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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
,
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

physics
, and
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

engineering
, a vector space (also called a linear space) is a set of objects called ''vectors'', which may be
added
added
together and
multiplied
multiplied
("scaled") by numbers called '' scalars''. Scalars are often
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, but some vector spaces have scalar multiplication by
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s or, generally, by a scalar from any mathematic field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called vector ''
axiom An axiom, postulate or assumption is a statement that is taken to be , to serve as a or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or 'that which comm ...

axiom
s'' (listed below in ). To specify whether the scalars in a particular vector space are real numbers or complex numbers, the terms real vector space and complex vector space are often used. Certain sets of
Euclidean vector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s are common examples of a vector space. They represent
physical Physical may refer to: *Physical examination, a regular overall check-up with a doctor *Physical (album), ''Physical'' (album), a 1981 album by Olivia Newton-John **Physical (Olivia Newton-John song), "Physical" (Olivia Newton-John song) *Physical ( ...

physical
quantities such as
force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

force
s, where any two forces of the same type can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same way (but in a more
geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, ...

geometric
sense), vectors representing displacements in the plane or
three-dimensional space Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point). This is the informal meaning of the ...
also form vector spaces. Vectors in vector spaces do not necessarily have to be arrow-like objects as they appear in the mentioned examples: vectors are regarded as abstract
mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and with which one may do deductive reasoning and mathematical proofs ...
s with particular properties, which in some cases can be visualized as arrows. Vector spaces are the subject of
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mat ...
and are well characterized by their
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
, which, roughly speaking, specifies the number of independent directions in the space. Infinite-dimensional vector spaces arise naturally in
mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathematics), series, and analytic ...
as
function space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s, whose vectors are
functions Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
. These vector spaces are generally endowed with some additional structure such as a
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

topology
, which allows the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a
norm Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy) Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...
or
inner product In mathematics, an inner product space or a Hausdorff space, Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a Scalar (mathematics), ...
are more commonly used (being equipped with a notion of
distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...
between two vectors). This is particularly the case of
Banach space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s and
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s, which are fundamental in mathematical analysis. Historically, the first ideas leading to vector spaces can be traced back as far as the 17th century's
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measur ...
,
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
, systems of
linear equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

linear equation
s, and Euclidean vectors. The modern, more abstract treatment, first formulated by
Giuseppe Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

Giuseppe Peano
in 1888, encompasses more general objects than
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
, but much of the theory can be seen as an extension of classical geometric ideas like
line Line, lines, The Line, or LINE may refer to: Arts, entertainment, and media Films * ''Lines'' (film), a 2016 Greek film * ''The Line'' (2017 film) * ''The Line'' (2009 film) * ''The Line'', a 2009 independent film by Nancy Schwartzman Lite ...

line
s,
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
s and their higher-dimensional analogs. Today, vector spaces are applied throughout
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
,
science Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity or awareness, of someone or something, such as facts A fact is something that is truth, true. The usual test for a statement of ...

science
and
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

engineering
. They are the appropriate linear-algebraic notion to deal with systems of linear equations. They offer a framework for
Fourier expansionFourier may refer to: People named Fourier *Joseph Fourier (1768–1830), French mathematician and physicist *Charles Fourier (1772–1837), French utopian socialist thinker *Peter Fourier (1565–1640), French saint in the Roman Catholic Church an ...
, which is employed in
image compression Image compression is a type of data compression In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image proc ...
routines, and they provide an environment that can be used for solution techniques for
partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s. Furthermore, vector spaces furnish an abstract,
coordinate-free A coordinate-free, or component-free, treatment of a scientific theory A scientific theory is an explanation of an aspect of the natural world and universe that can be repeatedly tested and verified in accordance with the scientific method ...
way of dealing with geometrical and physical objects such as
tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

tensor
s. This in turn allows the examination of local properties of
manifolds The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of suc ...
by linearization techniques. Vector spaces may be generalized in several ways, leading to more advanced notions in geometry and
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
. This article deals mainly with finite-dimensional vector spaces. However, many of the principles are also valid for infinite-dimensional vector spaces.


Motivating examples

Two typical vector space examples are described first.


First example: arrows in the plane

The first example of a vector space consists of
arrow An arrow is a fin-stabilized projectile launched by a bow and arrow, bow. A typical arrow usually consists of a long, stiff, straight ''shaft'' with a weighty (and usually sharp and pointed) ''arrowhead'' attached to the front end, multiple fi ...
s in a fixed
plane Plane or planes may refer to: * Airplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft A fixed-wing aircraft is a heavier-than-air flying machine Early flying machines include all forms of aircraft studied ...
, starting at one fixed point. This is used in physics to describe
force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

force
s or
velocities The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction o ...

velocities
. Given any two such arrows, and , the
parallelogram In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method c ...

parallelogram
spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the ''sum'' of the two arrows, and is denoted . In the special case of two arrows on the same line, their sum is the arrow on this line whose length is the sum or the difference of the lengths, depending on whether the arrows have the same direction. Another operation that can be done with arrows is scaling: given any positive
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
, the arrow that has the same direction as , but is dilated or shrunk by multiplying its length by , is called ''multiplication'' of by . It is denoted . When is negative, is defined as the arrow pointing in the opposite direction instead. The following shows a few examples: if , the resulting vector has the same direction as , but is stretched to the double length of (right image below). Equivalently, is the sum . Moreover, has the opposite direction and the same length as (blue vector pointing down in the right image).


Second example: ordered pairs of numbers

A second key example of a vector space is provided by pairs of real numbers and . (The order of the components and is significant, so such a pair is also called an
ordered pair In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

ordered pair
.) Such a pair is written as . The sum of two such pairs and multiplication of a pair with a number is defined as follows: : (x_1 , y_1) + (x_2 , y_2) = (x_1 + x_2, y_1 + y_2) and : a(x, y)=(ax, ay) . The first example above reduces to this example, if an arrow is represented by a pair of
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

Cartesian coordinates
of its endpoint.


Notation and definition

In this article, vectors are represented in boldface to distinguish them from scalars.It is also common, especially in physics, to denote vectors with an arrow on top: . A vector space over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
is a set  together with two operations that satisfy the eight axioms listed below. In the following, denotes the
Cartesian product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
of with itself, and denotes a
mapping Mapping may refer to: * Mapping (cartography), the process of making a map * Mapping (mathematics), a synonym for a mathematical function and its generalizations ** Mapping (logic), a synonym for functional predicate Types of mapping * Animated ...

mapping
from one set to another. * The first operation, called vector addition or simply addition , takes any two vectors  and and assigns to them a third vector which is commonly written as , and called the sum of these two vectors. (The resultant vector is also an element of the set .) * The second operation, called
scalar multiplication 250px, The scalar multiplications −a and 2a of a vector a In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ...

