View source for Vector space

{{Short description|Algebraic structure in linear algebra}}
{{Distinguish|Vector field}}
{{redirect|Linear space |a structure in incidence geometry|Linear space (geometry)}}
[[File: Vector add scale.svg|200px|thumb|right|Vector addition and scalar multiplication: a vector {{math|'''v'''}} (blue) is added to another vector {{math| '''w'''}} (red, upper illustration). Below, {{math| '''w'''}} is stretched by a factor of 2, yielding the sum {{math|'''v''' + 2'''w'''}}.]]

In [[mathematics]] and [[physics]], a '''vector space''' (also called a '''linear space''') is a [[set (mathematics)|set]] whose elements, often called [[vector (mathematics and physics)|''vectors'']], can be added together and multiplied ("scaled") by numbers called [[scalar (mathematics)|''scalars'']]. The operations of vector addition and [[scalar multiplication]] must satisfy certain requirements, called ''vector [[axiom]]s''. '''Real vector spaces''' and '''complex vector spaces''' are kinds of vector spaces based on different kinds of scalars: [[real numbers]] and [[complex numbers]]. Scalars can also be, more generally, elements of any [[field (mathematics)|field]].

Vector spaces generalize [[Euclidean vector]]s, which allow modeling of [[Physical quantity|physical quantities]] (such as [[force]]s and [[velocity]]) that have not only a [[Magnitude (mathematics)|magnitude]], but also a [[Orientation (geometry)|direction]]. The concept of vector spaces is fundamental for [[linear algebra]], together with the concept of [[matrix (mathematics)|matrices]], which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying [[systems of linear equations]].

Vector spaces are characterized by their [[dimension (vector space)|dimension]], which, roughly speaking, specifies the number of independent directions in the space. This means that, for two vector spaces over a given field and with the same dimension, the properties that depend only on the 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]]. Otherwise, it is ''infinite-dimensional'', and its dimension is an [[Transfinite number|infinite cardinal]]. Finite-dimensional vector spaces occur naturally in [[geometry]] and related areas. Infinite-dimensional vector spaces occur in many areas of mathematics. For example, [[polynomial ring]]s are [[countably infinite|countably]] infinite-dimensional vector spaces, and many [[function space]]s have the [[cardinality of the continuum]] as a dimension.

Many vector spaces that are considered in mathematics are also endowed with other [[mathematical structure|structures]]. This is the case of [[algebra over a field|algebras]], which include [[field extension]]s, polynomial rings, [[associative algebra]]s and [[Lie algebra]]s. This is also the case of [[topological vector space]]s, which include function spaces, [[inner product space]]s, [[Normed vector space|normed spaces]], [[Hilbert space]]s and [[Banach space]]s.

<noinclude>{{Algebraic structures |module}}</noinclude>

==Definition and basic properties==
In this article, vectors are represented in boldface to distinguish them from scalars.<ref group=nb>It is also common, especially in physics, to denote vectors with an arrow on top: <math>\vec v.</math> It is also common, especially in higher mathematics, to not use any typographical method for distinguishing vectors from other mathematical objects.</ref>{{sfn|Lang|2002}}

A vector space over a [[field (mathematics)|field]] {{mvar|F}} is a non-empty [[set (mathematics)|set]]&nbsp;{{mvar|V}} together with a [[binary operation]] and a [[binary function]] that satisfy the eight [[axiom]]s listed below. In this context, the elements of {{mvar|V}} are commonly called ''vectors'', and the elements of&nbsp;{{mvar|F}} are called ''scalars''.{{sfn|Brown|1991|p=86}}

* The binary operation, called ''vector addition'' or simply ''addition'' assigns to any two vectors&nbsp;{{math|'''v'''}} and {{math|'''w'''}} in {{mvar|V}} a third vector in {{mvar|V}} which is commonly written as {{math|'''v''' + '''w'''}}, and called the ''sum'' of these two vectors.

* The binary function, called ''[[scalar multiplication]]'', assigns to any scalar&nbsp;{{mvar|a}} in {{mvar|F}} and any vector&nbsp;{{math|'''v'''}} in {{mvar|V}} another vector in {{mvar|V}}, which is denoted&nbsp;{{math|''a'''''v'''}}.<ref group=nb>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]]s. Scalar multiplication is the 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.</ref>

To have a vector space, the eight following [[axiom]]s must be satisfied for every {{math|'''u'''}}, {{math|'''v'''}} and {{math|'''w'''}} in {{mvar|V}}, and {{mvar|a}} and {{mvar|b}} in {{mvar|F}}.{{sfn|Roman|2005|loc=ch. 1, p. 27}}



{| border="0" class="wikitable" style="max-width:50em"
|-
! Axiom
! Statement
|-
| [[Associativity]] of vector addition || {{math|1='''u''' + ('''v''' + '''w''') = ('''u''' + '''v''') + '''w'''}}
|-
| [[Commutativity]] of vector addition || {{math|1='''u''' + '''v''' = '''v''' + '''u'''}}
|-
| [[Identity element]] of vector addition || There exists an element {{math|'''0''' ∈ ''V''}}, called the ''[[zero vector]]'', such that {{math|1='''v''' + '''0''' = '''v'''}} for all {{math|'''v''' ∈ ''V''}}.
|-
| [[Inverse element]]s of vector addition || For every {{math|'''v''' ∈ ''V''}}, there exists an element {{math|−'''v''' ∈ ''V''}}, called the ''[[additive inverse]]'' of {{math|'''v'''}}, such that {{math|1='''v''' + (−'''v''') = '''0'''}}.
|-
| Compatibility of scalar multiplication with field multiplication || {{math|1=''a''(''b'''''v''') = (''ab'')'''v'''}} <ref group=nb>This axiom is not an [[associative property]], since it refers to two different operations, scalar multiplication and field multiplication. So, it is independent from the associativity of field multiplication, which is assumed by field axioms.</ref>
|-
| Identity element of scalar multiplication ||  {{math|1=1'''v''' = '''v'''}}, where {{math|1}} denotes the [[multiplicative identity]] in {{mvar|F}}.
|-
| [[Distributivity]] of scalar multiplication with respect to vector addition&emsp;&emsp;|| {{math|1=''a''('''u''' + '''v''') = ''a'''''u''' + ''a'''''v'''}}
|-
| Distributivity of scalar multiplication with respect to field addition || {{math|1=(''a'' + ''b'')'''v''' = ''a'''''v''' + ''b'''''v'''}}
|}

