major role in matrix theory


square matrices of a given dimension form a noncommutative ring which is one of the most common examples of a noncommutative ring


the determinant of a square matrix is a number associated to the matrix which is fundamental for the study of a square matrix for example a square matrix is invertible if and only if it has a nonzero determinant and the eigenvalues of a square matrix are the roots of a polynomial determinant


in geometry matrices are widely used for specifying and representing geometric transformations for example rotations and coordinate changes


in numerical analysis many computational problems are solved by reducing them to a matrix computation and this often involves computing with matrices of huge dimension


matrices are used in most areas of mathematics and most scientific fields either directly or through their use in geometry and numerical analysis


matrix theory is the branch of mathematics that focuses on the study of matrices


it was initially a subbranch of linear algebra but soon grew to include subjects related to graph theory algebra combinatorics and statistics


functional analysis is a branch of mathematical analysis the core of which is formed by the study of vector spaces endowed with some kind of limitrelated structure for example inner product norm or topology and the linear functions defined on these spaces and suitably respecting these structures


the historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the fourier transform as transformations defining for example continuous or unitary operators between function spaces


this point of view turned out to be particularly useful for the study of differential and integral equations


the usage of the word functional as a noun goes back to the calculus of variations implying a function whose argument is a function


the term was first used in hadamards  book on that subject


however the general concept of a functional had previously been introduced in  by the italian mathematician and physicist vito volterra


the theory of nonlinear functionals was continued by students of hadamard in particular fréchet and lévy


hadamard also founded the modern school of linear functional analysis further developed by riesz and the group of polish mathematicians around stefan banach


in modern introductory texts on functional analysis the subject is seen as the study of vector spaces endowed with a topology in particular infinitedimensional spaces


in contrast linear algebra deals mostly with finitedimensional spaces and does not use topology


an important part of functional analysis is the extension of the theories of measure integration and probability to infinite dimensional spaces also known as infinite dimensional analysis


in mathematics differential refers to several related notions derived from the early days of calculus put on a rigorous footing such as infinitesimal differences and the derivatives of functions


the term is used in various branches of mathematics such as calculus differential geometry algebraic geometry and algebraic topology


in mathematics and more specifically in linear algebra a linear map also called a linear mapping linear transformation vector space homomorphism or in some contexts linear function is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication


the same names and the same definition are also used for the more general case of modules over a ring see module homomorphism


f a linear map is a bijection then it is called a linear isomorphism


in the case where a linear map is called a linear endomorphism


sometimes the term linear operator refers to this case but the term linear operator can have different meanings for different conventions for example it can be used to emphasize that and are real vector spaces not necessarily with citation needed or it can be used to emphasize that  is a function space which is a common convention in functional analysis


sometimes the term linear function has the same meaning as linear map while in analysis it does not


in the language of category theory linear maps are the morphisms of vector spaces


a linear map from  to  always maps the origin of  to the origin of


moreover it maps linear subspaces in  onto linear subspaces in  possibly of a lower dimension for example it maps a plane through the origin in  to either a plane through the origin in  a line through the origin in  or just the origin in


linear maps can often be represented as matrices and simple examples include rotation and reflection linear transformations


a bijective linear map between two vector spaces that is every vector from the second space is associated with exactly one in the first is an isomorphism


because an isomorphism preserves linear structure two isomorphic vector spaces are essentially the same from the linear algebra point of view in the sense that they cannot be distinguished by using vector space properties


an essential question in linear algebra is testing whether a linear map is an isomorphism or not and if it is not an isomorphism finding its range or image and the set of elements that are mapped to the zero vector called the kernel of the map


all these questions can be solved by using gaussian elimination or some variant of this algorithm


in mathematics and more specifically in linear algebra a linear subspace or vector subspacenote  is a vector space that is a subset of some larger vector space


a linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces


as a corollary all vector spaces are equipped with at least two possibly different linear subspaces the zero vector space consisting of the zero vector alone and the entire vector space itself


these are called the trivial subspaces of the vector space


if v is a vector space over a field k and if w is a subset of v then w is a linear subspace of v if under the operations of v w is a vector space over k equivalently a nonempty subset w is a linear subspace of v if whenever w w are elements of w and α β are elements of k it follows that αw  βw is in w


in mathematics the linear span also called the linear hull or just span of a set s of vectors from a vector space denoted spans is defined as the set of all linear combinations of the vectors in s for example two linearly independent vectors span a plane


the linear span can be characterized either as the intersection of all linear subspaces that contain s or as the smallest subspace containing s the linear span of a set of vectors is therefore a vector space itself