scalar multiplication
, takes any scalar  and any vector  and gives another vector . (Similarly, the vector is an element of the set . Scalar multiplication is not to be confused with the
scalar product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
, also called ''inner product'' or ''dot product'', which is an additional structure present on some specific, but not all vector spaces. Scalar multiplication is a multiplication of a vector ''by'' a scalar; the other is a multiplication of two vectors ''producing'' a scalar.) Elements of are commonly called ''vectors''. Elements of  are commonly called ''scalars''. Common symbols for denoting vector spaces include , , and . In the two examples above, the field is the field of the real numbers, and the set of the vectors consists of the planar arrows with a fixed starting point and pairs of real numbers, respectively. To qualify as a vector space, the set  and the operations of vector addition and scalar multiplication must adhere to a number of requirements called
axiom An axiom, postulate or assumption is a statement that is taken to be , to serve as a or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or 'that which comm ...

axiom
s. These are listed in the table below, where , and denote arbitrary vectors in , and and denote scalars in . These axioms generalize properties of the vectors introduced in the above examples. Indeed, the result of addition of two ordered pairs (as in the second example above) does not depend on the order of the summands: :. Likewise, in the geometric example of vectors as arrows, since the parallelogram defining the sum of the vectors is independent of the order of the vectors. All other axioms can be verified in a similar manner in both examples. Thus, by disregarding the concrete nature of the particular type of vectors, the definition incorporates these two and many more examples in one notion of vector space. Subtraction of two vectors and division by a (non-zero) scalar can be defined as : \begin \mathbf - \mathbf &= \mathbf + (-\mathbf) \\ \frac &= \frac\mathbf. \end When the scalar field is the
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s , the vector space is called a ''real vector space''. When the scalar field is the
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

complex number
s , the vector space is called a ''complex vector space''. These two cases are the ones used most often in engineering. The general definition of a vector space allows scalars to be elements of any fixed
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
. The notion is then known as an -''vector space'' or a ''vector space over ''. A field is, essentially, a set of numbers possessing
addition Addition (usually signified by the plus symbol The plus and minus signs, and , are mathematical symbol A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object A mathematical object is an ...

addition
,
subtraction Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...

subtraction
,
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

multiplication
and
division Division or divider may refer to: Mathematics *Division (mathematics), the inverse of multiplication *Division algorithm, a method for computing the result of mathematical division Military *Division (military), a formation typically consisting o ...
operations.Some authors (such as ) restrict attention to the fields or , but most of the theory is unchanged for an arbitrary field. For example,
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s form a field. In contrast to the intuition stemming from vectors in the plane and higher-dimensional cases, in general vector spaces, there is no notion of nearness,
angle In Euclidean geometry Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematics , Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's method ...

angle
s or
distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or eve ...

distance
s. To deal with such matters, particular types of vector spaces are introduced; see § Vector spaces with additional structure below for more.


Alternative formulations and elementary consequences

Vector addition and scalar multiplication are operations, satisfying the closure property: and are in for all in , and , in . Some older sources mention these properties as separate axioms. In the parlance of
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
, the first four axioms are equivalent to requiring the set of vectors to be an
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
under addition. The remaining axioms give this group an -
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modula ...
structure. In other words, there is a
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function (mathematics), function between two ring (algebra), rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function s ...
from the field into the
endomorphism ringIn abstract algebra, the endomorphisms of an abelian group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometr ...
of the group of vectors. Then scalar multiplication is defined as . There are a number of direct consequences of the vector space axioms. Some of them derive from elementary group theory, applied to the additive group of vectors: for example, the zero vector of and the additive inverse of any vector are unique. Further properties follow by employing also the distributive law for the scalar multiplication, for example equals if and only if equals or equals .


History

Vector spaces stem from
affine geometry In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
, via the introduction of
coordinate In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...

coordinate
s in the plane or three-dimensional space. Around 1636, French mathematicians
René Descartes René Descartes ( or ; ; Latinized Latinisation or Latinization can refer to: * Latinisation of names, the practice of rendering a non-Latin name in a Latin style * Latinisation in the Soviet Union, the campaign in the USSR during the 1920s ...

René Descartes
and
Pierre de Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of suc ...

Pierre de Fermat
founded
analytic geometry In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measur ...
by identifying solutions to an equation of two variables with points on a plane
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

curve
. To achieve geometric solutions without using coordinates,
Bolzano Bolzano ( or ; german: Bozen (formerly ), ; bar, Bozn; lld, Balsan or ) is the capital city A capital or capital city is the municipality holding primary status in a Department (country subdivision), department, country, Constituent state, ...
introduced, in 1804, certain operations on points, lines and planes, which are predecessors of vectors. introduced the notion of barycentric coordinates. introduced the notion of a bipoint, i.e., an oriented segment one of whose ends is the origin and the other one a target. Vectors were reconsidered with the presentation of
complex numbers In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
by Argand and
HamiltonHamilton may refer to: * Alexander Hamilton (1755–1804), first American Secretary of the Treasury and one of the Founding Fathers of the United States **Hamilton (musical), ''Hamilton'' (musical), a 2015 Broadway musical written by Lin-Manuel Mira ...
and the inception of
quaternion In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

quaternion
s by the latter. They are elements in R2 and R4; treating them using
linear combination In mathematics, a linear combination is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be ...
s goes back to
Laguerre Edmond Nicolas Laguerre (9 April 1834, Bar-le-Duc Bar-le-Duc (), formerly known as Bar, is a commune in the Meuse département, of which it is the capital. The department is in Grand Est in northeastern France. The lower, more modern and busi ...

Laguerre
in 1867, who also defined systems of linear equations. In 1857,
Cayley
Cayley
introduced the
matrix notation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
which allows for a harmonization and simplification of
linear map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

linear map
s. Around the same time,
Grassmann Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath, known in his day as a linguistics, linguist and now also as a mathematics, mathematician. He was also a physics, physicist, gener ...

Grassmann
studied the barycentric calculus initiated by Möbius. He envisaged sets of abstract objects endowed with operations. In his work, the concepts of
linear independence In the theory of vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are con ...
and
dimension In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

dimension
, as well as
scalar product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
s are present. Actually Grassmann's 1844 work exceeds the framework of vector spaces, since his considering multiplication, too, led him to what are today called
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
s. Italian mathematician
Peano Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

Peano
was the first to give the modern definition of vector spaces and linear maps in 1888. An important development of vector spaces is due to the construction of function spaces by
Henri Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (a ...
. This was later formalized by
Banach
Banach
and
Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in man ...
, around 1920. At that time,
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
and the new field of
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
began to interact, notably with key concepts such as spaces of ''p''-integrable functions and
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
s. Also at this time, the first studies concerning infinite-dimensional vector spaces were done.


Examples


Coordinate space

The simplest example of a vector space over a field is the field itself (as it is an
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
for addition, a part of requirements to be a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
.), equipped with its addition (It becomes vector addition.) and multiplication (It becomes scalar multiplication.). More generally, all -tuples (sequences of length ) : of elements of form a vector space that is usually denoted and called a coordinate space. The case is the above-mentioned simplest example, in which the field is also regarded as a vector space over itself. The case and (so R2) was discussed in the introduction above.


