physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, a vector space (also called a linear space) is a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
whose elements, often called ''
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
s'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, but can be
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s or, more generally, elements of any
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 grass ...
. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or
complex coordinate space
In mathematics, the ''n''-dimensional complex coordinate space (or complex ''n''-space) is the set of all ordered ''n''-tuples of complex numbers. It is denoted \Complex^n, and is the ''n''-fold Cartesian product of the complex plane \Complex wi ...
.
Vector spaces generalize
Euclidean vector
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ...
s, which allow modeling of
physical quantities
A physical quantity is a physical property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For exam ...
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for
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 matrices ...
, together with the concept of
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying
systems of linear equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables.
For example,
:\begin
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\fracy-z=0
\end
is a system of three equations in th ...
.
Vector spaces are characterized by their
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
, which, roughly speaking, specifies the number of independent directions in the space. This means that, for two vector spaces with the same dimension, the properties that depend only on the vector-space structure are exactly the same (technically the vector spaces are isomorphic). A vector space is finite-dimensional if its dimension is a
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
and related areas. Infinite-dimensional vector spaces occur in many areas of mathematics. For example,
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
s are
countably
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
infinite-dimensional vector spaces, and many function spaces have the cardinality of the continuum as a dimension.
Many vector spaces that are considered in mathematics are also endowed with other
structures
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
. This is the case of
algebras
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
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: It is also common, especially in higher mathematics, to not use any typographical method for distinguishing vectors from other mathematical objects.
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 grass ...
is a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
together with two binary operations that satisfy the eight axioms listed below. In this context, the elements of are commonly called ''vectors'', and the elements of are called ''scalars''.
* The first operation, called ''vector addition'' or simply ''addition'' assigns to any two vectors and in a third vector in which is commonly written as , and called the ''sum'' of these two vectors.
* The second operation, called ''
scalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector b ...
'',assigns to any scalar in and any vector in another vector in , which is denoted .Scalar multiplication is not to be confused with the
scalar product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
, which is an additional operation on some specific vector spaces, called
inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
s. Scalar multiplication is a multiplication of a vector ''by'' a scalar that produces a vector, while the scalar product is a multiplication of two vectors that produces a scalar.
For having a vector space, the eight following axioms must be satisfied for every , and in , and and in .
When the scalar field is the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
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 extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s, the vector space is called a ''complex vector space''. These two cases are the most common ones, but vector spaces with scalars in an arbitrary field are also commonly considered. Such a vector space is called an -''vector space'' or a ''vector space over ''.
An equivalent definition of a vector space can be given, which is much more concise but less elementary: the first four axioms (related to vector addition) say that a vector space is an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
under addition, and the four remaining axioms (related to the scalar multiplication), say that this operation defines a
ring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is:
:addition preser ...
from the field into the
endomorphism ring
In mathematics, the endomorphisms of an abelian group ''X'' form a ring. This ring is called the endomorphism ring of ''X'', denoted by End(''X''); the set of all homomorphisms of ''X'' into itself. Addition of endomorphisms arises naturally in a ...
of this group.
Subtraction of two vectors can be defined as
:
Direct consequences of the axioms include that, for every and one has
*
*
*
* implies or
Related concepts and properties
; Linear combination
: Given a set of elements of a -vector space , a linear combination of elements of is an element of of the form where and The scalars are called the ''coefficients'' of the linear combination.
;
Linear independence
In the theory of vector spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
:The elements of a subset of a -vector space are said to be ''linearly independent'' if no element of can be written as a linear combination of the other elements of . Equivalently, they are linearly independent if two linear combinations of element of define the same element of if and only if they have the same coefficients. Also equivalently, they are linearly independent if a linear combination results in the zero vector if and only if all its coefficients are zero.
; Linear subspace
:A ''linear subspace'' or ''vector subspace'' of a vector space is a non-empty subset of that is closed under vector addition and scalar multiplication; that is, the sum of two elements of and the product of an element of by a scalar belong to . This implies that every linear combination of elements of belongs to . A linear subspace is a vector space for the induced addition and scalar multiplication; this means that the closure property implies that the axioms of a vector space are satisfied. The closure property also implies that ''every intersection of linear subspaces is a linear subspace.''
