In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and
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 ...
, 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 ''
vectors'', may be
added together and
multiplied ("scaled") by numbers called ''
scalars
Scalar may refer to:
* Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
* Scalar (physics), a physical quantity that can be described by a single element of a number field such ...
''. 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. 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.
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 ...
, such as
forces and
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 matrice ...
, 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'' (franchi ...
, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying
systems of linear equations.
Vector spaces are characterized by their
dimension
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 coord ...
, 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 ...
. Otherwise, it is
infinite-dimensional, and its dimension is an
infinite cardinal. Finite-dimensional vector spaces occur naturally in
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 c ...
and related areas. Infinite-dimensional vector spaces occur in many areas of mathematics. For example,
polynomial rings are
countably 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. This is the case of
algebras, which include
field extensions,
polynomial rings,
associative algebras and
Lie algebras. This is also the case of
topological vector spaces, which include
function spaces,
inner product 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,
normed spaces,
Hilbert spaces and
Banach spaces.
Definition and basic properties
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 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'',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, 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 from the field into the
endomorphism ring 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
: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
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
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'' or a ''generating set'' of .
;
Basis 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 coord ...
: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). 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 ra ...
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
In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimensio ...
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 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, 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 matrice ...
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,'' which provide more elementary formulations.
History
Vector spaces stem from
affine geometry, via the introduction of
coordinates in the plane or three-dimensional space. Around 1636, French mathematicians
René Descartes
René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Ma ...
and
Pierre de Fermat founded
analytic geometry by identifying solutions to an equation of two variables with points on a plane
curve. 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 ...
introduced, in 1804, certain operations on points, lines and planes, which are predecessors of vectors. introduced the notion of
barycentric coordinates. introduced the notion of a bipoint, i.e., an oriented segment one of whose ends is the origin and the other one a target. Vectors were reconsidered with the presentation of
complex numbers by
Argand and
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 Hamilto ...
and the inception of
quaternions by the latter. They are elements in R
2 and R
4; treating them using
linear combinations goes back to
Laguerre in 1867, who also defined
systems of linear equations.
In 1857,
Cayley introduced the
matrix notation 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 ...
s. Around the same time,
Grassmann studied the barycentric calculus initiated by Möbius. He envisaged sets of abstract objects endowed with operations. In his work, the concepts of
linear independence 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 coord ...
, as well as
scalar products 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 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 by
Henri Lebesgue
Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
. This was later formalized by
Banach and
Hilbert, 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 defi ...
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, starting at one fixed point. This is used in physics to describe
forces or
velocities. 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), 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 R
2) was discussed in the introduction above.
Complex numbers and other field extensions
The set of
complex numbers , 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
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s of . Many notions in topology and analysis, such as
continuity,
integrability or
differentiability 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 defi ...
.
Linear equations
Systems of
homogeneous linear equations are closely tied to vector spaces. For example, the solutions of
:
are given by triples with arbitrary , , and . They form a vector space: sums and scalar multiples of such triples still satisfy the same ratios of the three variables; thus they are solutions, too.
Matrices 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, 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 , which is a map such that the two possible
compositions
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 ...
and are
identity maps. Equivalently, is both one-to-one (
injective) and onto (
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
). 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), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* The Origin (Buffy comic), ''The Origin'' (Bu ...
of some (fixed)
coordinate system can be expressed as an ordered pair by considering the - and -component of the arrow, as shown in the image at the right. Conversely, given a pair , the arrow going by to the right (or to the left, if is negative), and up (down, if is negative) turns back the arrow .
Linear maps between two vector spaces form a vector space , also denoted , or . The space of linear maps from to is called the ''
dual vector space'', denoted . Via the injective
natural
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 ar ...
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
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred 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 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
Endomorphisms, linear maps , are particularly important since in this case vectors can be compared with their image under , . Any nonzero vector satisfying , where is a scalar, is called an ''eigenvector'' of with ''eigenvalue'' .
[The nomenclature derives from ]German
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", which means own or proper. Equivalently, is an element of the
kernel 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, unc ...
. 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, such as ) any linear map has at least one eigenvector. The vector space may or may not possess an
eigenbasis, a basis consisting of eigenvectors. This phenomenon is governed by the
Jordan canonical form of the map.
[See also Jordan–Chevalley decomposition.] The set of all eigenvectors corresponding to a particular eigenvalue of forms a vector space known as the ''eigenspace'' corresponding to the eigenvalue (and ) in question. To achieve the
spectral theorem
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 be ...