Complex numbers and other field extensions

The set of
complex numbers In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, that is, numbers that can be written in the form for
real numbers Real may refer to: * Reality, the state of things as they exist, rather than as they may appear or may be thought to be Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish col ...

real numbers
and where is the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad are ...
, form a vector space over the reals with the usual addition and multiplication: and for real numbers , , , and . The various axioms of a vector space follow from the fact that the same rules hold for complex number arithmetic. In fact, the example of complex numbers is essentially the same as (that is, it is ''isomorphic'' to) the vector space of ordered pairs of real numbers mentioned above: if we think of the complex number as representing the ordered pair in the
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
then we see that the rules for addition and scalar multiplication correspond exactly to those in the earlier example. More generally,
field extension In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s provide another class of examples of vector spaces, particularly in algebra and
algebraic number theory Algebraic number theory is a branch of number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is th ...
: a field containing a smaller field is an -vector space, by the given multiplication and addition operations of . For example, the complex numbers are a vector space over , and the field extension \mathbf(i\sqrt) is a vector space over .


Function spaces

Functions from any fixed set to a field also form vector spaces, by performing addition and scalar multiplication pointwise. That is, the sum of two functions and is the function given by :, and similarly for multiplication. Such function spaces occur in many geometric situations, when is the
real line In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
or an interval, or other
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

subset
s of . Many notions in topology and analysis, such as continuity,
integrability
integrability
or
differentiability In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. T ...
are well-behaved with respect to linearity: sums and scalar multiples of functions possessing such a property still have that property. Therefore, the set of such functions are vector spaces. They are studied in greater detail using the methods of
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
, see
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Fred Below (1926–1988), American blues drummer *Fritz von Below (1853 ...
. Algebraic constraints also yield vector spaces: the vector space is given by
polynomial function In mathematics, a polynomial is an expression (mathematics), expression consisting of variable (mathematics), variables (also called indeterminate (variable), indeterminates) and coefficients, that involves only the operations of addition, subtra ...
s: :, where the
coefficient In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s are in .


Linear equations

Systems of homogeneous linear equations are closely tied to vector spaces. For example, the solutions of : are given by triples with arbitrary , , and . They form a vector space: sums and scalar multiples of such triples still satisfy the same ratios of the three variables; thus they are solutions, too.
Matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...
can be used to condense multiple linear equations as above into one vector equation, namely :, where A = \begin 1 & 3 & 1 \\ 4 & 2 & 2\end is the matrix containing the coefficients of the given equations, is the vector , denotes the matrix product, and is the zero vector. In a similar vein, the solutions of homogeneous ''linear differential equations'' form vector spaces. For example, : yields , where and are arbitrary constants, and is the
natural exponential function
natural exponential function
.


Space of derived units

The following vector space, called the space of fundamental and derived physical units, is used in the
Buckingham π theorem In engineering, applied mathematics, and physics, the Buckingham theorem is a key theorem In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, stru ...
. Suppose that u_1, \ldots, u_n are fundamental units (for example,
gram The gram (alternative spelling: gramme; SI unit symbol: g) is a metric system The metric system is a that succeeded the decimalised system based on the introduced in France in the 1790s. The historical development of these systems culm ...
s g (
mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
),
meter The metre ( Commonwealth spelling) or meter (American spelling Despite the various English dialects spoken from country to country and within different regions of the same country, there are only slight regional variations in English ...

meter
s m (
length Length is a measure of distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be us ...

length
), and
second The second (symbol: s, also abbreviated: sec) is the base unit of time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, th ...
s s (
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
) are fundamental units since none can be defined in terms of the others). Let \mathbb be the rational \Q or real numbers \R, and let V denote the set of all possible derived units with exponents in \mathbb, which are units of the form u_1^ \cdots u_n^ where a_1, \ldots, a_n \in \mathbb (for example,
force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

force
has units g m/s^2 = g^1 m^1 s^ and
velocity The velocity of an object is the rate of change of its position with respect to a frame of reference In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical scie ...

velocity
has units m/s = m^1 s^ = g^0 m s^ where the zero exponent in g^0 indicates that grams g are not used and the omission of an exponent means that it is 1, for instance, m = m^1). Then V becomes a vector space over \mathbb when scalar multiplication of a vector u_1^ \cdots u_n^ \in V by a scalar c \in \mathbb is defined by c \cdot \left(u_1^ \cdots u_n^\right) ~=~ u_1^ \cdots u_n^ and when vector addition is defined by \left(u_1^ \cdots u_n^\right) \; + \; \left(u_1^ \cdots u_n^\right) ~=~ u_1^ \cdots u_n^. For example, the sum of the units of force g m/s^2 and the units of velocity m/s is g m s^ \;+\; m s^ ~=~ g^ m^ s^ ~=~ g m^2 s^, which are the units of force \times velocity = (gm/s^2) (m/s) = g m^2/s^3 = units of
power Power typically refers to: * Power (physics) In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal to one joule per second. In older works, p ...
. And the units of
acceleration In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approx ...

acceleration
m/s^2 = m s^ scalar multiplied by c = 1/2 is (1/2) \cdot \left(m s^\right) ~=~ m^ s^ ~=~ m^ s^. Consequently, units such as g m/s^2 can be expressed as the
linear combination In mathematics, a linear combination is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be ...
g m s^ = g + m -2 s and also as g m s^ = g m - 2 s = g m + s^ = g + m s^ = g s^ + m s^, and so on. The fundamental units u_1, \ldots, u_n form a
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other b ...
for V, which means that V is an n-dimensional vector space over \mathbb. The map V \to \mathbb^n defined by u_1^ \cdots u_n^ \mapsto \left(a_1, \ldots, a_n\right) is a vector space isomorphism. To only allow units with integer exponents (in order to disallow units such as m^), the field \mathbb can be replaced with the ring of integers \Z, although doing this will cause V to no longer be a vector space; it will, however, be a generalization of a vector space known as a (left) \Z-module. If g, m, s \in V then the three units/vectors g/m = g - m, \quad s/m = s - m, \quad \text \quad g^2/s^2 = 2g - 2s are linearly dependent because 2g - 2s = 2\left(g - m\right) - 2\left(s - m\right) (in the usual language of unit analysis, this is because g^2/s^2 = (g/m)^2 (s/m)^); in contrast, g m = g + m, \; s/m, \, \text \, g^2/s^2 are
linearly independent In the theory of vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change ...
and they even form a basis for the 3-dimensional vector subspace spanned by g, m, \text s. One consequence of this, stated in
plain English Plain English (or layman's terms) is language that is considered to be clear and concise. It may often attempt to avoid the use of uncommon vocabulary and lesser-known euphemisms in order to explain the subject matter. The wording is intended to be ...
, is that any unit made up of g, m, \text s (such as g or acceleration m/s^2) can also be expressed in terms of g m, s/m, \text g^2/s^2; however, some units (such as g for example) can be expressed in terms of g/m, s/m, \text g^2/s^2 (because their 2-dimensional span does not contain g).