spans can be generalized to matroids and modules


to express that a vector space v is a linear span of a subset s one commonly uses the following phraseseither s spans v s is a spanning set of v v is spannedgenerated by s or s is a generator or generator set of v


in mathematics a set b of vectors in a vector space v is called a basis pl


bases if every element of v may be written in a unique way as a finite linear combination of elements of b


the coefficients of this linear combination are referred to as components or coordinates of the vector with respect to b


the elements of a basis are called basis vectors


equivalently a set b is a basis if its elements are linearly independent and every element of v is a linear combination of elements of b


in other words a basis is a linearly independent spanning set


a vector space can have several bases however all the bases have the same number of elements called the dimension of the vector space


this article deals mainly with finitedimensional vector spaces


however many of the principles are also valid for infinitedimensional vector spaces


in mathematics a system of linear equations or linear system is a collection of one or more linear equations involving the same variables


a solution to a linear system is an assignment of values to the variables such that all the equations are simultaneously satisfied


in the example above a solution is given by the ordered triple  since it makes all three equations valid


the word system indicates that the equations should be considered collectively rather than individually


linear systems are the basis and a fundamental part of linear algebra a subject used in most modern mathematics


computational algorithms for finding the solutions are an important part of numerical linear algebra and play a prominent role in engineering physics chemistry computer science and economics


a system of nonlinear equations can often be approximated by a linear system see linearization a helpful technique when making a mathematical model or computer simulation of a relatively complex system


very often and in this article the coefficients of the equations are real or complex numbers and the solutions are searched in the same set of numbers but the theory and the algorithms apply for coefficients and solutions in any field


for solutions in an integral domain like the ring of the integers or in other algebraic structures other theories have been developed see linear equation over a ring


integer linear programming is a collection of methods for finding the best integer solution when there are many


gröbner basis theory provides algorithms when coefficients and unknowns are polynomials


tropical geometry is another example of linear algebra in a more exotic structure


a linear endomorphism is a linear map that maps a vector space v to itself


if v has a basis of n elements such an endomorphism is represented by a square matrix of size n


with respect to general linear maps linear endomorphisms and square matrices have some specific properties that make their study an important part of linear algebra which is used in many parts of mathematics including geometric transformations coordinate changes quadratic forms and many other part of mathematics


this allows all the language and theory of vector spaces or more generally modules to be brought to bear


for example the collection of all possible linear combinations of the vectors on the lefthand side is called their span and the equations have a solution just when the righthand vector is within that span


if every vector within that span has exactly one expression as a linear combination of the given lefthand vectors then any solution is unique


in any event the span has a basis of linearly independent vectors that do guarantee exactly one expression and the number of vectors in that basis its dimension cannot be larger than m or n but it can be smaller


this is important because if we have m independent vectors a solution is guaranteed regardless of the righthand side and otherwise not guaranteed


geometric interpretation


for a system involving two variables x and y each linear equation determines a line on the xyplane


because a solution to a linear system must satisfy all of the equations the solution set is the intersection of these lines and is hence either a line a single point or the empty set


for three variables each linear equation determines a plane in threedimensional space and the solution set is the intersection of these planes


thus the solution set may be a plane a line a single point or the empty set


for example as three parallel planes do not have a common point the solution set of their equations is empty the solution set of the equations of three planes intersecting at a point is single point if three planes pass through two points their equations have at least two common solutions in fact the solution set is infinite and consists in all the line passing through these points


for n variables each linear equation determines a hyperplane in ndimensional space


the solution set is the intersection of these hyperplanes and is a flat which may have any dimension lower than n


in general the behavior of a linear system is determined by the relationship between the number of equations and the number of unknowns


here in general means that a different behavior may occur for specific values of the coefficients of the equations


in general a system with fewer equations than unknowns has infinitely many solutions but it may have no solution


such a system is known as an underdetermined system


in general a system with the same number of equations and unknowns has a single unique solution


in general a system with more equations than unknowns has no solution


such a system is also known as an overdetermined system


in the first case the dimension of the solution set is in general equal to n  m where n is the number of variables and m is the number of equations


the following pictures illustrate this trichotomy in the case of two variables


the first system has infinitely many solutions namely all of the points on the blue line


the second system has a single unique solution namely the intersection of the two lines


the third system has no solutions since the three lines share no common point


it must be kept in mind that the pictures above show only the most common case the general case


it is possible for a system of two equations and two unknowns to have no solution if the two lines are parallel or for a system of three equations and two unknowns to be solvable if the three lines intersect at a single point


a system of linear equations behave differently from the general case if the equations are linearly dependent or if it is inconsistent and has no more equations than unknowns