; Linear span
:Given a subset of a vector space , the ''linear span'' or simply the ''span'' of is the smallest linear subspace of that contains , in the sense that it is the intersection of all linear subspaces that contain . The span of is also the set of all linear combinations of elements of . If is the span of , one says that ''spans'' or ''generates'' , and that is a ''
spanning set
In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characterized ...
'' or a ''generating set'' of .
;
Basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
and
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
:A subset of a vector space is a ''basis'' if its elements are linearly independent and span the vector space. Every vector space has at least one basis, generally many (see ). Moreover, all bases of a vector space have the same cardinality, which is called the ''dimension'' of the vector space (see
Dimension theorem for vector spaces
In mathematics, the dimension theorem for vector spaces states that all 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 defines the dimension ...
). This is a fundamental property of vector spaces, which is detailed in the remainder of the section.
''Bases'' are a fundamental tool for the study of vector spaces, especially when the dimension is finite. In the infinite-dimensional case, the existence of infinite bases, often called Hamel bases, depend on 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 of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
. It follows that, in general, no base can be explicitly described. For example, the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s form an infinite-dimensional vector space over the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
s, for which no specific basis is known.
Consider a basis of a vector space of dimension over a field . The definition of a basis implies that every may be written
:
with in , and that this decomposition is unique. The scalars are called the ''coordinates'' of on the basis. They are also said to be the ''coefficients'' of the decomposition of on the basis. One also says that the -
tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
of the coordinates is the coordinate vector of on the basis, since the set of the -tuples of elements of is a vector space for componentwise addition and scalar multiplication, whose dimension is .
The
one-to-one correspondence
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
between vectors and their coordinate vectors maps vector addition to vector addition and scalar multiplication to scalar multiplication. It is thus a vector space isomorphism, which allows translating reasonings and computations on vectors into reasonings and computations on their coordinates. If, in turn, these coordinates are arranged as
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
, these reasonings and computations on coordinates can be expressed concisely as reasonings and computations on matrices. Moreover, a linear equation relating matrices can be expanded into a system of linear equations, and, conversely, every such system can be compacted into a linear equation on matrices.
So, in summary, finite-dimensional
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 matrices ...
may be expressed in three equivalent languages:
*''Vector spaces'', which provide concise and coordinate-free statements,
*''Matrices'', which are convenient for expressing concisely explicit computations,
*''
Systems of linear equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables.
For example,
:\begin
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\fracy-z=0
\end
is a system of three equations in th ...
,'' which provide more elementary formulations.
History
Vector spaces stem from
affine geometry
In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle.
As the notion of '' parallel lines'' is one of the main properties that is ...
, via the introduction of
coordinate
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
s in the plane or three-dimensional space. Around 1636, French mathematicians
Pierre de Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
founded analytic geometry 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, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
. To achieve geometric solutions without using coordinates,
Bolzano
Bolzano ( or ; german: Bozen, (formerly ); bar, Bozn; lld, Balsan or ) is the capital city of the province of South Tyrol in northern Italy. With a population of 108,245, Bolzano is also by far the largest city in South Tyrol and the third la ...
introduced, in 1804, certain operations on points, lines and planes, which are predecessors of vectors. introduced the notion of
barycentric coordinates
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
. 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, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
Hamilton Hamilton may refer to:
People
* Hamilton (name), a common British surname and occasional given name, usually of Scottish origin, including a list of persons with the surname
** The Duke of Hamilton, the premier peer of Scotland
** Lord Hamilt ...
and the inception of quaternions by the latter. They are elements in R2 and R4; treating them using linear combinations goes back to
systems of linear equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables.
For example,
:\begin
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\fracy-z=0
\end
is a system of three equations in th ...
matrix notation
In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object.