Basis and dimension

''Bases'' allow one to represent vectors by a
sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

sequence
of scalars called ''coordinates'' or ''components''. A basis is a set of vectors , for convenience often indexed by some
index set In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a Set (mathematics), set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. Th ...
, that spans the whole space and is
linearly independent In the theory of vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change ...
. "Spanning the whole space" means that any vector can be expressed as a finite sum (called a ''
linear combination In mathematics, a linear combination is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be ...
'') of the basis elements: where the are scalars, called the coordinates (or the components) of the vector with respect to the basis , and elements of . Linear independence means that the coordinates are uniquely determined for any vector in the vector space. For example, the
coordinate vector In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers that describes the vector in terms of a particular ordered basis. Coordinates are always specified relative to an ordered basis. Bases and their a ...
s , , to , form a basis of , called the
standard basis In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
, since any vector can be uniquely expressed as a linear combination of these vectors: :. The corresponding coordinates , , , are just the
Cartesian coordinates A Cartesian coordinate system (, ) in a plane Plane or planes may refer to: * Airplane or aeroplane or informally plane, a powered, fixed-wing aircraft Arts, entertainment and media *Plane (Dungeons & Dragons), Plane (''Dungeons & Dragons'') ...

Cartesian coordinates
of the vector. Every vector space has a basis. This follows from
Zorn's lemma Zorn's lemma, also known as the Kuratowski–Zorn lemma, after mathematicians Max August Zorn, Max Zorn and Kazimierz Kuratowski, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (ord ...
, an equivalent formulation of the
Axiom of Choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...

Axiom of Choice
. Given the other axioms of
Zermelo–Fraenkel set theory In set theory illustrating the intersection of two sets Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set ...
, the existence of bases is equivalent to the axiom of choice. The ultrafilter lemma, which is weaker than the axiom of choice, implies that all bases of a given vector space have the same number of elements, or
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(cf. ''
Dimension theorem for vector spaces In mathematics, the dimension theorem for vector spaces states that all Basis (linear algebra), bases of a vector space have equally many elements. This number of elements may be finite or infinite (in the latter case, it is a cardinal number), and ...
''). It is called the ''dimension'' of the vector space, denoted by dim ''V''. If the space is spanned by finitely many vectors, the above statements can be proven without such fundamental input from set theory. The dimension of the coordinate space is , by the basis exhibited above. The dimension of the polynomial ring ''F'' 'x''introduced above is countably infinite, a basis is given by , , , A fortiori, the dimension of more general function spaces, such as the space of functions on some (bounded or unbounded) interval, is infinite.The indicator functions of intervals (of which there are infinitely many) are linearly independent, for example. Under suitable regularity assumptions on the coefficients involved, the dimension of the solution space of a homogeneous ordinary differential equation equals the degree of the equation. For example, the solution space for the #equation1, above equation is generated by . These two functions are linearly independent over , so the dimension of this space is two, as is the degree of the equation. A field extension over the rationals can be thought of as a vector space over (by defining vector addition as field addition, defining scalar multiplication as field multiplication by elements of , and otherwise ignoring the field multiplication). The dimension (or degree of a field extension, degree) of the field extension over depends on . If satisfies some polynomial equation q_n \alpha^n + q_ \alpha^ + \cdots + q_0 = 0 with rational coefficients (in other words, if ''α'' is algebraic number, algebraic), the dimension is finite. More precisely, it equals the degree of the minimal polynomial (field theory), minimal polynomial having α as a root of a function, root. For example, the complex numbers C are a two-dimensional real vector space, generated by 1 and the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad are ...
''i''. The latter satisfies ''i''2 + 1 = 0, an equation of degree two. Thus, C is a two-dimensional R-vector space (and, as any field, one-dimensional as a vector space over itself, C). If ''α'' is not algebraic, the dimension of Q(''α'') over Q is infinite. For example, for ''α'' = pi, π there is no such equation. That is, π is transcendental number, transcendental.


Linear maps and matrices

The relation of two vector spaces can be expressed by ''linear map'' or ''linear transformation''. They are
functions Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
that reflect the vector space structure, that is, they preserve sums and scalar multiplication: :f(\mathbf v + \mathbf w) = f(\mathbf v) + f(\mathbf w) and for all and in , all in . An ''isomorphism'' is a linear map such that there exists an inverse map , which is a map such that the two possible function composition, compositions and are Identity function, identity maps. Equivalently, is both one-to-one (injective) and onto (surjective). If there exists an isomorphism between and , the two spaces are said to be ''isomorphic''; they are then essentially identical as vector spaces, since all identities holding in are, via , transported to similar ones in , and vice versa via . For example, the "arrows in the plane" and "ordered pairs of numbers" vector spaces in the introduction are isomorphic: a planar arrow departing at the origin (mathematics), origin of some (fixed) coordinate system can be expressed as an ordered pair by considering the - and -component of the arrow, as shown in the image at the right. Conversely, given a pair , the arrow going by to the right (or to the left, if is negative), and up (down, if is negative) turns back the arrow . Linear maps between two vector spaces form a vector space , also denoted , or . The space of linear maps from to is called the ''dual vector space'', denoted . Via the injective natural (category theory), natural map , any vector space can be embedded into its ''bidual''; the map is an isomorphism if and only if the space is finite-dimensional. Once a basis of is chosen, linear maps are completely determined by specifying the images of the basis vectors, because any element of is expressed uniquely as a linear combination of them. If , a bijection, 1-to-1 correspondence between fixed bases of and gives rise to a linear map that maps any basis element of to the corresponding basis element of . It is an isomorphism, by its very definition. Therefore, two vector spaces are isomorphic if their dimensions agree and vice versa. Another way to express this is that any vector space is ''completely classified'' (up to isomorphism) by its dimension, a single number. In particular, any ''n''-dimensional -vector space is isomorphic to . There is, however, no "canonical" or preferred isomorphism; actually an isomorphism is equivalent to the choice of a basis of , by mapping the standard basis of to , via . The freedom of choosing a convenient basis is particularly useful in the infinite-dimensional context; see #basis in inf dim context, below.


Matrices

''Matrices'' are a useful notion to encode linear maps. They are written as a rectangular array of scalars as in the image at the right. Any -by- matrix gives rise to a linear map from to , by the following :\mathbf x = (x_1, x_2, \ldots, x_n) \mapsto \left(\sum_^n a_x_j, \sum_^n a_x_j, \ldots, \sum_^n a_x_j \right), where \sum denotes summation, or, using the matrix multiplication of the matrix with the coordinate vector : :. Moreover, after choosing bases of and , ''any'' linear map is uniquely represented by a matrix via this assignment. The determinant of a square matrix is a scalar that tells whether the associated map is an isomorphism or not: to be so it is sufficient and necessary that the determinant is nonzero. The linear transformation of corresponding to a real ''n''-by-''n'' matrix is Orientation (vector space), orientation preserving if and only if its determinant is positive.