, the corresponding statement in the infinite-dimensional case, the machinery of functional analysis is needed, see
below
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred 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
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
''W'' of a vector space ''V'' that is closed under addition and scalar multiplication (and therefore contains the 0-vector of ''V'') is called a ''linear subspace'' of ''V'', or simply a ''subspace'' of ''V'', when the ambient space is unambiguously a vector space.
[This is typically the case when a vector space is also considered as an affine space. In this case, a linear subspace contains the zero vector, while an affine subspace does not necessarily contain it.] Subspaces of ''V'' are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set ''S'' of vectors is called its
span, and it is the smallest subspace of ''V'' containing the set ''S''. Expressed in terms of elements, the span is the subspace consisting of all the
linear 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
In computing, the modulo operation returns the remainder or signed remainder of a division, after one number is divided by another (called the '' modulus'' of the operation).
Given two positive numbers and , modulo (often abbreviated as ) is ...
''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 bic ...
the difference of v
1 and v
2 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 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 (over a fixed field ''F'') is an
abelian category, that is, a corpus of mathematical objects and structure-preserving maps between them (a
category) that behaves much like the
category of abelian groups. Because of this, many statements such as the
first isomorphism theorem (also called
rank–nullity theorem in matrix-related terms)
:''V'' / ker(''f'') ≡ im(''f'').
and the second and third isomorphism theorem can be formulated and proven in a way very similar to the corresponding statements for
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 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 ''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 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 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
tensors
:v
1 ⊗ w
1 + v
2 ⊗ w
2 + ⋯ + v
''n'' ⊗ w
''n'',
subject to the rules
: ''a'' · (v ⊗ w) = (''a'' · v) ⊗ w = v ⊗ (''a'' · w), where ''a'' is a scalar,
:(v
1 + v
2) ⊗ w = v
1 ⊗ w + v
2 ⊗ w, and
:v ⊗ (w
1 + w
2) = v ⊗ w
1 + v ⊗ w
2.
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 with ''f'' equals ''g'': . This is called the
universal property of the tensor product, an instance of the method—much used in advanced abstract algebra—to indirectly define objects by specifying maps from or to this object.
Vector spaces with additional structure
From the point of view of linear algebra, vector spaces are completely understood insofar as any vector space is characterized, up to isomorphism, by its dimension. However, vector spaces ''per se'' do not offer a framework to deal with the question—crucial to analysis—whether a sequence of functions
converges to another function. Likewise, linear algebra is not adapted to deal with
infinite series, since the addition operation allows only finitely many terms to be added.
Therefore, the needs of functional analysis
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 defi ...
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 binary ...
≤, under which some vectors can be compared. For example, ''n''-dimensional real space R
''n'' can be ordered by comparing its vectors componentwise.
Ordered vector spaces, for example
Riesz spaces, are fundamental to
Lebesgue integration
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 Le ...
, 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, 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 alg ...
:
:
In R
2, this reflects the common notion of the angle between two vectors x and y, by the
law of cosines:
:
Because of this, two vectors satisfying
are called
orthogonal. An important variant of the standard dot product is used in
Minkowski space: R
4 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 fu ...
:
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 law ...
.
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 maps. Roughly, if x and y in ''V'', and ''a'' in ''F'' vary by a bounded amount, then so do and .
[This requirement implies that the topology gives rise to a uniform structure, ] To make sense of specifying the amount a scalar changes, the field ''F'' also has to carry a topology in this context; a common choice are the reals or the complex numbers.
In such ''topological vector spaces'' one can consider
series of vectors. The
infinite sum
:
denotes the
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
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. 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 an ...
and
uniform convergence are two prominent examples.
A way to ensure the existence of limits of certain infinite series is to restrict attention to spaces where any
Cauchy sequence has a limit; such a vector space is called
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
. 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 is not complete because any continuous function on
,1can be uniformly approximated by a sequence of polynomials, by the
Weierstrass approximation theorem. 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 defi ...
—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 R
2: as the unit "balls" enclose each other, a sequence converges to zero in one norm if and only if it so does in the other norm. In the infinite-dimensional case, however, there will generally be inequivalent topologies, which makes the study of topological vector spaces richer than that of vector spaces without additional data.