For example,
\begin ...
which allows for a harmonization and simplification of
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
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 linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mat ...
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 spaces, a set of vectors is said to be if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be . These concepts are ...
and
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
, as well as
scalar product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
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 () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
s. Italian mathematician
Peano
Giuseppe Peano (; ; 27 August 1858 – 20 April 1932) was an Italian mathematician and glottologist. The author of over 200 books and papers, he was a founder of mathematical logic and set theory, to which he contributed much notation. The sta ...
was the first to give the modern definition of vector spaces and linear maps in 1888, although he called them "linear systems".
An important development of vector spaces is due to the construction of
function spaces
In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vector ...
Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
, around 1920. At that time,
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
and the new field of
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
began to interact, notably with key concepts such as spaces of ''p''-integrable functions and Hilbert spaces. Also at this time, the first studies concerning infinite-dimensional vector spaces were done.
Examples
Arrows in the plane
The first example of a vector space consists of arrows in a fixed
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
, starting at one fixed point. This is used in physics to describe forces or
velocities
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
. Given any two such arrows, and , the 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, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
, 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.) Such a pair is written as . The sum of two such pairs and multiplication of a pair with a number is defined as follows:
:
and
:
The first example above reduces to this example, if an arrow is represented by a pair of Cartesian coordinates of its endpoint.
Coordinate space
The simplest example of a vector space over a field is the field itself (as it is an
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
for addition, a part of the 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 grass ...
), 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, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
, that is, numbers that can be written in the form for real numbers and where is the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
, 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 then we see that the rules for addition and scalar multiplication correspond exactly to those in the earlier example.
More generally, field extensions provide another class of examples of vector spaces, particularly in algebra and algebraic number theory: 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 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 or an interval, or other subsets of . Many notions in topology and analysis, such as continuity, integrability or
differentiability
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
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, whose study belongs to
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
.
Linear equations
Systems of
homogeneous linear equation
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables.
For example,
:\begin
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\fracy-z=0
\end
is a system of three equations in th ...
s 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 most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
can be used to condense multiple linear equations as above into one vector equation, namely
:,
where is the matrix containing the coefficients of the given equations, is the vector , denotes the
matrix product
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
, 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.
Linear maps and matrices
The relation of two vector spaces can be expressed by ''linear map'' or ''linear transformation''. They are functions that reflect the vector space structure, that is, they preserve sums and scalar multiplication:
: and for all and in , all in .
An ''
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
'' is a linear map such that there exists an
inverse map
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ .
For a function f\colon X ...
, which is a map such that the two possible compositions and are 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
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sl ...
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
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
'', denoted . Via the injective
natural
Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are ...
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 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 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
:, where denotes summation,
or, using the
matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
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
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
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 preserving if and only if its determinant is positive.
Eigenvalues and eigenvectors
Endomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a gr ...
s, 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
German(s) may refer to:
* Germany (of or related to)
** Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ge ...
"
eigen Eigen may refer to:
* Eigen (C++ library), computer programming library for matrix and linear algebra operations
* Eigen Technologies, the Document AI software company
* Eigen, Schwyz, settlement in the municipality of Alpthal in the canton of S ...
", which means own or proper. Equivalently, is an element of the
kernel
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learn ...
of the difference (where Id is the
identity map
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
. 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
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
, such as ) any linear map has at least one eigenvector. The vector space may or may not possess an
eigenbasis
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
, a basis consisting of eigenvectors. This phenomenon is governed by the
Jordan canonical form
In linear algebra, a Jordan normal form, also known as a Jordan canonical form (JCF),
is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to so ...
of the map.See also
Jordan–Chevalley decomposition In mathematics, the Jordan–Chevalley decomposition, named after Camille Jordan and Claude Chevalley, expresses a linear operator as the sum of its commuting semisimple part and its nilpotent part. The multiplicative decomposition expresses an inve ...
. 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
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful ...
, the corresponding statement in the infinite-dimensional case, the machinery of functional analysis is needed, see 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 properties, which determine an object by specifying the linear maps from to any other vector space.