Eigenvalues and eigenvectors

Endomorphisms, linear maps , are particularly important since in this case vectors can be compared with their image under , . Any nonzero vector satisfying , where is a scalar, is called an ''eigenvector'' of with ''eigenvalue'' .The nomenclature derives from German language, German "wikt:eigen, eigen", which means own or proper. Equivalently, is an element of the Kernel (linear algebra), kernel of the difference (where Id is the identity function, identity map . If is finite-dimensional, this can be rephrased using determinants: having eigenvalue is equivalent to :. By spelling out the definition of the determinant, the expression on the left hand side can be seen to be a polynomial function in , called the characteristic polynomial of . If the field is large enough to contain a zero of this polynomial (which automatically happens for algebraically closed field, algebraically closed, such as ) any linear map has at least one eigenvector. The vector space may or may not possess an eigenbasis, a basis consisting of eigenvectors. This phenomenon is governed by the Jordan canonical form of the map.See also Jordan–Chevalley decomposition. The set of all eigenvectors corresponding to a particular eigenvalue of forms a vector space known as the ''eigenspace'' corresponding to the eigenvalue (and ) in question. To achieve the spectral theorem, the corresponding statement in the infinite-dimensional case, the machinery of functional analysis is needed, see #labelSpectralTheorem, below.


Basic constructions

In addition to the above concrete examples, there are a number of standard linear algebraic constructions that yield vector spaces related to given ones. In addition to the definitions given below, they are also characterized by universal property, universal properties, which determine an object by specifying the linear maps from to any other vector space.


Subspaces and quotient spaces

A nonempty
subset In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

subset
''W'' of a vector space ''V'' that is closed under addition and scalar multiplication (and therefore contains the 0-vector of ''V'') is called a ''linear subspace'' of ''V'', or simply a ''subspace'' of ''V'', when the ambient space is unambiguously a vector space.This is typically the case when a vector space is also considered as an affine space. In this case, a linear subspace contains the zero vector, while an affine subspace does not necessarily contain it. Subspaces of ''V'' are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set ''S'' of vectors is called its linear span, span, and it is the smallest subspace of ''V'' containing the set ''S''. Expressed in terms of elements, the span is the subspace consisting of all the
linear combination In mathematics, a linear combination is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be ...
s of elements of ''S''. A linear subspace of dimension 1 is a vector line. A linear subspace of dimension 2 is a vector plane. A linear subspace that contains all elements but one of a basis of the ambient space is a vector hyperplane. In a vector space of finite dimension , a vector hyperplane is thus a subspace of dimension . The counterpart to subspaces are ''quotient vector spaces''. Given any subspace , the quotient space ''V''/''W'' ("''V'' modular arithmetic, modulo ''W''") is defined as follows: as a set, it consists of where v is an arbitrary vector in ''V''. The sum of two such elements and is and scalar multiplication is given by . The key point in this definition is that if and only if the difference of v1 and v2 lies in ''W''.Some authors (such as ) choose to start with this equivalence relation and derive the concrete shape of ''V''/''W'' from this. This way, the quotient space "forgets" information that is contained in the subspace ''W''. The kernel (algebra), kernel ker(''f'') of a linear map consists of vectors v that are mapped to 0 in ''W''. The kernel and the image (mathematics), image are subspaces of ''V'' and ''W'', respectively. The existence of kernels and images is part of the statement that the category of vector spaces (over a fixed field ''F'') is an abelian category, that is, a corpus of mathematical objects and structure-preserving maps between them (a category (mathematics), category) that behaves much like the category of abelian groups. Because of this, many statements such as the first isomorphism theorem (also called rank–nullity theorem in matrix-related terms) :''V'' / ker(''f'') ≡ im(''f''). and the second and third isomorphism theorem can be formulated and proven in a way very similar to the corresponding statements for group (mathematics), groups. An important example is the kernel of a linear map for some fixed matrix ''A'', as #equation2, above. The kernel of this map is the subspace of vectors x such that , which is precisely the set of solutions to the system of homogeneous linear equations belonging to ''A''. This concept also extends to linear differential equations :a_0 f + a_1 \frac + a_2 \frac + \cdots + a_n \frac = 0, where the coefficients ''a''''i'' are functions in ''x'', too. In the corresponding map :f \mapsto D(f) = \sum_^n a_i \frac, the derivatives of the function ''f'' appear linearly (as opposed to ''f''′′(''x'')2, for example). Since differentiation is a linear procedure (that is, and for a constant ) this assignment is linear, called a linear differential operator. In particular, the solutions to the differential equation form a vector space (over or ).


Direct product and direct sum

The ''direct product'' of vector spaces and the ''direct sum'' of vector spaces are two ways of combining an indexed family of vector spaces into a new vector space. The ''direct product'' \textstyle of a family of vector spaces ''V''''i'' consists of the set of all tuples (, which specify for each index ''i'' in some
index set In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a Set (mathematics), set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. Th ...
''I'' an element v''i'' of ''V''''i''. Addition and scalar multiplication is performed componentwise. A variant of this construction is the ''direct sum'' \bigoplus_ V_i (also called coproduct and denoted \coprod_V_i), where only tuples with finitely many nonzero vectors are allowed. If the index set ''I'' is finite, the two constructions agree, but in general they are different.


Tensor product

The ''tensor product'' , or simply , of two vector spaces ''V'' and ''W'' is one of the central notions of multilinear algebra which deals with extending notions such as linear maps to several variables. A map is called bilinear map, bilinear if ''g'' is linear in both variables v and w. That is to say, for fixed w the map is linear in the sense above and likewise for fixed v. The tensor product is a particular vector space that is a ''universal'' recipient of bilinear maps ''g'', as follows. It is defined as the vector space consisting of finite (formal) sums of symbols called
tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

tensor
s :v1 ⊗ w1 + v2 ⊗ w2 + ⋯ + v''n'' ⊗ w''n'', subject to the rules : ''a'' · (v ⊗ w) = (''a'' · v) ⊗ w = v ⊗ (''a'' · w), where ''a'' is a scalar, :(v1 + v2) ⊗ w = v1 ⊗ w + v2 ⊗ w, and :v ⊗ (w1 + w2) = v ⊗ w1 + v ⊗ w2. These rules ensure that the map ''f'' from the to that maps a tuple to is bilinear. The universality states that given ''any'' vector space ''X'' and ''any'' bilinear map , there exists a unique map ''u'', shown in the diagram with a dotted arrow, whose function composition, composition with ''f'' equals ''g'': . This is called the universal property of the tensor product, an instance of the method—much used in advanced abstract algebra—to indirectly define objects by specifying maps from or to this object.


Vector spaces with additional structure

From the point of view of linear algebra, vector spaces are completely understood insofar as any vector space is characterized, up to isomorphism, by its dimension. However, vector spaces ''per se'' do not offer a framework to deal with the question—crucial to analysis—whether a sequence of functions Limit of a sequence, converges to another function. Likewise, linear algebra is not adapted to deal with infinite series, since the addition operation allows only finitely many terms to be added. Therefore, the needs of
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
require considering additional structures.
A vector space may be given a partial order ≤, under which some vectors can be compared. For example, ''n''-dimensional real space R''n'' can be ordered by comparing its vectors componentwise. Ordered vector spaces, for example Riesz spaces, are fundamental to Lebesgue integration, which relies on the ability to express a function as a difference of two positive functions :f = f^ - f^, where f^ denotes the positive part of f and f^ the negative part.