From a conceptual point of view, all notions related to topological vector spaces should match the topology. For example, instead of considering all linear maps (also called
functional
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
** Form follows function
* Functional group, combination of atoms within molecules
* Medical conditions without currently visible organic basis:
** Functional sy ...
s) , maps between topological vector spaces are required to be continuous. In particular, the
(topological) dual space consists of continuous functionals (or to ). The fundamental
Hahn–Banach theorem is concerned with separating subspaces of appropriate topological vector spaces by continuous functionals.
Banach spaces
''Banach spaces'', introduced by
Stefan Banach, are complete normed vector spaces.
A first example is
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 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
:
The space of
integrable functions on a given
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
** Domain of definition of a partial function
** Natural domain of a partial function
**Domain of holomorphy of a function
* ...
(for example an interval) satisfying
,
and equipped with this norm are called
Lebesgue spaces, denoted
.
[The triangle inequality for
is provided by the Minkowski inequality. For technical reasons, in the context of functions one has to identify functions that agree almost everywhere to get a norm, and not only a ]seminorm 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 a ...
.
These spaces are complete. (If one uses the
Riemann integral instead, the space is ''not'' complete, which may be seen as a justification for Lebesgue's integration theory.
[
"Many functions in
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 ...
s.
Hilbert spaces
Complete inner product spaces are known as ''Hilbert spaces'', in honor of
David Hilbert. The Hilbert space ''L''
2(Ω), with inner product given by
:
where
denotes the
complex conjugate of ''g''(''x''),
[For ''p'' ≠2, ''L''''p''(Ω) is not a Hilbert space.] is a key case.
By definition, in a Hilbert space any Cauchy sequence converges to a limit. Conversely, finding a sequence of functions ''f''
''n'' with desirable properties that approximates a given limit function, is equally crucial. Early analysis, in the guise of the
Taylor approximation, established an approximation of
differentiable 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 ...
s ''f'' by polynomials. By the
Stone–Weierstrass theorem
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are among the ...
, every continuous function on can be approximated as closely as desired by a polynomial. A similar approximation technique by
trigonometric functions is commonly called
Fourier expansion, and is much applied in engineering, see
below
Below may refer to:
*Earth
* Ground (disambiguation)
*Soil
*Floor
* Bottom (disambiguation)
*Less than
*Temperatures below freezing
*Hell or underworld
People with the surname
*Ernst von Below (1863–1955), German World War I general
*Fred 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.] Not only does the theorem exhibit suitable basis functions as sufficient for approximation purposes, but also together with the
Gram–Schmidt process, it enables one to construct a
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 Euclidea ...
.
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, ...
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 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 chemistry, ...
describes the change of physical properties in time by means of a
partial differential equation, whose solutions are called
wavefunctions. 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 denote ...
s of a certain (linear)
differential operator and the associated wavefunctions are called
eigenstates. 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 be ...
decomposes a linear compact operator acting on functions in terms of these eigenfunctions and their eigenvalues.
Algebras over fields
General vector spaces do not possess a multiplication between vectors. A vector space equipped with an additional
bilinear operator defining the multiplication of two vectors is an ''algebra over a field''. Many algebras stem from functions on some geometrical object: since functions with values in a given field can be multiplied pointwise, these entities form algebras. The Stone–Weierstrass theorem, for example, relies on
Banach algebras which are both Banach spaces and algebras.
Commutative algebra
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 and
associative. These rings and their
quotients form the basis of
algebraic geometry, because they are
rings of functions of algebraic geometric objects.
Another crucial example are ''Lie algebras'', which are neither commutative nor associative, but the failure to be so is limited by the constraints ( denotes the product of and ):
* (
anticommutativity), and
* (
Jacobi identity).
Examples include the vector space of ''n''-by-''n'' matrices, with , the
commutator of two matrices, and , endowed with the
cross product.
The
tensor algebra T(''V'') is a formal way of adding products to any vector space ''V'' to obtain an algebra. As a vector space, it is spanned by symbols, called simple
tensors
:, where the
degree varies.
The multiplication is given by concatenating such symbols, imposing the
distributive law under addition, and requiring that scalar multiplication commute with the tensor product ⊗, much the same way as with the tensor product of two vector spaces introduced
above. In general, there are no relations between and . Forcing two such elements to be equal leads to the
symmetric algebra, whereas forcing yields the
exterior algebra.
When a field, is explicitly stated, a common term used is -algebra.
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 poin ...
''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 incorporate ...
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 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 isomorphic
[That 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 isom ...