Subspaces and quotient spaces
A nonempty 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 mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
. In this case, a linear subspace contains the
zero vector
In mathematics, a zero element is one of several generalizations of 0, the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context.
Additive identities
An additive iden ...
, 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
span
Span may refer to:
Science, technology and engineering
* Span (unit), the width of a human hand
* Span (engineering), a section between two intermediate supports
* Wingspan, the distance between the wingtips of a bird or aircraft
* Sorbitan ester ...
, 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 combinations 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'' 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
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
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
Kernel may refer to:
Computing
* Kernel (operating system), the central component of most operating systems
* Kernel (image processing), a matrix used for image convolution
* Compute kernel, in GPGPU programming
* Kernel method, in machine learn ...
ker(''f'') of a linear map consists of vectors v that are mapped to 0 in ''W''. The kernel and the 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
In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ring ...
(over a fixed field ''F'') is an
abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
, that is, a corpus of mathematical objects and structure-preserving maps between them (a
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
first isomorphism theorem
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist fo ...
(also called
rank–nullity theorem
The rank–nullity theorem is a theorem in linear algebra, which asserts that the dimension of the domain of a linear map is the sum of its rank (the dimension of its image) and its ''nullity'' (the dimension of its kernel). p. 70, §2.1, Theo ...
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
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
.
An important example is the kernel of a linear map for some fixed matrix ''A'', as 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
:, where the coefficients ''a''''i'' are functions in ''x'', too.
In the corresponding map
:,
the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s 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 mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
. 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'' 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 may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consists ...
''I'' an element v''i'' of ''V''''i''. Addition and scalar multiplication is performed componentwise. A variant of this construction is the ''direct sum'' (also called
coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduc ...
and denoted ), 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
Multilinear algebra is a subfield of mathematics that extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of ''p' ...
which deals with extending notions such as linear maps to several variables. A map is called 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, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other 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
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
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
composition
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
*Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
with ''f'' equals ''g'': . This is called the
universal property
In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fr ...
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 converges to another function. Likewise, linear algebra is not adapted to deal with
infinite series
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
, since the addition operation allows only finitely many terms to be added. Therefore, the needs of
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
require considering additional structures.
A vector space may be given a
partial order
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a bina ...
≤, 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 space
In mathematics, an ordered vector space or partially ordered vector space is a vector space equipped with a partial order that is compatible with the vector space operations.
Definition
Given a vector space ''X'' over the real numbers R and a pr ...
s, for example
Riesz space
In mathematics, a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice.
Riesz spaces are named after Frigyes Riesz who first defined them in his 1928 paper ''Su ...
s, are fundamental to
Lebesgue integration
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Leb ...
, which relies on the ability to express a function as a difference of two positive functions
:,
where denotes the positive part of and the negative part.
Normed vector spaces and inner product spaces
"Measuring" vectors is done by specifying a
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
, a datum which measures lengths of vectors, or by an
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
, which measures angles between vectors. Norms and inner products are denoted 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
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
:
:
In R2, this reflects the common notion of the angle between two vectors x and y, by the
law of cosines
In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...
:
:
Because of this, two vectors satisfying are called orthogonal. An important variant of the standard dot product is used in
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
: R4 endowed with the Lorentz product
:
In contrast to the standard dot product, it is not
positive definite In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive-definite. See, in particular:
* Positive-definite bilinear form
* Positive-definite f ...
: also takes negative values, for example, for . Singling out the fourth coordinate— corresponding to time, as opposed to three space-dimensions—makes it useful for the mathematical treatment of
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws ...
.
Topological vector spaces
Convergence questions are treated by considering vector spaces ''V'' carrying a compatible
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, a structure that allows one to talk about elements being close to each other. Compatible here means that addition and scalar multiplication have to be
continuous map
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
s. 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
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifor ...
