HOME

TheInfoList



OR:

In 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 rel ...
, a vector space (also called a linear space) is a set 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 measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
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 for ...
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 In mathematics, the real coordinate space of dimension , denoted ( ) or is the set of the -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real vecto ...
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 examp ...
, such as
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
s 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 matric ...
, 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 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 ...
. 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 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 variable ...
s are
countably In mathematics, a Set (mathematics), set is countable if either it is finite set, 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 fro ...
infinite-dimensional vector spaces, and many function spaces have the
cardinality of the continuum In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \ma ...
as a dimension. Many vector spaces that are considered in mathematics are also endowed with other structures. 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 additio ...
, which include
field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
s,
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 variable ...
s, associative algebras and
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi iden ...
s. This is also the case of
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 al ...
s, 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 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 "lengt ...
s,
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
s and
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...
s.


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: \vec v. 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  together with two
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...
s 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 ...
'',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
axiom An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy o ...
s 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 measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
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 for ...
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 com ...
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 prese ...
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 ...
of this group. Subtraction of two vectors can be defined as : \mathbf - \mathbf = \mathbf + (-\mathbf). Direct consequences of the axioms include that, for every s\in F and \mathbf v\in V, one has *0\mathbf v = \mathbf 0, *s\mathbf 0=\mathbf 0, *(-1)\mathbf v = -\mathbf v, *s\mathbf v = \mathbf 0 implies s=0 or \mathbf v= \mathbf 0.


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 a_1 \mathbf_1 + a_2 \mathbf_2 + \cdots + a_k \mathbf_k, where a_1, \ldots, a_k\in F and \mathbf_1, \ldots, \mathbf_k\in G. The scalars a_1, \ldots, a_k 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 a ...
: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 In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
of linear subspaces is a linear subspace.'' ;
Linear span 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 characteri ...
: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 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 In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalized ...
, 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 measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
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 (\mathbf_1, \mathbf_2 , \ldots, \mathbf_n) of a vector space of dimension over a field . The definition of a basis implies that every \mathbf v \in V may be written :\mathbf v = a_1\mathbf b_1 +\cdots +a_n \mathbf b_n, with a_1,\dots, a_n in , and that this decomposition is unique. The scalars a_1, \ldots, a_n 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 F^n 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, these reasonings and computations on coordinates can be expressed concisely as reasonings and computations on matrices. Moreover, a
linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coeffici ...
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 matric ...
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
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 si ...
s 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. Mathe ...
and
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 ...
introduced, in 1804, certain operations on points, lines and planes, which are predecessors of vectors. introduced the notion of barycentric coordinates. introduced the notion of a bipoint, i.e., an oriented segment one of whose ends is the origin and the other one a target. Vectors were reconsidered with the presentation of
complex numbers In mathematics, 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 for ...
by Argand and Hamilton and the inception of
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quat ...
s by the latter. They are elements in R2 and R4; 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 pr ...
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 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 a ...
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 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 areas of mathematics, 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 mathem ...
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. This was later formalized by
Banach Banach (pronounced in German, in Slavic Languages, and or in English) is a Jewish surname of Ashkenazi origin believed to stem from the translation of the phrase " son of man", combining the Hebrew word ''ben'' ("son of") and Arameic ''nash ...
and Hilbert, around 1920. At that time,
algebra Algebra () is one of the areas of mathematics, 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 mathem ...
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 ...
began to interact, notably with key concepts such as spaces of ''p''-integrable functions and
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natu ...
s. 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
arrow An arrow is a fin-stabilized projectile launched by a bow. A typical arrow usually consists of a long, stiff, straight shaft with a weighty (and usually sharp and pointed) arrowhead attached to the front end, multiple fin-like stabilizers ...
s in a fixed plane, starting at one fixed point. This is used in physics to describe
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
s or velocities. Given any two such arrows, and , the
parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of eq ...
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 measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
, the arrow that has the same direction as , but is dilated or shrunk by multiplying its length by , is called ''multiplication'' of by . It is denoted . When is negative, is defined as the arrow pointing in the opposite direction instead. The following shows a few examples: if , the resulting vector has the same direction as , but is stretched to the double length of (right image below). Equivalently, is the sum . Moreover, has the opposite direction and the same length as (blue vector pointing down in the right image).