When the scalar field is the [[real number]]s, the vector space is called a ''real vector space'', and when the scalar field is the [[complex number]]s, the vector space is called a ''complex vector space''.{{sfn|Brown|1991|p=87}} These two cases are the most common ones, but vector spaces with scalars in an arbitrary field {{mvar|F}} are also commonly considered. Such a vector space is called an {{nowrap|1={{mvar|F}}-}}''vector space'' or a ''vector space over {{mvar|F}}''.{{sfnm
 | 1a1 = Springer | 1y = 2000 | 1p = [https://books.google.com/books?id=Ces-AAAAQBAJ&pg=PA185 185]
 | 2a1 =  Brown | 2y = 1991 | 2p = 86
}}

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]] under addition, and the four remaining axioms (related to the scalar multiplication) say that this operation defines a [[ring homomorphism]] from the field {{math|''F''}} into the [[endomorphism ring]] of this group.{{sfn|Atiyah|Macdonald|1969|p=17}}

Subtraction of two vectors can be defined as
<math display=block>\mathbf{v} - \mathbf{w} = \mathbf{v} + (-\mathbf{w}).</math>

Direct consequences of the axioms include that, for every <math>s\in F</math> and <math>\mathbf v\in V,</math> one has
*<math>0\mathbf v = \mathbf 0,</math>
*<math>s\mathbf 0=\mathbf 0,</math>
*<math>(-1)\mathbf v = -\mathbf v,</math>
*<math>s\mathbf v = \mathbf 0</math> implies <math>s=0</math> or <math>\mathbf v= \mathbf 0.</math>

Even more concisely, a vector space is a [[Module (mathematics)|module]] over a [[Field (mathematics)|field]].{{sfn|Bourbaki|1998|p=|loc=§1.1, Definition 2}}

== Bases, vector coordinates, and subspaces ==
[[File:Vector components and base change.svg|A vector {{math|'''v'''}} in {{math|'''R'''<sup>2</sup>}} (blue) expressed in terms of different bases: using the [[standard basis]] of {{Math|'''R'''<sup>2</sup>}}: {{math|1='''v''' = ''x'''''e'''<sub>1</sub> + ''y'''''e'''<sub>2</sub>}} (black), and using a different, non-[[orthogonal vector|orthogonal]] basis: {{math|1='''v''' = '''f'''<sub>1</sub> + '''f'''<sub>2</sub>}} (red).|thumb|200px]]

;[[Linear combination]]
: Given a set {{mvar|G}} of elements of a {{mvar|F}}-vector space {{mvar|V}}, a linear combination of elements of {{mvar|G}} is an element of {{mvar|V}} of the form <math display=block> a_1 \mathbf{g}_1 + a_2 \mathbf{g}_2 + \cdots + a_k \mathbf{g}_k,</math> where <math>a_1, \ldots, a_k\in F</math> and <math>\mathbf{g}_1, \ldots, \mathbf{g}_k\in G.</math> The scalars <math>a_1, \ldots, a_k</math> are called the ''coefficients'' of the linear combination.{{sfn|Brown|1991|p=94}}
;[[Linear independence]]
:The elements of a subset {{mvar|G}} of a {{mvar|F}}-vector space {{mvar|V}} are said to be ''linearly independent'' if no element of {{mvar|G}} can be written as a linear combination of the other elements of {{mvar|G}}. Equivalently, they are linearly independent if two linear combinations of elements of {{mvar|G}} define the same element of {{mvar|V}} 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.{{sfn|Brown|1991|pages=99-101}}
;[[Linear subspace]]
:A ''linear subspace'' or ''vector subspace'' {{mvar|W}} of a vector space {{mvar|V}} is a non-empty subset of {{mvar|V}} that is [[closure (mathematics)|closed]] under vector addition and scalar multiplication; that is, the sum of two elements of {{mvar|W}} and the product of an element of {{mvar|W}} by a scalar belong to {{mvar|W}}.{{sfn|Brown|1991|p=92}} This implies that every linear combination of elements of {{mvar|W}} belongs to {{mvar|W}}. 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.{{sfn|Stoll|Wong|1968|p=[https://books.google.com/books?id=gLbiBQAAQBAJ&pg=PA14 14]}}<br>The closure property also implies that ''every [[intersection (set theory)|intersection]] of linear subspaces is a linear subspace.''{{sfn|Stoll|Wong|1968|p=[https://books.google.com/books?id=gLbiBQAAQBAJ&pg=PA14 14]}}
;[[Linear span]]
:Given a subset {{mvar|G}} of a vector space {{mvar|V}}, the ''linear span'' or simply the ''span'' of {{mvar|G}} is the smallest linear subspace of {{mvar|V}} that contains {{mvar|G}}, in the sense that it is the intersection of all linear subspaces that contain {{mvar|G}}. The span of {{mvar|G}} is also the set of all linear combinations of elements of {{mvar|G}}.<br> If {{mvar|W}} is the span of {{mvar|G}}, one says that {{mvar|G}} ''spans'' or ''generates'' {{mvar|W}}, and that {{mvar|G}} is a ''[[spanning set]]'' or a ''generating set'' of {{mvar|W}}.{{sfn|Roman|2005|pages=41-42}}
;[[Basis (linear algebra)|Basis]] and [[dimension (vector space)|dimension]]
:A subset of a vector space is a ''basis'' if its elements are linearly independent and span the vector space.{{sfnm
 | 1a1 = Lang | 1y = 1987 | 1p = 10–11
 | 2a1 = Anton | 2a2 = Rorres | 2y = 2010 | 2p = [https://books.google.com/books?id=1PJ-WHepeBsC&pg=PA212 212]}} Every vector space has at least one basis, or many in general (see {{slink|Basis (linear algebra)|Proof that every vector space has a basis}}).{{sfn|Blass|1984}} Moreover, all bases of a vector space have the same [[cardinality]], which is called the ''dimension'' of the vector space (see [[Dimension theorem for vector spaces]]).{{sfn|Joshi|1989|p=[https://books.google.com/books?id=RM1D3mFw2u0C&pg=PA450 450]}} This is a fundamental property of vector spaces, which is detailed in the remainder of the section.