, 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
Series may refer to:
People with the name
* Caroline Series (born 1951), English mathematician, daughter of George Series
* George Series (1920–1995), English physicist
Arts, entertainment, and media
Music
* Series, the ordered sets used in ...
of vectors. The
infinite sum
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...
:
denotes the 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 ''V'', in which case the series is a
function series
In calculus, a function series is a series, where the summands are not just real or complex numbers but functions.
Examples
Examples of function series include power series, Laurent series, Fourier series, etc.
Convergence
There exist many type ...
. The mode of convergence of the series depends on the topology imposed on the function space. In such cases,
pointwise convergence
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared.
Definition
Suppose that X is a set and ...
and
uniform convergence
In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily ...
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
In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
has a limit; such a vector space is called 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 ,1 equipped with the
topology of uniform convergence In mathematics, particularly functional analysis, spaces of linear maps between two vector spaces can be endowed with a variety of topologies. Studying space of linear maps and these topologies can give insight into the spaces themselves.
The arti ...
is not complete because any continuous function on ,1can be uniformly approximated by a sequence of polynomials, by the
Weierstrass approximation theorem
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
. In contrast, the space of ''all'' continuous functions on ,1with 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
:
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
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
—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 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
The Hahn–Banach theorem is a central tool in functional analysis.
It allows the extension of bounded linear functionals defined on a subspace of some vector space to the whole space, and it also shows that there are "enough" continuous linear f ...
is concerned with separating subspaces of appropriate topological vector spaces by continuous functionals.
Banach spaces
''Banach spaces'', introduced by
Stefan Banach
Stefan Banach ( ; 30 March 1892 – 31 August 1945) was a Polish mathematician who is generally considered one of the 20th century's most important and influential mathematicians. He was the founder of modern functional analysis, and an origina ...
, are complete normed vector spaces.
A first example is the vector space consisting of infinite vectors with real entries
whose -norm
given by
: for and .
The topologies on the infinite-dimensional space
are inequivalent for different . For example, the sequence of vectors
,
in which the first components are and the following ones are , converges to the
zero vector
In mathematics, a zero element is one of several generalizations of 0, the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context.
Additive identities
An additive iden ...
for
,
but does not for
:
:, but
More generally than sequences of real numbers, functions
are endowed with a norm that replaces the above sum by the
Lebesgue integral
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...
:
The space of
integrable function
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with d ...
s on a given domain
(for example an interval) satisfying
,
and equipped with this norm are called Lebesgue spaces, denoted .The
triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
This statement permits the inclusion of degenerate triangles, but ...
for
is provided by the
Minkowski inequality
In mathematical analysis, the Minkowski inequality establishes that the L''p'' spaces are normed vector spaces. Let ''S'' be a measure space, let and let ''f'' and ''g'' be elements of L''p''(''S''). Then is in L''p''(''S''), and we have the t ...
. For technical reasons, in the context of functions one has to identify functions that agree
almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
to get a norm, and not only a
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk ...
.
These spaces are complete. (If one uses the
Riemann integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of GÃ ...
instead, the space is ''not'' complete, which may be seen as a justification for Lebesgue's integration theory.
"Many functions in
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
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
with
,
satisfying the condition
:
there exists a function belonging to the vector space such that
:
Imposing boundedness conditions not only on the function, but also on its
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s leads to
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
complex conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
Taylor approximation
In calculus, Taylor's theorem gives an approximation of a ''k''-times differentiable function around a given point by a polynomial of degree ''k'', called the ''k''th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the t ...
, established an approximation of
differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in it ...
s ''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 function
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
s is commonly called
Fourier expansion
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
, and is much applied in engineering, see 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'' 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 above. For distinction, the latter is then called a
Hamel basis
In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components ...
. Not only does the theorem exhibit suitable basis functions as sufficient for approximation purposes, but also together with the
Gram–Schmidt process
In mathematics, particularly linear algebra and numerical analysis, the Gram–Schmidt process is a method for orthonormalizing a set of vectors in an inner product space, most commonly the Euclidean space equipped with the standard inner produ ...
, it enables one to construct a 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 geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
.