a differential equation can be homogeneous in either of two respects


a first order differential equation is said to be homogeneous if it may be written


where f and g are homogeneous functions of the same degree of x and y


in this case the change of variable y  ux leads to an equation of the form


which is easy to solve by integration of the two members


otherwise a differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives


in the case of linear differential equations this means that there are no constant terms


the solutions of any linear ordinary differential equation of any order may be deduced by integration from the solution of the homogeneous equation obtained by removing the constant term


there is a strong relationship between linear algebra and geometry which started with the introduction by rené descartes in  of cartesian coordinates


in this new at that time geometry now called cartesian geometry points are represented by cartesian coordinates which are sequences of three real numbers in the case of the usual threedimensional space


the basic objects of geometry which are lines and planes are represented by linear equations


thus computing intersections of lines and planes amounts to solving systems of linear equations


this was one of the main motivations for developing linear algebra


most geometric transformation such as translations rotations reflections rigid motions isometries and projections transform lines into lines


it follows that they can be defined specified and studied in terms of linear maps


this is also the case of homographies and möbius transformations when considered as transformations of a projective space


until the end of the th century geometric spaces were defined by axioms relating points lines and planes synthetic geometry


around this date it appeared that one may also define geometric spaces by constructions involving vector spaces see for example projective space and affine space


it has been shown that the two approaches are essentially equivalent


in classical geometry the involved vector spaces are vector spaces over the reals but the constructions may be extended to vector spaces over any field allowing considering geometry over arbitrary fields including finite fields


presently most textbooks introduce geometric spaces from linear algebra and geometry is often presented at elementary level as a subfield of linear algebra


linear algebra is used in almost all areas of mathematics thus making it relevant in almost all scientific domains that use mathematics


these applications may be divided into several wide categories


functional analysis studies function spaces


these are vector spaces with additional structure such as hilbert spaces


linear algebra is thus a fundamental part of functional analysis and its applications which include in particular quantum mechanics wave functions and fourier analysis orthogonal basis


nearly all scientific computations involve linear algebra


consequently linear algebra algorithms have been highly optimized


blas and lapack are the best known implementations


for improving efficiency some of them configure the algorithms automatically at run time for adapting them to the specificities of the computer cache size number of available cores


some processors typically graphics processing units gpu are designed with a matrix structure for optimizing the operations of linear algebra


citation needed


the modeling of ambient space is based on geometry


sciences concerned with this space use geometry widely


this is the case with mechanics and robotics for describing rigid body dynamics geodesy for describing earth shape perspectivity computer vision and computer graphics for describing the relationship between a scene and its plane representation and many other scientific domains


in all these applications synthetic geometry is often used for general descriptions and a qualitative approach but for the study of explicit situations one must compute with coordinates


this requires the heavy use of linear algebra


most physical phenomena are modeled by partial differential equations


to solve them one usually decomposes the space in which the solutions are searched into small mutually interacting cells


for linear systems this interaction involves linear functions


for nonlinear systems this interaction is often approximated by linear functions


bthis is called a linear model or firstorder approximation


linear models are frequently used for complex nonlinear realworld systems because it makes parametrization more manageable


in both cases very large matrices are generally involved


weather forecasting or more specifically parametrization for atmospheric modeling is a typical example of a realworld application where the whole earth atmosphere is divided into cells of say  km of width and  km of height


fluid mechanics fluid dynamics and thermal energy systems


linear algebra a branch of mathematics dealing with vector spaces and linear mappings between these spaces plays a critical role in various engineering disciplines including fluid mechanics fluid dynamics and thermal energy systems


its application in these fields is multifaceted and indispensable for solving complex problems


in fluid mechanics linear algebra is integral to understanding and solving problems related to the behavior of fluids


it assists in the modeling and simulation of fluid flow providing essential tools for the analysis of fluid dynamics problems


for instance linear algebraic techniques are used to solve systems of differential equations that describe fluid motion


these equations often complex and nonlinear can be linearized using linear algebra methods allowing for simpler solutions and analyses


in the field of fluid dynamics linear algebra finds its application in computational fluid dynamics cfd a branch that uses numerical analysis and data structures to solve and analyze problems involving fluid flows


cfd relies heavily on linear algebra for the computation of fluid flow and heat transfer in various applications


for example the navierstokes equations fundamental in fluid dynamics are often solved using techniques derived from linear algebra


this includes the use of matrices and vectors to represent and manipulate fluid flow fields


furthermore linear algebra plays a crucial role in thermal energy systems particularly in power systems analysis


it is used to model and optimize the generation transmission and distribution of electric power