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 ...
, because the latter is
orientable whereas the former is not.
Properties of certain vector bundles provide information about the underlying topological space. For example, the
tangent bundle consists of the collection of
tangent spaces parametrized by the points of a differentiable manifold. The tangent bundle of the circle ''S''
1 is globally isomorphic to , since there is a global nonzero
vector field on ''S''
1.
[A line bundle, such as the tangent bundle of ''S''1 is trivial if and only if there is a section that vanishes nowhere, see . The sections of the tangent bundle are just vector fields.] In contrast, by the
hairy ball theorem, there is no (tangent) vector field on the
2-sphere ''S''
2 which is everywhere nonzero.
K-theory studies the isomorphism classes of all vector bundles over some topological space. In addition to deepening topological and geometrical insight, it has purely algebraic consequences, such as the classification of finite-dimensional real
division algebras: R, C, the
quaternions H and the
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
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. Th ...
of a differentiable manifold consists, at every point of the manifold, of the dual of the tangent space, the
cotangent space.
Sections
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 ...
of that bundle are known as
differential one-forms.
Modules
''Modules'' are to
rings what vector spaces are to fields: the same axioms, applied to a ring ''R'' instead of a field ''F'', yield modules. The theory of modules, compared to that of vector spaces, is complicated by the presence of ring elements that do not have
multiplicative 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
* Modul ...
over a
ring which is a
field, with the elements being called vectors. Some authors use the term ''vector space'' to mean modules over a
division ring. The algebro-geometric interpretation of commutative rings via their
spectrum 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. In particular, a vector space is an affine space over itself, by the map
:.
If ''W'' is a vector space, then an affine subspace is a subset of ''W'' obtained by translating a linear subspace ''V'' by a fixed vector ; this space is denoted by (it is a
coset of ''V'' in ''W'') and consists of all vectors of the form for An important example is the space of solutions of a system of inhomogeneous linear equations
:
generalizing the homogeneous case
above, which can be found by setting in this equation. The space of solutions is the affine subspace where x is a particular solution of the equation, and ''V'' is the space of solutions of the homogeneous equation (the
nullspace of ''A'').
The set of one-dimensional subspaces of a fixed finite-dimensional vector space ''V'' is known as ''projective space''; it may be used to formalize the idea of
parallel
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 o ...
lines intersecting at infinity.
Grassmannians and
flag manifolds generalize this by parametrizing linear subspaces of fixed dimension ''k'' and
flags of subspaces, respectively.
Related concepts
;Specific vectors in a vector space
*
Zero vector (sometimes also called ''null vector'' and denoted by
), the
additive identity in a vector space. In a
normed vector space, it is the unique vector of norm zero. In a
Euclidean vector space, it is the unique vector of length zero.
*
Basis vector, an element of a given
basis of a vector space.
*
Unit vector, a vector in a normed vector space whose
norm 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 ...
of length one.
*
Isotropic vector or
null vector, 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 ...
.
;Vectors in specific vector spaces
*
Column vector, a matrix with only one column. The column vectors with a fixed number of rows form a vector space.
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Row vector, a matrix with only one row. The row vectors with a fixed number of columns form a vector space.
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Coordinate vector
In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimensio ...
, the
-tuple of the
coordinates
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 sig ...
of a vector on a
basis of elements. For a vector space over a
field , these -tuples form the vector space
(where the operation are pointwise addition and scalar multiplication).
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Displacement vector, 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.
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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 Euclidea ...
or an
affine space.
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Velocity vector, 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.
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Pseudovector, also called ''axial vector''
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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).
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Tangent vector, an element of the
tangent space of a
curve, a
surface or, more generally, a
differential manifold at a given point (these tangent spaces are naturally endowed with a structure of vector space)
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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.
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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 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.
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Four-vector, in the
theory of relativity, a vector in a four-dimensional real vector space called
Minkowski space
See also
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Vector (mathematics and physics), for a list of various kinds of vectors
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Cartesian coordinate system
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Graded vector space
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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 setti ...
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P-vector
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Riesz–Fischer theorem
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Space (mathematics)
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Ordered vector space
Notes
Citations
References
Algebra
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Analysis
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Historical references
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* Peano, G. (1901)
Formulario mathematicovct axiomsvia
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, ...
Further references
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External links
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{{DEFAULTSORT:Vector Space
Concepts in physics
Group theory
Mathematical structures
Vectors (mathematics and physics)