Normed vector spaces and inner product spaces

"Measuring" vectors is done by specifying a
norm Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy) Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...
, a datum which measures lengths of vectors, or by an
inner product In mathematics, an inner product space or a Hausdorff space, Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a Scalar (mathematics), ...
, which measures angles between vectors. Norms and inner products are denoted , \mathbf v, and respectively. The datum of an inner product entails that lengths of vectors can be defined too, by defining the associated norm Vector spaces endowed with such data are known as ''normed vector spaces'' and ''inner product spaces'', respectively. Coordinate space ''F''''n'' can be equipped with the standard dot product: :\lang \mathbf x , \mathbf y \rang = \mathbf x \cdot \mathbf y = x_1 y_1 + \cdots + x_n y_n. In R2, this reflects the common notion of the angle between two vectors x and y, by the law of cosines: :\mathbf x \cdot \mathbf y = \cos\left(\angle (\mathbf x, \mathbf y)\right) \cdot , \mathbf x, \cdot , \mathbf y, . Because of this, two vectors satisfying \lang \mathbf x , \mathbf y \rang = 0 are called orthogonal. An important variant of the standard dot product is used in Minkowski space: R4 endowed with the Lorentz product :\lang \mathbf x , \mathbf y \rang = x_1 y_1 + x_2 y_2 + x_3 y_3 - x_4 y_4. In contrast to the standard dot product, it is not positive definite bilinear form, positive definite: \lang \mathbf x , \mathbf x \rang also takes negative values, for example, for \mathbf x = (0, 0, 0, 1). Singling out the fourth coordinate—timelike, corresponding to time, as opposed to three space-dimensions—makes it useful for the mathematical treatment of special relativity.


Topological vector spaces

Convergence questions are treated by considering vector spaces ''V'' carrying a compatible topological space, topology, a structure that allows one to talk about elements being neighborhood (topology), close to each other. Compatible here means that addition and scalar multiplication have to be continuous maps. Roughly, if x and y in ''V'', and ''a'' in ''F'' vary by a bounded amount, then so do and .This requirement implies that the topology gives rise to a uniform structure, To make sense of specifying the amount a scalar changes, the field ''F'' also has to carry a topology in this context; a common choice are the reals or the complex numbers. In such ''topological vector spaces'' one can consider series (mathematics), series of vectors. The infinite sum :\sum_^ f_i denotes the limit of a sequence, limit of the corresponding finite partial sums of the sequence (''f''''i'')''i''∈N of elements of ''V''. For example, the ''f''''i'' could be (real or complex) functions belonging to some
function space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
''V'', in which case the series is a function series. The modes of convergence, mode of convergence of the series depends on the topology imposed on the function space. In such cases, pointwise convergence and uniform convergence are two prominent examples. A way to ensure the existence of limits of certain infinite series is to restrict attention to spaces where any Cauchy sequence has a limit; such a vector space is called Completeness (topology), complete. Roughly, a vector space is complete provided that it contains all necessary limits. For example, the vector space of polynomials on the unit interval [0,1], equipped with the topology of uniform convergence is not complete because any continuous function on [0,1] can be uniformly approximated by a sequence of polynomials, by the Weierstrass approximation theorem. In contrast, the space of ''all'' continuous functions on [0,1] with the same topology is complete. A norm gives rise to a topology by defining that a sequence of vectors v''n'' converges to v if and only if :\lim_ , \mathbf v_n - \mathbf v, = 0. Banach and Hilbert spaces are complete topological vector spaces whose topologies are given, respectively, by a norm and an inner product. Their study—a key piece of
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
—focuses on infinite-dimensional vector spaces, since all norms on finite-dimensional topological vector spaces give rise to the same notion of convergence. The image at the right shows the equivalence of the 1-norm and ∞-norm on R2: as the unit "balls" enclose each other, a sequence converges to zero in one norm if and only if it so does in the other norm. In the infinite-dimensional case, however, there will generally be inequivalent topologies, which makes the study of topological vector spaces richer than that of vector spaces without additional data. From a conceptual point of view, all notions related to topological vector spaces should match the topology. For example, instead of considering all linear maps (also called functional (mathematics), functionals) , maps between topological vector spaces are required to be continuous. In particular, the (topological) dual space consists of continuous functionals (or to ). The fundamental Hahn–Banach theorem is concerned with separating subspaces of appropriate topological vector spaces by continuous functionals.


Banach spaces

''Banach spaces'', introduced by Stefan Banach, are complete normed vector spaces. A first example is Lp space, the vector space \ell^p consisting of infinite vectors with real entries \mathbf = \left(x_1, x_2, \ldots, x_n, \ldots\right) whose p-norm, p-norm (1\leq\leq\infty) given by :\left\, \mathbf\right\, _p := \left(\sum_i \left\vert x_i\right\vert^p\right)^\frac for p < \infty   and   \left\, \mathbf x\right\, _ := \sup_i \left, x_i \. The topologies on the infinite-dimensional space \ell^p are inequivalent for different p. For example, the sequence of vectors \mathbf_n = \left(2^, 2^, \ldots, 2^, 0, 0, \ldots\right), in which the first 2^n components are 2^ and the following ones are 0, converges to the zero vector for p = \infty, but does not for p = 1: : \left\Vert\mathbf_\right\Vert_ = \sup (2^, 0) = 2^ \rightarrow 0 , but \left\Vert\mathbf_\right\Vert_ = \sum_^ 2^ = 2^n \cdot 2^ = 1. More generally than sequences of real numbers, functions f\colon \Omega \to \mathbb are endowed with a norm that replaces the above sum by the Lebesgue integral : \left\Vert\right\Vert_ := \left( \int_ \left\vert\left(x\right)\right\vert^ \, \right)^\frac. The space of integrable functions on a given domain of a function, domain \Omega (for example an interval) satisfying \left\Vert\right\Vert_ < \infty, and equipped with this norm are called Lp space, Lebesgue spaces, denoted L^\left(\Omega\right).The triangle inequality for \left\Vert\right\Vert_ \leq \left\Vert\right\Vert_ + \left\Vert\right\Vert_ is provided by the Minkowski inequality. For technical reasons, in the context of functions one has to identify functions that agree almost everywhere to get a norm, and not only a seminorm. These spaces are complete. (If one uses the Riemann integral instead, the space is ''not'' complete, which may be seen as a justification for Lebesgue's integration theory. "Many functions in L^ of Lebesgue measure, being unbounded, cannot be integrated with the classical Riemann integral. So spaces of Riemann integrable functions would not be complete in the L^ norm, and the orthogonal decomposition would not apply to them. This shows one of the advantages of Lebesgue integration.", ) Concretely this means that for any sequence of Lebesgue-integrable functions   f_, f_, \ldots, f_, \ldots   with \left\Vert_\right\Vert_<\infty, satisfying the condition :\lim_\int_ \left\vert_(x) - _(x)\right\vert^ \, = 0 there exists a function \left(x\right) belonging to the vector space L^\left(\Omega\right) such that :\lim_\int_ \left\vert\left(x\right) - _\left(x\right)\right\vert^ \, = 0. Imposing boundedness conditions not only on the function, but also on its derivatives leads to Sobolev spaces.