<span id=label1>''Bases''</span> 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]], depends on the [[axiom of choice]]. It follows that, in general, no base can be explicitly described.{{sfn|Heil|2011|p=[https://books.google.com/books?id=prfuUT0Sw-AC&pg=PA126 126]}} For example, the [[real number]]s form an infinite-dimensional vector space over the [[rational number]]s, for which no specific basis is known. 

Consider a basis <math>(\mathbf{b}_1, \mathbf{b}_2 , \ldots, \mathbf{b}_n)</math> of a vector space {{mvar|V}} of dimension {{mvar|n}} over a field {{mvar|F}}. The definition of a basis implies that every <math>\mathbf v \in V</math> may be written 
<math display=block>\mathbf v = a_1 \mathbf b_1 + \cdots + a_n \mathbf b_n,</math> 
with <math>a_1,\dots, a_n</math> in {{mvar|F}}, and that this decomposition is unique. The scalars <math>a_1, \ldots, a_n</math> are called the ''coordinates'' of {{math|'''v'''}} on the basis. They are also said to be the ''coefficients'' of the decomposition of {{math|'''v'''}} on the basis. One also says that the {{mvar|n}}-[[tuple]] of the coordinates is the [[coordinate vector]] of {{math|'''v'''}} on the basis, since the set <math>F^n</math> of the {{mvar|n}}-tuples of elements of {{mvar|F}} is a vector space for [[componentwise operation|componentwise]] addition and scalar multiplication, whose dimension is {{mvar|n}}.

The [[one-to-one correspondence]] between vectors and their coordinate vectors maps vector addition to vector addition and scalar multiplication to scalar multiplication. It is thus a [[vector space isomorphism]], which allows translating reasonings and computations on vectors into reasonings and computations on their coordinates.{{sfn|Halmos|1948|p=[https://books.google.com/books?id=1hzYCwAAQBAJ&pg=PA12 12]}}

==History==

Vector spaces stem from [[affine geometry]], via the introduction of [[coordinate]]s in the plane or three-dimensional space. Around 1636, French mathematicians [[René Descartes]] and [[Pierre de Fermat]] founded [[analytic geometry]] by identifying solutions to an equation of two variables with points on a plane [[curve]].{{sfn|Bourbaki|1969|loc = ch. "Algèbre linéaire et algèbre multilinéaire", pp. 78–91}} To achieve geometric solutions without using coordinates, [[Bernhard Bolzano|Bolzano]] introduced, in 1804, certain operations on points, lines, and planes, which are predecessors of vectors.{{sfn|Bolzano|1804}} {{harvtxt|Möbius|1827}} introduced the notion of [[Barycentric coordinates (mathematics)|barycentric coordinates]].{{sfn|Möbius|1827}} {{harvtxt|Bellavitis|1833}} introduced an [[equivalence relation]] on directed line segments that share the same length and direction which he called [[equipollence (geometry)|equipollence]].{{sfn|Bellavitis|1833}} A [[Euclidean vector]] is then an [[equivalence class]] of that relation.{{sfn|Dorier|1995}} 

Vectors were reconsidered with the presentation of [[complex numbers]] by [[Jean-Robert Argand|Argand]] and [[William Rowan Hamilton|Hamilton]] and the inception of [[quaternion]]s by the latter.{{sfn|Hamilton|1853}} They are elements in '''R'''<sup>2</sup> and '''R'''<sup>4</sup>; treating them using [[linear combination]]s goes back to [[Laguerre]] in 1867, who also defined [[system of linear equations|systems of linear equations]].

In 1857, [[Arthur Cayley|Cayley]] introduced the [[matrix notation]] which allows for harmonization and simplification of [[linear map]]s. Around the same time, [[Grassmann]] studied the barycentric calculus initiated by Möbius. He envisaged sets of abstract objects endowed with operations.{{sfn|Grassmann|2000}} In his work, the concepts of [[linear independence]] and [[dimension]], as well as [[scalar product]]s are present. Grassmann's 1844 work exceeds the framework of vector spaces as well since his considering multiplication led him to what are today called [[Algebras over a field|algebras]]. Italian mathematician [[Giuseppe Peano|Peano]] was the first to give the modern definition of vector spaces and linear maps in 1888,{{sfn|Peano|1888|loc = ch. IX}} although he called them "linear systems".{{sfn|Guo|2021}} Peano's axiomatization allowed for vector spaces with infinite dimension, but Peano did not develop that theory further. In 1897, [[Salvatore Pincherle]] adopted Peano's axioms and made initial inroads into the theory of infinite-dimensional vector spaces.{{sfn|Moore|1995|pages=268-271}} 

An important development of vector spaces is due to the construction of [[function spaces]] by [[Henri Lebesgue]]. This was later formalized by [[Stefan Banach|Banach]] and [[David Hilbert|Hilbert]], around 1920.{{sfn|Banach|1922}} At that time, [[algebra]] and the new field of [[functional analysis]] began to interact, notably with key concepts such as [[Lp space|spaces of ''p''-integrable functions]] and [[Hilbert space]]s.{{sfnm
 | 1a1 = Dorier | 1y = 1995
 | 2a1 = Moore | 2y = 1995
}}

==Examples==
{{main|Examples of vector spaces}}

===Arrows in the plane===
{{multiple image
 | total_width = 200
 | direction = vertical
 | image1 = Vector addition3.svg
 | caption1 = Vector addition: the sum {{math|'''v''' + '''w'''}} (black) of the vectors {{math|'''v'''}} (blue) and {{math|'''w'''}} (red) is shown.
 | image2 = Scalar multiplication.svg
 | caption2 = Scalar multiplication: the multiples {{math|−'''v'''}} and {{math|2'''w'''}} are shown.
}}
The first example of a vector space consists of [[arrow (symbol)|arrow]]s in a fixed [[plane (geometry)|plane]], starting at one fixed point. This is used in physics to describe [[force]]s or [[velocity|velocities]].{{sfn|Kreyszig|2020|p=[https://books.google.com/books?id=w4T3DwAAQBAJ&pg=PA355 355]}} Given any two such arrows, {{math|'''v'''}} and {{math|'''w'''}}, the [[parallelogram]] spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the ''sum'' of the two arrows, and is denoted {{math|'''v''' + '''w'''}}. 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]] {{math|''a''}}, the arrow that has the same direction as {{math|'''v'''}}, but is dilated or shrunk by multiplying its length by {{math|''a''}}, is called ''multiplication'' of {{math|'''v'''}} by {{math|''a''}}. It is denoted {{math|''a'''''v'''}}. When {{math|''a''}} is negative, {{math|''a'''''v'''}} is defined as the arrow pointing in the opposite direction instead.{{sfn|Kreyszig|2020|p=[https://books.google.com/books?id=w4T3DwAAQBAJ&pg=PA358 358&ndash;359]}}