linear algebraic concepts such as matrix operations and eigenvalue problems are employed to enhance the efficiency reliability and economic performance of power systems


the application of linear algebra in this context is vital for the design and operation of modern power systems including renewable energy sources and smart grids


overall the application of linear algebra in fluid mechanics fluid dynamics and thermal energy systems is an example of the profound interconnection between mathematics and engineering


it provides engineers with the necessary tools to model analyze and solve complex problems in these domains leading to advancements in technology and industry


the existence of multiplicative inverses in fields is not involved in the axioms defining a vector space


one may thus replace the field of scalars by a ring r and this gives the structure called a module over r or rmodule


the concepts of linear independence span basis and linear maps also called module homomorphisms are defined for modules exactly as for vector spaces with the essential difference that if r is not a field there are modules that do not have any basis


the modules that have a basis are the free modules and those that are spanned by a finite set are the finitely generated modules


module homomorphisms between finitely generated free modules may be represented by matrices


the theory of matrices over a ring is similar to that of matrices over a field except that determinants exist only if the ring is commutative and that a square matrix over a commutative ring is invertible only if its determinant has a multiplicative inverse in the ring


vector spaces are completely characterized by their dimension up to an isomorphism


in general there is not such a complete classification for modules even if one restricts oneself to finitely generated modules


however every module is a cokernel of a homomorphism of free modules


modules over the integers can be identified with abelian groups since the multiplication by an integer may be identified to a repeated addition


most of the theory of abelian groups may be extended to modules over a principal ideal domain


in particular over a principal ideal domain every submodule of a free module is free and the fundamental theorem of finitely generated abelian groups may be extended straightforwardly to finitely generated modules over a principal ring


there are many rings for which there are algorithms for solving linear equations and systems of linear equations


however these algorithms have generally a computational complexity that is much higher than the similar algorithms over a field


for more details see linear equation over 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 dual space the vector space v consisting of linear maps f  v  f where f is the field of scalars


multilinear maps t  vn  f can be described via tensor products of elements of v


if in addition to vector addition and scalar multiplication there is a bilinear vector product v  v  v the vector space is called an algebra for instance associative algebras are algebras with an associate vector product like the algebra of square matrices or the algebra of polynomials


vector spaces that are not finite dimensional often require additional structure to be tractable


a normed vector space is a vector space along with a function called a norm which measures the size of elements


the norm induces a metric which measures the distance between elements and induces a topology which allows for a definition of continuous maps


the metric also allows for a definition of limits and completeness  a metric space that is complete is known as a banach space


a complete metric space along with the additional structure of an inner product a conjugate symmetric sesquilinear form is known as a hilbert space which is in some sense a particularly wellbehaved banach space


functional analysis applies the methods of linear algebra alongside those of mathematical analysis to study various function spaces the central objects of study in functional analysis are lp spaces which are banach spaces and especially the l space of square integrable functions which is the only hilbert space among them


functional analysis is of particular importance to quantum mechanics the theory of partial differential equations digital signal processing and electrical engineering


it also provides the foundation and theoretical framework that underlies the fourier transform and related methods


analysis is the branch of mathematics dealing with continuous functions limits and related theories such as differentiation integration measure infinite sequences series and analytic functions


these theories are usually studied in the context of real and complex numbers and functions


analysis evolved from calculus which involves the elementary concepts and techniques of analysis


analysis may be distinguished from geometry however it can be applied to any space of mathematical objects that has a definition of nearness a topological space or specific distances between objects a metric space


in mathematics physics and engineering a euclidean vector or simply a vector sometimes called a geometric vector or spatial vector is a geometric object that has magnitude or length and direction


vectors can be added to other vectors according to vector algebra


a euclidean vector is frequently represented by a directed line segment or graphically as an arrow connecting an initial point a with a terminal point b and denoted by


a vector is what is needed to carry the point a to the point b the latin word vector means carrier


it was first used by th century astronomers investigating planetary revolution around the sun


the magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement from a to b


many algebraic operations on real numbers such as addition subtraction multiplication and negation have close analogues for vectors operations which obey the familiar algebraic laws of commutativity associativity and distributivity


these operations and associated laws qualify euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space


vectors play an important role in physics the velocity and acceleration of a moving object and the forces acting on it can all be described with vectors


many other physical quantities can be usefully thought of as vectors


although most of them do not represent distances except for example position or displacement their magnitude and direction can still be represented by the length and direction of an arrow


the mathematical representation of a physical vector depends on the coordinate system used to describe it


other vectorlike objects that describe physical quantities and transform in a similar way under changes of the coordinate system include pseudovectors and tensors