Hilbert spaces

Complete inner product spaces are known as ''Hilbert spaces'', in honor of David Hilbert. The Hilbert space ''L''2(Ω), with inner product given by : \langle f\ , \ g \rangle = \int_\Omega f(x) \overline \, dx, where \overline denotes the complex conjugate of ''g''(''x''),For ''p'' ≠2, ''L''''p''(Ω) is not a Hilbert space. is a key case. By definition, in a Hilbert space any Cauchy sequence converges to a limit. Conversely, finding a sequence of functions ''f''''n'' with desirable properties that approximates a given limit function, is equally crucial. Early analysis, in the guise of the Taylor approximation, established an approximation of differentiable functions ''f'' by polynomials. By the Stone–Weierstrass theorem, every continuous function on can be approximated as closely as desired by a polynomial. A similar approximation technique by trigonometric functions is commonly called Fourier expansion, and is much applied in engineering, see #labelFourier, below. More generally, and more conceptually, the theorem yields a simple description of what "basic functions", or, in abstract Hilbert spaces, what basic vectors suffice to generate a Hilbert space ''H'', in the sense that the ''closure (topology), closure'' of their span (that is, finite linear combinations and limits of those) is the whole space. Such a set of functions is called a ''basis'' of ''H'', its cardinality is known as the Hilbert space dimension.A basis of a Hilbert space is not the same thing as a basis in the sense of linear algebra #label1, above. For distinction, the latter is then called a Hamel basis. Not only does the theorem exhibit suitable basis functions as sufficient for approximation purposes, but also together with the Gram–Schmidt process, it enables one to construct a orthogonal basis, basis of orthogonal vectors. Such orthogonal bases are the Hilbert space generalization of the coordinate axes in finite-dimensional
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
. The solutions to various differential equations can be interpreted in terms of Hilbert spaces. For example, a great many fields in physics and engineering lead to such equations and frequently solutions with particular physical properties are used as basis functions, often orthogonal. As an example from physics, the time-dependent Schrödinger equation in quantum mechanics describes the change of physical properties in time by means of a
partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, whose solutions are called wavefunctions. Definite values for physical properties such as energy, or momentum, correspond to eigenvalues of a certain (linear) differential operator and the associated wavefunctions are called eigenstates. The spectral theorem decomposes a linear compact operator acting on functions in terms of these eigenfunctions and their eigenvalues.


Algebras over fields

General vector spaces do not possess a multiplication between vectors. A vector space equipped with an additional bilinear operator defining the multiplication of two vectors is an ''algebra over a field''. Many algebras stem from functions on some geometrical object: since functions with values in a given field can be multiplied pointwise, these entities form algebras. The Stone–Weierstrass theorem, for example, relies on Banach algebras which are both Banach spaces and algebras. Commutative algebra makes great use of polynomial ring, rings of polynomials in one or several variables, introduced above. Their multiplication is both commutative and associative. These rings and their quotient ring, quotients form the basis of algebraic geometry, because they are coordinate ring, rings of functions of algebraic geometric objects. Another crucial example are ''Lie algebras'', which are neither commutative nor associative, but the failure to be so is limited by the constraints ( denotes the product of and ): * (anticommutativity), and * (Jacobi identity). Examples include the vector space of ''n''-by-''n'' matrices, with , the commutator of two matrices, and , endowed with the cross product. The tensor algebra T(''V'') is a formal way of adding products to any vector space ''V'' to obtain an algebra. As a vector space, it is spanned by symbols, called simple
tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...

tensor
s :, where the rank of a tensor, degree varies. The multiplication is given by concatenating such symbols, imposing the distributive law under addition, and requiring that scalar multiplication commute with the tensor product ⊗, much the same way as with the tensor product of two vector spaces introduced #Tensor product, above. In general, there are no relations between and . Forcing two such elements to be equal leads to the symmetric algebra, whereas forcing yields the exterior algebra. When a field, is explicitly stated, a common term used is -algebra.


Applications

Vector spaces have many applications as they occur frequently in common circumstances, namely wherever functions with values in some field are involved. They provide a framework to deal with analytical and geometrical problems, or are used in the Fourier transform. This list is not exhaustive: many more applications exist, for example in optimization (mathematics), optimization. The minimax theorem of game theory stating the existence of a unique payoff when all players play optimally can be formulated and proven using vector spaces methods. Representation theory fruitfully transfers the good understanding of linear algebra and vector spaces to other mathematical domains such as group theory.


Distributions

A ''distribution'' (or ''generalized function'') is a linear map assigning a number to each test function, "test" function, typically a smooth function with compact support, in a continuous way: in the #label2, above terminology the space of distributions is the (continuous) dual of the test function space. The latter space is endowed with a topology that takes into account not only ''f'' itself, but also all its higher derivatives. A standard example is the result of integrating a test function ''f'' over some domain Ω: :I(f) = \int_\Omega f(x)\,dx. When , the set consisting of a single point, this reduces to the Dirac distribution, denoted by ''δ'', which associates to a test function ''f'' its value at the . Distributions are a powerful instrument to solve differential equations. Since all standard analytic notions such as derivatives are linear, they extend naturally to the space of distributions. Therefore, the equation in question can be transferred to a distribution space, which is bigger than the underlying function space, so that more flexible methods are available for solving the equation. For example, Green's functions and fundamental solutions are usually distributions rather than proper functions, and can then be used to find solutions of the equation with prescribed boundary conditions. The found solution can then in some cases be proven to be actually a true function, and a solution to the original equation (for example, using the Lax–Milgram theorem, a consequence of the Riesz representation theorem).


Fourier analysis

Resolving a periodic function into a sum of trigonometric functions forms a ''Fourier series'', a technique much used in physics and engineering.Although the Fourier series is periodic, the technique can be applied to any ''L''2 function on an interval by considering the function to be continued periodically outside the interval. See The underlying vector space is usually the
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
''L''2(0, 2π), for which the functions sin(''mx'') and cos(''mx'') (where ''m'' is an integer) form an orthogonal basis. The Fourier expansion of an ''L''2 function ''f'' is : \frac + \sum_^\left[a_m\cos\left(mx\right)+b_m\sin\left(mx\right)\right]. The coefficients ''a''''m'' and ''b''''m'' are called Fourier coefficients of ''f'', and are calculated by the formulas :a_m = \frac \int_0^ f(t) \cos (mt) \, dt, b_m = \frac \int_0^ f(t) \sin (mt) \, dt. In physical terms the function is represented as a Superposition principle, superposition of sine waves and the coefficients give information about the function's frequency spectrum. A complex-number form of Fourier series is also commonly used. The concrete formulae above are consequences of a more general duality (mathematics), mathematical duality called Pontryagin duality. Applied to the group (mathematics), group R, it yields the classical Fourier transform; an application in physics are reciprocal lattices, where the underlying group is a finite-dimensional real vector space endowed with the additional datum of a lattice (group), lattice encoding positions of atoms in crystals. Fourier series are used to solve boundary value problems in
partial differential equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s. In 1822, Joseph Fourier, Fourier first used this technique to solve the heat equation. A discrete version of the Fourier series can be used in Sampling (signal processing), sampling applications where the function value is known only at a finite number of equally spaced points. In this case the Fourier series is finite and its value is equal to the sampled values at all points. The set of coefficients is known as the discrete Fourier transform (DFT) of the given sample sequence. The DFT is one of the key tools of digital signal processing, a field whose applications include radar, speech encoding,
image compression Image compression is a type of data compression In signal processing Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and synthesizing signals such as audio signal processing, sound, image proc ...
. The JPEG image format is an application of the closely related discrete cosine transform. The fast Fourier transform is an algorithm for rapidly computing the discrete Fourier transform. It is used not only for calculating the Fourier coefficients but, using the convolution theorem, also for computing the convolution of two finite sequences. They in turn are applied in digital filters and as a rapid multiplication algorithm for polynomials and large integers (Schönhage–Strassen algorithm).