The following shows a few examples: if {{math|1=''a'' = 2}}, the resulting vector {{math|''a'''''w'''}} has the same direction as {{math|'''w'''}}, but is stretched to the double length of {{math|'''w'''}} (the second image). Equivalently, {{math|2'''w'''}} is the sum {{math|'''w''' + '''w'''}}. Moreover, {{math|1=(−1)'''v''' = −'''v'''}} has the opposite direction and the same length as {{math|'''v'''}} (blue vector pointing down in the second image).

===Ordered pairs of numbers===
A second key example of a vector space is provided by pairs of real numbers {{mvar|x}} and {{mvar|y}}. The order of the components {{mvar|x}} and {{mvar|y}} is significant, so such a pair is also called an [[ordered pair]]. Such a pair is written as {{math|(''x'', ''y'')}}. The sum of two such pairs and the multiplication of a pair with a number is defined as follows:{{sfn|Jain|2001|p=[https://books.google.com/books?id=-lzAee3uQtIC&pg=PA11 11]}}
<math display="block">
 \begin{align}
 (x_1 , y_1) + (x_2 , y_2) &= (x_1 + x_2, y_1 + y_2), \\
 a(x, y) &= (ax, ay).
\end{align}
</math>

The first example above reduces to this example if an arrow is represented by a pair of [[Cartesian coordinates]] of its endpoint.

===Coordinate space===
The simplest example of a vector space over a field {{math|''F''}} is the field {{math|''F''}} itself with its addition viewed as vector addition and its multiplication viewed as scalar multiplication. More generally, all [[tuple|{{math|''n''}}-tuples]] (sequences of length {{math|''n''}}) 
<math display=block>(a_1, a_2, \dots, a_n)</math>
of elements {{math|''a''<sub>''i''</sub>}} of {{math|''F''}} form a vector space that is usually denoted {{math|''F''<sup>''n''</sup>}} and called a '''coordinate space'''.{{sfn|Lang|1987|loc = ch. I.1}} 
The case {{math|1=''n'' = 1}} is the above-mentioned simplest example, in which the field {{math|''F''}} is also regarded as a vector space over itself. The case {{math|1=''F'' = '''R'''}} and {{math|1=''n'' = 2}} (so '''R'''<sup>2</sup>) reduces to the previous example.

===Complex numbers and other field extensions===
The set of [[complex numbers]] {{math|'''C'''}}, numbers that can be written in the form {{math|1=''x'' + ''iy''}} for [[real numbers]] {{math|''x''}} and {{math|''y''}} where {{math|''i''}} is the [[imaginary unit]], form a vector space over the reals with the usual addition and multiplication: {{math|1=(''x'' + ''iy'') + (''a'' + ''ib'') = (''x'' + ''a'') + ''i''(''y'' + ''b'')}} and {{math|1=''c'' ⋅ (''x'' + ''iy'') = (''c'' ⋅ ''x'') + ''i''(''c'' ⋅ ''y'')}} for real numbers {{math|''x''}}, {{math|''y''}}, {{math|''a''}}, {{math|''b''}} and {{math|''c''}}.  The various axioms of a vector space follow from the fact that the same rules hold for complex number arithmetic. 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 {{math|''x'' + ''i'' ''y''}} as representing the ordered pair {{math|(''x'', ''y'')}} in the [[complex plane]] then we see that the rules for addition and scalar multiplication correspond exactly to those in the earlier example.

More generally, [[field extension]]s provide another class of examples of vector spaces, particularly in algebra and [[algebraic number theory]]: a field {{math|''F''}} containing a [[Field extension|smaller field]] {{math|''E''}} is an {{math|''E''}}-vector space, by the given multiplication and addition operations of {{math|''F''}}.{{sfn|Lang|2002|loc = ch. V.1}} For example, the complex numbers are a vector space over {{math|'''R'''}}, and the field extension <math>\mathbf{Q}(i\sqrt{5})</math> is a vector space over {{math|'''Q'''}}.  <!--A particularly interesting type of field extension in [[number theory]] is {{math|'''Q'''(''α'')}}, the extension of the rational numbers {{math|'''Q'''}} by a fixed complex number {{math|''α''}}. {{math|'''Q'''(''α'')}} is the smallest field containing the rationals and a fixed complex number ''α''. Its dimension as a vector space over {{math|'''Q'''}} depends on the choice of {{math|''α''}}.-->

===Function spaces===
{{Main|Function space}}
[[File:Example for addition of functions.svg|thumb|Addition of functions: the sum of the sine and the exponential function is <math>\sin+\exp:\R\to\R</math> with <math>(\sin+\exp)(x)=\sin(x)+\exp(x)</math>.]]

Functions from any fixed set {{math|Ω}} to a field {{math|''F''}} also form vector spaces, by performing addition and scalar multiplication pointwise. That is, the sum of two functions {{math|''f''}} and {{math|''g''}} is the function <math>(f + g)</math> given by
<math display=block>(f + g)(w) = f(w) + g(w),</math>
and similarly for multiplication. Such function spaces occur in many geometric situations, when {{math|Ω}} is the [[real line]] or an [[interval (mathematics)|interval]], or other [[subset]]s of {{math|'''R'''}}. Many notions in topology and analysis, such as [[continuous function|continuity]], [[integral|integrability]] or [[differentiability]] are well-behaved with respect to linearity: sums and scalar multiples of functions possessing such a property still have that property.{{sfn|Lang|1993|loc = ch. XII.3., p. 335}} Therefore, the set of such functions are vector spaces, whose study belongs to [[functional analysis]].