Second example: ordered pairs of numbers

A second key example of a vector space is provided by pairs of real numbers and . (The order of the components and is significant, so such a pair is also called an
ordered pair In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In co ...
.) Such a pair is written as . The sum of two such pairs and multiplication of a pair with a number is defined as follows: : (x_1 , y_1) + (x_2 , y_2) = (x_1 + x_2, y_1 + y_2) and : a(x, y)=(ax, ay) . The first example above reduces to this example, if an arrow is represented by a pair of
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured i ...
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 com ...
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 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 for ...
, that is, numbers that can be written in the form for
real numbers 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 ...
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 a ...
, form a vector space over the reals with the usual addition and multiplication: and for real numbers , , , and . The various axioms of a vector space follow from the fact that the same rules hold for complex number arithmetic. In fact, the example of complex numbers is essentially the same as (that is, it is ''isomorphic'' to) the vector space of ordered pairs of real numbers mentioned above: if we think of the complex number as representing the ordered pair in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...
then we see that the rules for addition and scalar multiplication correspond exactly to those in the earlier example. More generally,
field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
s 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 \mathbf(i\sqrt) is a vector space over .


Function spaces

Functions from any fixed set to a field also form vector spaces, by performing addition and scalar multiplication pointwise. That is, the sum of two functions and is the function given by :, and similarly for multiplication. Such function spaces occur in many geometric situations, when is the
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
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 o ...
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 defined ...
.


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 A = \begin 1 & 3 & 1 \\ 4 & 2 & 2\end is the matrix containing the coefficients of the given equations, is the vector , denotes the matrix product, and is the zero vector. In a similar vein, the solutions of homogeneous ''linear differential equations'' form vector spaces. For example, : yields , where and are arbitrary constants, and is the natural exponential function.


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: :f(\mathbf v + \mathbf w) = f(\mathbf v) + f(\mathbf w) 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 ...
and are identity maps. Equivalently, is both one-to-one (
injective In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contraposi ...
) 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 con ...
'', 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 Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
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 * Fr ...
.


Matrices

''Matrices'' are a useful notion to encode linear maps. They are written as a rectangular array of scalars as in the image at the right. Any -by- matrix gives rise to a linear map from to , by the following :\mathbf x = (x_1, x_2, \ldots, x_n) \mapsto \left(\sum_^n a_x_j, \sum_^n a_x_j, \ldots, \sum_^n a_x_j \right), where \sum denotes
summation In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, m ...
, 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 ...
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 ...
of a
square matrix In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are ofte ...
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 ...
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 **Ger ...
" eigen", 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 lea ...
of the difference (where Id is the identity map . If is finite-dimensional, this can be rephrased using determinants: having eigenvalue is equivalent to :. By spelling out the definition of the determinant, the expression on the left hand side can be seen to be a polynomial function in , called the
characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...
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 b ...
, 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 * Fr ...
.


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 o ...
''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 relat ...
. 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 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 es ...
, 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 t ...
''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 bi ...
the difference of v1 and v2 lies in ''W''.Some authors (such as ) choose to start with this
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
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 lea ...
ker(''f'') of a linear map consists of vectors v that are mapped to 0 in ''W''. The kernel and the
image An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
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 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) ...
) that behaves much like the
category of abelian groups In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab. Properties The zero object of ...
. Because of this, many statements such as the first isomorphism theorem (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, Th ...
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. 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 :a_0 f + a_1 \frac + a_2 \frac + \cdots + a_n \frac = 0, where the coefficients ''a''''i'' are functions in ''x'', too. In the corresponding map :f \mapsto D(f) = \sum_^n a_i \frac, the
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'' \textstyle of a family of vector spaces ''V''''i'' consists of the set of all tuples (, which specify for each index ''i'' in some
index set In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consis ...
''I'' an element v''i'' of ''V''''i''. Addition and scalar multiplication is performed componentwise. A variant of this construction is the ''direct sum'' \bigoplus_ V_i (also called
coproduct 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 cop ...
and denoted \coprod_V_i), where only tuples with finitely many nonzero vectors are allowed. If the index set ''I'' is finite, the two constructions agree, but in general they are different.