Differential geometry

The tangent plane to a surface at a point is naturally a vector space whose origin is identified with the point of contact. The tangent plane is the best linear approximation, or linearization, of a surface at a point.That is to say , the plane passing through the point of contact ''P'' such that the distance from a point ''P''1 on the surface to the plane is little o, infinitesimally small compared to the distance from ''P''1 to ''P'' in the limit as ''P''1 approaches ''P'' along the surface. Even in a three-dimensional Euclidean space, there is typically no natural way to prescribe a basis of the tangent plane, and so it is conceived of as an abstract vector space rather than a real coordinate space. The ''tangent space'' is the generalization to higher-dimensional differentiable manifolds. Riemannian manifolds are manifolds whose tangent spaces are endowed with a Riemannian metric, suitable inner product. Derived therefrom, the Riemann curvature tensor encodes all curvature (mathematics), curvatures of a manifold in one object, which finds applications in general relativity, for example, where the Einstein curvature tensor describes the matter and energy content of space-time. The tangent space of a Lie group can be given naturally the structure of a Lie algebra and can be used to classify compact Lie groups.


Generalizations


Vector bundles

A ''vector bundle'' is a family of vector spaces parametrized continuously by a topological space ''X''. More precisely, a vector bundle over ''X'' is a topological space ''E'' equipped with a continuous map :π : ''E'' → ''X'' such that for every ''x'' in ''X'', the fiber (mathematics), fiber π−1(''x'') is a vector space. The case dim is called a line bundle. For any vector space ''V'', the projection makes the product into a trivial bundle, "trivial" vector bundle. Vector bundles over ''X'' are required to be locally a product of ''X'' and some (fixed) vector space ''V'': for every ''x'' in ''X'', there is a neighborhood (topology), neighborhood ''U'' of ''x'' such that the restriction of π to π−1(''U'') is isomorphicThat is, there is a homeomorphism from π−1(''U'') to which restricts to linear isomorphisms between fibers. to the trivial bundle . Despite their locally trivial character, vector bundles may (depending on the shape of the underlying space ''X'') be "twisted" in the large (that is, the bundle need not be (globally isomorphic to) the trivial bundle ). For example, the Möbius strip can be seen as a line bundle over the circle ''S''1 (by homeomorphism#Examples, identifying open intervals with the real line). It is, however, different from the cylinder (geometry), cylinder , because the latter is orientable manifold, orientable whereas the former is not. Properties of certain vector bundles provide information about the underlying topological space. For example, the tangent bundle consists of the collection of tangent spaces parametrized by the points of a differentiable manifold. The tangent bundle of the circle ''S''1 is globally isomorphic to , since there is a global nonzero vector field on ''S''1.A line bundle, such as the tangent bundle of ''S''1 is trivial if and only if there is a section (fiber bundle), section that vanishes nowhere, see . The sections of the tangent bundle are just vector fields. In contrast, by the hairy ball theorem, there is no (tangent) vector field on the 2-sphere ''S''2 which is everywhere nonzero. K-theory studies the isomorphism classes of all vector bundles over some topological space. In addition to deepening topological and geometrical insight, it has purely algebraic consequences, such as the classification of finite-dimensional real division algebras: R, C, the
quaternion In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

quaternion
s H and the octonions O. The cotangent bundle of a differentiable manifold consists, at every point of the manifold, of the dual of the tangent space, the cotangent space. Section (fiber bundle), Sections of that bundle are known as differential form, differential one-forms.


Modules

''Modules'' are to ring (mathematics), rings what vector spaces are to fields: the same axioms, applied to a ring ''R'' instead of a field ''F'', yield modules. The theory of modules, compared to that of vector spaces, is complicated by the presence of ring elements that do not have multiplicative inverses. For example, modules need not have bases, as the Z-module (that is,
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
) Modular arithmetic, Z/2Z shows; those modules that do (including all vector spaces) are known as free modules. Nevertheless, a vector space can be compactly defined as a
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Modula ...
over a ring which is a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
, with the elements being called vectors. Some authors use the term ''vector space'' to mean modules over a division ring. The algebro-geometric interpretation of commutative rings via their spectrum of a ring, spectrum allows the development of concepts such as locally free modules, the algebraic counterpart to vector bundles.


Affine and projective spaces

Roughly, ''affine spaces'' are vector spaces whose origins are not specified. More precisely, an affine space is a set with a transitive group action, free transitive vector space Group action (mathematics), action. In particular, a vector space is an affine space over itself, by the map :. If ''W'' is a vector space, then an affine subspace is a subset of ''W'' obtained by translating a linear subspace ''V'' by a fixed vector ; this space is denoted by (it is a coset of ''V'' in ''W'') and consists of all vectors of the form for An important example is the space of solutions of a system of inhomogeneous linear equations : generalizing the homogeneous case #equation3, above, which can be found by setting in this equation. The space of solutions is the affine subspace where x is a particular solution of the equation, and ''V'' is the space of solutions of the homogeneous equation (the nullspace of ''A''). The set of one-dimensional subspaces of a fixed finite-dimensional vector space ''V'' is known as ''projective space''; it may be used to formalize the idea of parallel (geometry), parallel lines intersecting at infinity. Grassmannian manifold, Grassmannians and flag manifolds generalize this by parametrizing linear subspaces of fixed dimension ''k'' and flag (linear algebra), flags of subspaces, respectively.


See also

*Vector (mathematics and physics), for a list of various kinds of vectors *Cartesian coordinate system *Graded vector space *Metric space *P-vector *Riesz–Fischer theorem *Space (mathematics) *Ordered vector space


Notes


Citations


References


Algebra

* * * * * * * * * *


Analysis

* * * * * * * * * * * * * * * * * * * * *


Historical references

* * * . * * * * , reprint: * * * * * Peano, G. (1901) Formulario mathematico
vct axioms
via Internet Archive


Further references

* * * * * * * * * * * * * * * * * * * * * * * * * *


External links

* {{DEFAULTSORT:Vector Space Concepts in physics Group theory Mathematical structures Vectors (mathematics and physics) Vector spaces,