===Linear equations===
{{Main|Linear equation|Linear differential equation|Systems of linear equations}}
Systems of [[homogeneous linear equation]]s are closely tied to vector spaces.{{sfn|Lang|1987|loc = ch. VI.3.}} For example, the solutions of 
<math display=block>\begin{alignat}{9}
  && a \,&&+\, 3 b \,&\, + &\,   & c & \,= 0 \\
4 && a \,&&+\, 2 b \,&\, + &\, 2 & c & \,= 0 \\
\end{alignat}</math>
are given by triples with arbitrary <math>a,</math> <math>b = a / 2,</math> and <math>c = -5 a / 2.</math> 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. [[matrix (mathematics)|Matrices]] can be used to condense multiple linear equations as above into one vector equation, namely
<div id=equation3><math display=block>A \mathbf{x} = \mathbf{0},</math></div>
where <math>A = \begin{bmatrix}
1 & 3 & 1 \\
4 & 2 & 2\end{bmatrix}</math> is the matrix containing the coefficients of the given equations, <math>\mathbf{x}</math> is the vector <math>(a, b, c),</math> <math>A \mathbf{x}</math> denotes the [[matrix product]], and <math>\mathbf{0} = (0, 0)</math> is the zero vector. In a similar vein, the solutions of homogeneous ''linear differential equations'' form vector spaces. For example,
<div id=equation1><math display=block>f^{\prime\prime}(x) + 2 f^\prime(x) + f(x) = 0</math></div>
yields <math>f(x) = a e^{-x} + b x e^{-x},</math> where <math>a</math> and <math>b</math> are arbitrary constants, and <math>e^x</math> is the [[natural exponential function]].

==Linear maps and matrices==
{{Main|Linear map}}
The relation of two vector spaces can be expressed by ''linear map'' or ''[[linear transformation]]''. They are [[function (mathematics)|functions]] that reflect the vector space structure, that is, they preserve sums and scalar multiplication:
<math display=block>
\begin{align}
 f(\mathbf{v} + \mathbf{w}) &= f(\mathbf{v}) + f(\mathbf{w}), \\ 
 f(a \cdot \mathbf{v}) &= a \cdot f(\mathbf{v})
\end{align}
</math> 
for all <math>\mathbf{v}</math> and <math>\mathbf{w}</math> in <math>V,</math> all <math>a</math> in <math>F.</math>{{sfn|Roman|2005|loc=ch. 2, p. 45}}

An ''[[isomorphism]]'' is a linear map {{math|''f'' : ''V'' → ''W''}} such that there exists an [[inverse map]] {{math|''g'' : ''W'' → ''V''}}, which is a map such that the two possible [[function composition|compositions]] {{math|''f'' ∘ ''g'' : ''W'' → ''W''}} and {{math|''g'' ∘ ''f'' : ''V'' → ''V''}} are [[Identity function|identity maps]]. Equivalently, {{math|''f''}} is both one-to-one ([[injective]]) and onto ([[surjective]]).{{sfn|Lang|1987|loc=ch. IV.4, Corollary, p. 106}} If there exists an isomorphism between {{math|''V''}} and {{math|''W''}}, the two spaces are said to be ''isomorphic''; they are then essentially identical as vector spaces, since all identities holding in {{math|''V''}} are, via {{math|''f''}}, transported to similar ones in {{math|''W''}}, and vice versa via {{math|''g''}}.

[[File:Vector components.svg|180px|right|thumb|Describing an arrow vector {{math|'''v'''}} by its coordinates {{math|''x''}} and {{math|''y''}} yields an isomorphism of vector spaces.]]
For example, the arrows in the plane and the ordered pairs of numbers vector spaces in the introduction above (see {{slink||Examples}}) are isomorphic: a planar arrow {{math|'''v'''}} departing at the [[origin (mathematics)|origin]] of some (fixed) [[coordinate system]] can be expressed as an ordered pair by considering the {{math|''x''}}- and {{math|''y''}}-component of the arrow, as shown in the image at the right. Conversely, given a pair {{math|(''x'', ''y'')}}, the arrow going by {{math|''x''}} to the right (or to the left, if {{math|''x''}} is negative), and {{math|''y''}} up (down, if {{math|''y''}} is negative) turns back the arrow {{math|'''v'''}}.{{sfn|Nicholson|2018|loc=ch. 7.3}}

Linear maps {{math|''V'' → ''W''}} between two vector spaces form a vector space {{math|Hom<sub>''F''</sub>(''V'', ''W'')}}, also denoted {{math|L(''V'', ''W'')}}, or {{math|𝓛(''V'', ''W'')}}.{{sfn|Lang|1987|loc=Example IV.2.6}} The space of linear maps from {{math|''V''}} to {{math|''F''}} is called the ''[[dual vector space]]'', denoted {{math|''V''<sup>∗</sup>}}.{{sfn|Lang|1987|loc=ch. VI.6}} Via the injective [[natural (category theory)|natural]] map {{math|''V'' → ''V''<sup>∗∗</sup>}}, any vector space can be embedded into its ''bidual''; the map is an isomorphism if and only if the space is finite-dimensional.{{sfn|Halmos|1974|loc=p. 28, Ex. 9}}

Once a basis of {{math|''V''}} is chosen, linear maps {{math|''f'' : ''V'' → ''W''}} are completely determined by specifying the images of the basis vectors, because any element of {{math|''V''}} is expressed uniquely as a linear combination of them.{{sfn|Lang|1987|loc=Theorem IV.2.1, p. 95}} If {{math|1=dim ''V'' = dim ''W''}}, a [[bijection|1-to-1 correspondence]] between fixed bases of {{math|''V''}} and {{math|''W''}} gives rise to a linear map that maps any basis element of {{math|''V''}} to the corresponding basis element of {{math|''W''}}. It is an isomorphism, by its very definition.{{sfn|Roman|2005|loc=Th. 2.5 and 2.6, p. 49}} Therefore, two vector spaces over a given field are isomorphic if their dimensions agree and vice versa. Another way to express this is that any vector space over a given field is ''completely classified'' ([[up to]] isomorphism) by its dimension, a single number. In particular, any ''n''-dimensional {{math|''F''}}-vector space {{math|''V''}} is isomorphic to {{math|''F''<sup>''n''</sup>}}. However, there is no "canonical" or preferred isomorphism; an isomorphism {{math|''φ'' : ''F''<sup>''n''</sup> → ''V''}} is equivalent to the choice of a basis of {{math|''V''}}, by mapping the standard basis of {{math|''F''<sup>''n''</sup>}} to {{math|''V''}}, via {{math|''φ''}}. 