Tensor product

The ''tensor product'' , or simply , of two vector spaces ''V'' and ''W'' is one of the central notions of multilinear algebra which deals with extending notions such as linear maps to several variables. A map is called bilinear 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 tens ...
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 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 ...
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 defined ...
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, which relies on the ability to express a function as a difference of two positive functions :f = f^ - f^, where f^ denotes the positive part of f and f^ the negative part.


Normed vector spaces and inner product spaces

"Measuring" vectors is done by specifying a norm, 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 , \mathbf v, and respectively. The datum of an inner product entails that lengths of vectors can be defined too, by defining the associated norm Vector spaces endowed with such data are known as ''normed vector spaces'' and ''inner product spaces'', respectively. Coordinate space ''F''''n'' can be equipped with the standard
dot product 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 ...
: :\lang \mathbf x , \mathbf y \rang = \mathbf x \cdot \mathbf y = x_1 y_1 + \cdots + x_n y_n. In R2, this reflects the common notion of the angle between two vectors x and y, by the
law of cosines 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 stat ...
: :\mathbf x \cdot \mathbf y = \cos\left(\angle (\mathbf x, \mathbf y)\right) \cdot , \mathbf x, \cdot , \mathbf y, . Because of this, two vectors satisfying \lang \mathbf x , \mathbf y \rang = 0 are called
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
. 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 ...
: R4 endowed with the Lorentz product :\lang \mathbf x , \mathbf y \rang = x_1 y_1 + x_2 y_2 + x_3 y_3 - x_4 y_4. In contrast to the standard dot product, it is not positive definite: \lang \mathbf x , \mathbf x \rang also takes negative values, for example, for \mathbf x = (0, 0, 0, 1). 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 ho ...
, 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 va ...
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, 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 :\sum_^ f_i 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 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 arbitra ...
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 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 :\lim_ , \mathbf v_n - \mathbf v, = 0. Banach and Hilbert spaces are complete topological vector spaces whose topologies are given, respectively, by a norm and an inner product. Their study—a key piece of
functional analysis 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 ...
—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 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 \ell^p consisting of infinite vectors with real entries \mathbf = \left(x_1, x_2, \ldots, x_n, \ldots\right) whose p-norm (1\leq\leq\infty) given by :\left\, \mathbf\right\, _p := \left(\sum_i \left\vert x_i\right\vert^p\right)^\frac for p < \infty   and   \left\, \mathbf x\right\, _ := \sup_i \left, x_i \. The topologies on the infinite-dimensional space \ell^p are inequivalent for different p. For example, the sequence of vectors \mathbf_n = \left(2^, 2^, \ldots, 2^, 0, 0, \ldots\right), in which the first 2^n components are 2^ and the following ones are 0, converges to the zero vector for p = \infty, but does not for p = 1: : \left\Vert\mathbf_\right\Vert_ = \sup (2^, 0) = 2^ \rightarrow 0 , but \left\Vert\mathbf_\right\Vert_ = \sum_^ 2^ = 2^n \cdot 2^ = 1. More generally than sequences of real numbers, functions f\colon \Omega \to \mathbb are endowed with a norm that replaces the above sum by the
Lebesgue integral 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 ...
: \left\Vert\right\Vert_ := \left( \int_ \left\vert\left(x\right)\right\vert^ \, \right)^\frac. 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 ...
s 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 *Do ...
\Omega (for example an interval) satisfying \left\Vert\right\Vert_ < \infty, and equipped with this norm are called Lebesgue spaces, denoted L^\left(\Omega\right).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, bu ...
for \left\Vert\right\Vert_ \leq \left\Vert\right\Vert_ + \left\Vert\right\Vert_ is provided by the Minkowski inequality. For technical reasons, in the context of functions one has to identify functions that agree
almost everywhere 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 t ...
to get a norm, and not only a seminorm.
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 ...
instead, the space is ''not'' complete, which may be seen as a justification for Lebesgue's integration theory. "Many functions in L^ of Lebesgue measure, being unbounded, cannot be integrated with the classical Riemann integral. So spaces of Riemann integrable functions would not be complete in the L^ norm, and the orthogonal decomposition would not apply to them. This shows one of the advantages of Lebesgue integration.", ) Concretely this means that for any sequence of Lebesgue-integrable functions   f_, f_, \ldots, f_, \ldots   with \left\Vert_\right\Vert_<\infty, satisfying the condition :\lim_\int_ \left\vert_(x) - _(x)\right\vert^ \, = 0 there exists a function \left(x\right) belonging to the vector space L^\left(\Omega\right) such that :\lim_\int_ \left\vert\left(x\right) - _\left(x\right)\right\vert^ \, = 0. Imposing boundedness conditions not only on the function, but also on its
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 : \langle f\ , \ g \rangle = \int_\Omega f(x) \overline \, dx, where \overline denotes the
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 - ...
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 si ...
, 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 ...
s 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 * Fr ...
. 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 In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally ...
.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 componen ...
.
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 Euclidean sp ...
. 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, a ...
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 chemistry, q ...