The solutions to various
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s 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
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
in
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistr ...
describes the change of physical properties in time by means of a partial differential equation, whose solutions are called
wavefunction
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
s. Definite values for physical properties such as energy, or momentum, correspond to
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful ...
decomposes a linear
compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact c ...
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
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.
Definition
Vector spaces
Let V, W ...
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 algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
s which are both Banach spaces and algebras.
Commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
makes great use of rings of polynomials in one or several variables, introduced above. Their multiplication is both
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
anticommutativity
In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswapped ...
), and
* (
Jacobi identity
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
).
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
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
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, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
s
:, where the degree varies.
The multiplication is given by concatenating such symbols, imposing the
distributive law
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmetic, ...
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 above. In general, there are no relations between and . Forcing two such elements to be equal leads to the
symmetric algebra
In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ...
, whereas forcing yields the
exterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
.
When a field, is explicitly stated, a common term used is -algebra.
Related structures
Vector bundles
A ''vector bundle'' is a family of vector spaces parametrized continuously by a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
''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
Fiber or fibre (from la, fibra, links=no) is a natural or artificial substance that is significantly longer than it is wide. Fibers are often used in the manufacture of other materials. The strongest engineering materials often incorpora ...
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" vector bundle. Vector bundles over ''X'' are required to be
locally In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points).
P ...
a product of ''X'' and some (fixed) vector space ''V'': for every ''x'' in ''X'', there is a neighborhood ''U'' of ''x'' such that the restriction of π to π−1(''U'') is isomorphicThat is, there is a
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
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 identifying open intervals with the real line). It is, however, different from the
cylinder
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base.
A cylinder may also be defined as an infin ...
, because the latter is
orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
whereas the former is not.
Properties of certain vector bundles provide information about the underlying topological space. For example, the
tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
consists of the collection of
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
s 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
Section, Sectioning or Sectioned may refer to:
Arts, entertainment and media
* Section (music), a complete, but not independent, musical idea
* Section (typography), a subdivision, especially of a chapter, in books and documents
** Section sig ...
that vanishes nowhere, see . The sections of the tangent bundle are just vector fields. In contrast, by the
hairy ball theorem
The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous tangent vector field on even-dimensional ''n''-spheres. For the ordinary sphere, or 2‑sphere, if ...
, there is no (tangent) vector field on the
2-sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
''S''2 which is everywhere nonzero.
K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ...
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 algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
Definitions
Formally, we start with a non-zero algebra ''D'' over a fie ...
octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions hav ...
s 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
In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T ...
rings
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
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 inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a fraction ''a''/ ...
s. For example, modules need not have bases, as the Z-module (that is,
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
) 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
* Mo ...
over a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
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 grass ...
, with the elements being called vectors. Some authors use the term ''vector space'' to mean modules over a
division ring
In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
. The algebro-geometric interpretation of commutative rings via their
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors ...
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 free transitive vector space
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
. 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
In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint, equal-size subsets called cosets. There are ''left cosets'' and ''right cosets''. Cosets (both left and right) ...
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 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
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kernel of ...
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
Parallel is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster of ...
flag manifold In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space ''V'' over a field F. When F is the real or complex numbers, a generalized flag variety is a smo ...
s generalize this by parametrizing linear subspaces of fixed dimension ''k'' and
flags
A flag is a piece of textile, fabric (most often rectangular or quadrilateral) with a distinctive design and colours. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic desi ...
of subspaces, respectively.
Related concepts
;Specific vectors in a vector space
*
Zero vector
In mathematics, a zero element is one of several generalizations of 0, the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context.
Additive identities
An additive iden ...
(sometimes also called ''null vector'' and denoted by ), the
additive identity In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element ''x'' in the set, yields ''x''. One of the most familiar additive identities is the number 0 from elemen ...
in a vector space. In a
normed vector space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
, it is the unique vector of norm zero. In a
Euclidean vector space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
, it is the unique vector of length zero.
*
Basis vector
In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components ...