===Matrices===
{{Main|Matrix (mathematics)|l1=Matrix|Determinant}}
[[Image:Matrix.svg|right|thumb|200px|A typical matrix]]
''Matrices'' are a useful notion to encode linear maps.{{sfn|Lang|1987|loc=ch. V.1}} They are written as a rectangular array of scalars as in the image at the right. Any {{math|''m''}}-by-{{math|''n''}} matrix <math>A</math> gives rise to a linear map from {{math|''F''<sup>''n''</sup>}} to {{math|''F''<sup>''m''</sup>}}, by the following
<math display=block>\mathbf x = (x_1, x_2, \ldots, x_n) \mapsto \left(\sum_{j=1}^n a_{1j}x_j, \sum_{j=1}^n a_{2j}x_j, \ldots, \sum_{j=1}^n a_{mj}x_j \right),</math> 
where <math display="inline">\sum</math> denotes [[summation]], or by using the [[matrix multiplication]] of the matrix <math>A</math> with the coordinate vector <math>\mathbf{x}</math>:
<div id=equation2><math display=block>\mathbf{x} \mapsto A \mathbf{x}.</math></div>
Moreover, after choosing bases of {{math|''V''}} and {{math|''W''}}, ''any'' linear map {{math|''f'' : ''V'' → ''W''}} is uniquely represented by a matrix via this assignment.{{sfn|Lang|1987|loc=ch. V.3., Corollary, p. 106}}

[[Image:Determinant parallelepiped.svg|200px|right|thumb|The volume of this [[parallelepiped]] is the absolute value of the determinant of the 3-by-3 matrix formed by the vectors {{math|'''r'''<sub>1</sub>}}, {{math|'''r'''<sub>2</sub>}}, and {{math|'''r'''<sub>3</sub>}}.]]
The [[determinant]] {{math|det (''A'')}} of a [[square matrix]] {{math|''A''}} 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.{{sfn|Lang|1987|loc=Theorem VII.9.8, p. 198}} The linear transformation of {{math|'''R'''<sup>''n''</sup>}} corresponding to a real ''n''-by-''n'' matrix is [[Orientation (vector space)|orientation preserving]] if and only if its determinant is positive.

===Eigenvalues and eigenvectors===
{{Main|Eigenvalues and eigenvectors}}
[[Endomorphism]]s, linear maps {{math|''f'' : ''V'' → ''V''}}, are particularly important since in this case vectors {{math|'''v'''}} can be compared with their image under {{math|''f''}}, {{math|''f''('''v''')}}. Any nonzero vector {{math|'''v'''}} satisfying {{math|1=''λ'''''v''' = ''f''('''v''')}}, where {{math|''λ''}} is a scalar, is called an ''eigenvector'' of {{math|''f''}} with ''eigenvalue'' {{math|''λ''}}.{{sfn|Roman|2005||loc=ch. 8, p. 135–156}} Equivalently, {{math|'''v'''}} is an element of the [[Kernel (linear algebra)|kernel]] of the difference {{math|''f'' − ''λ'' · Id}} (where Id is the [[identity function|identity map]] {{math|''V'' → ''V'')}}. If {{math|''V''}} is finite-dimensional, this can be rephrased using determinants: {{math|''f''}}  having eigenvalue {{math|''λ''}} is equivalent to
<math display=block>\det(f - \lambda \cdot \operatorname{Id}) = 0.</math>
By spelling out the definition of the determinant, the expression on the left hand side can be seen to be a polynomial function in {{math|''λ''}}, called the [[characteristic polynomial]] of {{math|''f''}}.{{sfn||Lang|1987|loc=ch. IX.4}} If the field {{math|''F''}} is large enough to contain a zero of this polynomial (which automatically happens for {{math|''F''}} [[algebraically closed field|algebraically closed]], such as {{math|1=''F'' = '''C'''}}) any linear map has at least one eigenvector. The vector space {{math|''V''}} may or may not possess an [[eigenbasis]], a basis consisting of eigenvectors. This phenomenon is governed by the [[Jordan canonical form]] of the map.{{sfn|Roman|2005|loc=ch. 8, p. 140}} The set of all eigenvectors corresponding to a particular eigenvalue of {{math|''f''}} forms a vector space known as the ''eigenspace'' corresponding to the eigenvalue (and {{math|''f''}}) in question.

==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. 

===Subspaces and quotient spaces===
{{Main|Linear subspace|Quotient vector space}}
[[File:Linear subspaces with shading.svg|thumb|250px|right|A line passing through the [[origin (mathematics)|origin]] (blue, thick) in {{math|[[Euclidean space|'''R'''<sup>3</sup>]]}} is a linear subspace. It is the intersection of two [[plane (mathematics)|planes]] (green and yellow).]]

A nonempty [[subset]] <math>W</math> of a vector space <math>V</math> that is closed under addition and scalar multiplication (and therefore contains the <math>\mathbf{0}</math>-vector of <math>V</math>) is called a ''linear subspace'' of <math> V </math>, or simply a ''subspace'' of <math> V </math>, when the ambient space is unambiguously a vector space.{{sfn|Roman|2005|loc=ch. 1, p. 29}}<ref group=nb>This is typically the case when a vector space is also considered as an [[affine space]]. In this case, a linear subspace contains the [[zero vector]], while an affine subspace does not necessarily contain it.</ref> Subspaces of <math>V</math> are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set <math>S</math> of vectors is called its [[linear span|span]], and it is the smallest subspace of <math>V</math> containing the set <math>S</math>. Expressed in terms of elements, the span is the subspace consisting of all the [[linear combination]]s of elements of <math>S</math>.{{sfn|Roman|2005|loc=ch. 1, p. 35}}