describes the change of physical properties in time by means of a
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
, 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 ma ...
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 denot ...
s of a certain (linear) differential operator and the associated wavefunctions are called
eigenstate In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...
s. 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 b ...
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 ...
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 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. Promi ...
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 o ...
and
associative In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
. 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 In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
of two matrices, and , endowed with the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and i ...
. 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 bein ...
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 tens ...
s :, where the
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathemati ...
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 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 ...
. 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 In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organisin ...
. 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 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 In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and A ...
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 infi ...
, 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 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 ...
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 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 sign ...
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, ...
, 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 ...
''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 geom ...
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 f ...
s: R, C, the
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quat ...
s H and the octonions 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. This ...
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''/''b ...
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 com ...
) Z/2Z shows; those modules that do (including all vector spaces) are known as
free module In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a fie ...
s. Nevertheless, a vector space can be compactly defined as a module 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 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 color ...
allows the development of concepts such as
locally free module In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. Various equivalent characterizati ...
s, 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 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 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 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 \mathbf), the additive identity 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 "leng ...
, 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 Euclidea ...
, 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 componen ...
, an element of a given basis 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 ve ...
, 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 In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...
, a non-zero vector for which the form is zero. If a null vector exists, the quadratic form is said an isotropic quadratic form. ;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, ...
, a matrix with only one column. The column vectors with a fixed number of rows form a vector space. * Row vector, a matrix with only one row. The row vectors with a fixed number of columns form a vector space. *
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 si ...
of a vector on a basis of elements. For a vector space over a field , these -tuples form the vector space F^n (where the operation are pointwise addition and scalar multiplication). * 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 Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
. *
Position vector In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point ''P'' in space in relation to an arbitrary reference origin ''O''. Usually denoted x, r, or ...
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 sp ...
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 relat ...
. * 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. * Pseudovector, also called ''axial vector'' * Covector, an element of the
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...
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, an element of the tangent space 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 or, more generally, a differential manifold at a given point (these tangent spaces are naturally endowed with a structure of vector space) *
Normal vector In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve ...
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 functi ...
. In a Euclidean space the gradient gives the magnitude and direction of maximum increase of a
scalar field In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical quantit ...
. The gradient is a covector that is normal to a level curve. *
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 ...
, 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 ...


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 physi ...
, for a list of various kinds of vectors *
Cartesian coordinate system A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured ...
*
Graded vector space In mathematics, a graded vector space is a vector space that has the extra structure of a '' grading'' or a ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces. Integer gradation Let \mathbb be ...
*
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 sett ...
* P-vector * Riesz–Fischer theorem * Space (mathematics) * Ordered vector space


Notes


Citations


References


Algebra

* * * * * * * * * *


Analysis

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


Historical references

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

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


External links

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