, an element of a given
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
of a vector space.
*
Unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction v ...
, a vector in a normed vector space whose
norm
Naturally occurring radioactive materials (NORM) and technologically enhanced naturally occurring radioactive materials (TENORM) consist of materials, usually industrial wastes or by-products enriched with radioactive elements found in the envi ...
is 1, or a
Euclidean vector
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ...
null vector
In mathematics, given a vector space ''X'' with an associated quadratic form ''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which .
In the theory of real bilinear forms, definite quadratic forms an ...
, in a vector space with a quadratic form, a non-zero vector for which the form is zero. If a null vector exists, the quadratic form is said an
isotropic quadratic form
In mathematics, a quadratic form over a field ''F'' is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More precisely, if ''q'' is a quadratic form on a vector s ...
.
;Vectors in specific vector spaces
*
Column vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, c ...
, a matrix with only one column. The column vectors with a fixed number of rows form a vector space.
*
Row vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, c ...
, a matrix with only one row. The row vectors with a fixed number of columns form a vector space.
* Coordinate vector, the -tuple of the coordinates of a vector on a
basis
Basis may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
* Basis trading, a trading strategy consisting ...
of elements. For 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 grass ...
, these -tuples form the vector space (where the operation are pointwise addition and scalar multiplication).
*
Displacement vector
In geometry and mechanics, a displacement is a vector whose length is the shortest distance from the initial to the final position of a point P undergoing motion. It quantifies both the distance and direction of the net or total motion along a ...
, a vector that specifies the change in position of a point relative to a previous position. Displacement vectors belong to the vector space of
translations
Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transl ...
.
* Position vector of a point, the displacement vector from a reference point (called the ''origin'') to the point. A position vector represents the position of a point in a
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
or an
affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
.
*
Velocity vector
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
, the derivative, with respect to time, of the position vector. It does not depend of the choice of the origin, and, thus belongs to the vector space of translations.
* Pseudovector, also called ''axial vector''
* Covector, an element of the dual of a vector space. In an
inner product space
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
, the inner product defines an isomorphism between the space and its dual, which may make difficult to distinguish a covector from a vector. The distinction becomes apparent when one changes coordinates (non-orthogonally).
*
Tangent vector
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are e ...
, an element of the
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
of a
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
, a
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
or, more generally, a
differential manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
at a given point (these tangent spaces are naturally endowed with a structure of vector space)
* Normal vector or simply ''normal'', in a Euclidean space or, more generally, in an inner product space, a vector that is perpendicular to a tangent space at a point.
*
Gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
, the coordinates vector of the partial derivatives of a
function of several real variables
In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function o ...
. In a Euclidean space the gradient gives the magnitude and direction of maximum increase of a scalar field. The gradient is a covector that is normal to a
level curve
In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is:
: L_c(f) = \left\~,
When the number of independent variables is two, a level set is cal ...
.
*
Four-vector
In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
, in the
theory of relativity
The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...
, a vector in a four-dimensional real vector space called
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
See also
*
Vector (mathematics and physics)
In mathematics and physics, vector is a term that refers colloquially to some quantities that cannot be expressed by a single number (a scalar), or to elements of some vector spaces.
Historically, vectors were introduced in geometry and physic ...
Metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
Riesz–Fischer theorem
In mathematics, the Riesz–Fischer theorem in real analysis is any of a number of closely related results concerning the properties of the space ''L''2 of square integrable functions. The theorem was proven independently in 1907 by Frigyes Rie ...
Ordered vector space
In mathematics, an ordered vector space or partially ordered vector space is a vector space equipped with a partial order that is compatible with the vector space operations.
Definition
Given a vector space ''X'' over the real numbers R and a pr ...
Formulario mathematico
''Formulario Mathematico'' (Latino sine flexione: ''Formulary for Mathematics'') is a book
There are many editions. Here are two:
* (French) Published 1901 by Gauthier-Villars, Paris. 230p.OpenLibrary OL15255022W vct axioms via
Internet Archive
The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...