{{anchor|vector line|vector plane|vector hyperplane}}Linear subspace of dimension 1 and 2 are referred to as a ''line'' (also ''vector line''), and a ''plane'' respectively. If ''W'' is an ''n''-dimensional vector space, any subspace of dimension 1 less, i.e., of dimension <math>n-1</math> is called a ''[[hyperplane]]''.{{sfn|Nicholson|2018|loc=ch. 10.4}}

The counterpart to subspaces are ''quotient vector spaces''.{{sfn|Roman|2005|loc=ch. 3, p. 64}} Given any subspace <math>W \subseteq V</math>, the quotient space <math>V / W</math> ("<math>V</math> [[modular arithmetic|modulo]] <math>W</math>") is defined as follows: as a set, it consists of
<math display="block">\mathbf{v} + W = \{\mathbf{v} + \mathbf{w} : \mathbf{w} \in W\},</math>
where <math>\mathbf{v}</math> is an arbitrary vector in <math>V</math>. The sum of two such elements <math>\mathbf{v}_1 + W</math> and <math>\mathbf{v}_2 + W</math> is <math>\left(\mathbf{v}_1 + \mathbf{v}_2\right) + W</math>, and scalar multiplication is given by <math>a \cdot (\mathbf{v} + W) = (a \cdot \mathbf{v}) + W</math>. The key point in this definition is that <math>\mathbf{v}_1 + W = \mathbf{v}_2 + W</math> [[if and only if]] the difference of <math>\mathbf{v}_1</math> and <math>\mathbf{v}_2</math> lies in <math>W</math>.<ref group=nb>Some authors, such as {{harvtxt|Roman|2005}}, choose to start with this [[equivalence relation]] and derive the concrete shape of <math>V / W</math> from this.</ref> This way, the quotient space "forgets" information that is contained in the subspace <math>W</math>.

The [[kernel (algebra)|kernel]] <math>\ker(f)</math> of a linear map <math>f : V \to W</math> consists of vectors <math>\mathbf{v}</math> that are mapped to <math>\mathbf{0}</math> in <math>W</math>.{{sfn|Lang|1987|loc=ch. IV.3.}} The kernel and the [[image (mathematics)|image]] <math>\operatorname{im}(f) = \{f(\mathbf{v}) : \mathbf{v} \in V\}</math> are subspaces of <math>V</math> and <math>W</math>, respectively.{{sfn|Roman|2005|loc=ch. 2, p. 48}} 

An important example is the kernel of a linear map <math>\mathbf{x} \mapsto A \mathbf{x}</math> for some fixed matrix <math>A</math>. The kernel of this map is the subspace of vectors <math>\mathbf{x}</math> such that <math>A \mathbf{x} = \mathbf{0}</math>, which is precisely the set of solutions to the system of homogeneous linear equations belonging to <math>A</math>. This concept also extends to linear differential equations
<math display=block>a_0 f + a_1 \frac{d f}{d x} + a_2 \frac{d^2 f}{d x^2} + \cdots + a_n \frac{d^n f}{d x^n} = 0,</math>
where the coefficients <math>a_i</math> are functions in <math>x,</math> too.
In the corresponding map
<math display=block>f \mapsto D(f) = \sum_{i=0}^n a_i \frac{d^i f}{d x^i},</math>
the [[derivative]]s of the function <math>f</math> appear linearly (as opposed to <math>f^{\prime\prime}(x)^2</math>, for example). Since differentiation is a linear procedure (that is, <math>(f + g)^\prime = f^\prime + g^\prime</math> and <math>(c \cdot f)^\prime = c \cdot f^\prime</math> for a constant <math>c</math>) this assignment is linear, called a [[linear differential operator]]. In particular, the solutions to the differential equation <math>D(f) = 0</math> form a vector space (over {{math|'''R'''}} or {{math|'''C'''}}).{{sfn|Nicholson|2018|loc=ch. 7.4}}

The existence of kernels and images is part of the statement that the [[category of vector spaces]] (over a fixed field <math>F</math>) is an [[abelian category]], that is, a corpus of mathematical objects and structure-preserving maps between them (a [[category (mathematics)|category]]) that behaves much like the [[category of abelian groups]].{{sfn|Mac Lane|1998}} Because of this, many statements such as the [[first isomorphism theorem]] (also called [[rank–nullity theorem]] in matrix-related terms)
<math display=block>V / \ker(f) \; \equiv \; \operatorname{im}(f)</math>
and the second and third isomorphism theorem can be formulated and proven in a way very similar to the corresponding statements for [[group (mathematics)|groups]].

===Direct product and direct sum===
{{Main|Direct product|Direct sum of modules}}
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''<!--explain direct--> <math>\textstyle{\prod_{i \in I} V_i}</math> of a family of vector spaces <math>V_i</math> consists of the set of all tuples <math>\left(\mathbf{v}_i\right)_{i \in I}</math>, which specify for each index <math>i</math> in some [[index set]] <math>I</math> an element <math>\mathbf{v}_i</math> of <math>V_i</math>.{{sfn|Roman|2005|loc=ch. 1, pp. 31–32}} Addition and scalar multiplication is performed componentwise. A variant of this construction is the ''direct sum'' <math display="inline">\bigoplus_{i \in I} V_i</math> (also called [[coproduct]] and denoted <math display="inline">\coprod_{i \in I}V_i</math>), where only tuples with finitely many nonzero vectors are allowed. If the index set <math>I</math> is finite, the two constructions agree, but in general they are different.

===Tensor product===
{{Main|Tensor product of vector spaces}}
The ''tensor product'' <math>V \otimes_F W,</math> or simply <math>V \otimes W,</math> of two vector spaces <math>V</math> and <math>W</math> is one of the central notions of [[multilinear algebra]] which deals with extending notions such as linear maps to several variables. A map <math>g : V \times W \to X</math> from the [[Cartesian product]] <math>V \times W</math> is called [[bilinear map|bilinear]] if <math>g</math> is linear in both variables <math>\mathbf{v}</math> and <math>\mathbf{w}.</math>  That is to say, for fixed <math>\mathbf{w}</math> the map <math>\mathbf{v} \mapsto g(\mathbf{v}, \mathbf{w})</math> is linear in the sense above and likewise for fixed <math>\mathbf{v}.</math>

[[Image:Universal tensor prod.svg|right|thumb|200px|[[Commutative diagram]] depicting the universal property of the tensor product]]
The tensor product is a particular vector space that is a ''universal'' recipient of bilinear maps <math>g,</math> as follows. It is defined as the vector space consisting of finite (formal) sums of symbols called [[tensor]]s
<math display=block>\mathbf{v}_1 \otimes \mathbf{w}_1 + \mathbf{v}_2 \otimes \mathbf{w}_2 + \cdots + \mathbf{v}_n \otimes \mathbf{w}_n,</math>
subject to the rules{{sfn|Lang|2002|loc = ch. XVI.1}}
<math display=block>\begin{alignat}{6}
a \cdot (\mathbf{v} \otimes \mathbf{w}) ~&=~ (a \cdot \mathbf{v}) \otimes \mathbf{w} ~=~ \mathbf{v} \otimes (a \cdot \mathbf{w}), && ~~\text{ where } a \text{ is a scalar} \\
(\mathbf{v}_1 + \mathbf{v}_2) \otimes \mathbf{w} ~&=~ \mathbf{v}_1 \otimes \mathbf{w} + \mathbf{v}_2 \otimes \mathbf{w}  &&  \\
\mathbf{v} \otimes (\mathbf{w}_1 + \mathbf{w}_2) ~&=~ \mathbf{v} \otimes \mathbf{w}_1 + \mathbf{v} \otimes \mathbf{w}_2.  &&  \\
\end{alignat}</math>
These rules ensure that the map <math>f</math> from the <math>V \times W</math> to <math>V \otimes W</math> that maps a [[tuple]] <math>(\mathbf{v}, \mathbf{w})</math> to <math>\mathbf{v} \otimes \mathbf{w}</math> is bilinear. The universality states that given ''any'' vector space <math>X</math> and ''any'' bilinear map <math>g : V \times W \to X,</math> there exists a unique map <math>u,</math> shown in the diagram with a dotted arrow, whose [[function composition|composition]] with <math>f</math> equals <math>g:</math> <math>u(\mathbf{v} \otimes \mathbf{w}) = g(\mathbf{v}, \mathbf{w}).</math><ref>{{harvtxt|Roman|2005}}, Th. 14.3. See also [[Yoneda lemma]].</ref> This is called the [[universal property]] of the tensor product, an instance of the method—much used in advanced abstract algebra—to indirectly define objects by specifying maps from or to this object.

==Vector spaces with additional structure==
From the point of view of linear algebra, vector spaces are completely understood insofar as any vector space over a given field is characterized, up to isomorphism, by its dimension. However, vector spaces ''per se'' do not offer a framework to deal with the question—crucial to analysis—whether a sequence of functions [[Limit of a sequence|converges]] to another function. Likewise, linear algebra is not adapted to deal with [[infinite series]], since the addition operation allows only finitely many terms to be added. <span id=labelFunctionalAnalysis>Therefore, the needs of [[functional analysis]] require considering additional structures.</span>{{sfn|Rudin|1991|loc=p.3}}

A vector space may be given a [[partial order]] <math>\,\leq,\,</math> under which some vectors can be compared.{{sfn|Schaefer|Wolff|1999|loc = pp. 204–205}} For example, <math>n</math>-dimensional real space <math>\mathbf{R}^n</math> can be ordered by comparing its vectors componentwise. [[Ordered vector space]]s, for example [[Riesz space]]s, are fundamental to [[Lebesgue integration]], which relies on the ability to express a function as a difference of two positive functions
<math display=block>f = f^+ - f^-.</math>
where <math>f^+</math> denotes the positive part of <math>f</math> and <math>f^-</math> the negative part.{{sfn|Bourbaki|2004|loc=ch. 2, p. 48}}

===Normed vector spaces and inner product spaces===
{{Main|Normed vector space|Inner product space}}
"Measuring" vectors is done by specifying a [[norm (mathematics)|norm]], a datum which measures lengths of vectors, or by an [[inner product]], which measures angles between vectors. Norms and inner products are denoted <math>| \mathbf v|</math> and {{nowrap|<math>\lang \mathbf v , \mathbf w \rang,</math>}} respectively. The datum of an inner product entails that lengths of vectors can be defined too, by defining the associated norm {{nowrap|<math display="inline">|\mathbf v| := \sqrt {\langle \mathbf v , \mathbf v \rangle}.</math>}} Vector spaces endowed with such data are known as ''normed vector spaces'' and ''inner product spaces'', respectively.{{sfn|Roman|2005|loc=ch. 9}}

Coordinate space <math>F^n</math> can be equipped with the standard [[dot product]]:
<math display=block>\lang \mathbf x , \mathbf y \rang = \mathbf x \cdot \mathbf y = x_1 y_1 + \cdots + x_n y_n.</math>
In <math>\mathbf{R}^2,</math> this reflects the common notion of the angle between two vectors <math>\mathbf{x}</math> and <math>\mathbf{y},</math> by the [[law of cosines]]:
<math display=block>\mathbf x \cdot \mathbf y = \cos\left(\angle (\mathbf x, \mathbf y)\right) \cdot |\mathbf x| \cdot |\mathbf y|.</math>
Because of this, two vectors satisfying <math>\lang \mathbf x , \mathbf y \rang = 0</math> are called [[orthogonal]]. An important variant of the standard dot product is used in [[Minkowski space]]: <math>\mathbf{R}^4</math> endowed with the Lorentz product{{sfn|Naber|2003|loc=ch. 1.2}}
<math display=block>\lang \mathbf x | \mathbf y \rang = x_1 y_1 + x_2 y_2 + x_3 y_3 - x_4 y_4.</math>
In contrast to the standard dot product, it is not [[positive definite bilinear form|positive definite]]: <math>\lang \mathbf x | \mathbf x \rang</math> also takes negative values, for example, for <math>\mathbf x = (0, 0, 0, 1).</math> Singling out the fourth coordinate—[[timelike|corresponding to time]], as opposed to three space-dimensions—makes it useful for the mathematical treatment of [[special relativity]]. Note that in other conventions time is often written as the first, or "zeroeth" component so that the Lorentz product is written <math display=block>\lang \mathbf x | \mathbf y \rang =  - x_0 y_0+x_1 y_1 + x_2 y_2 + x_3 y_3.</math>

===Topological vector spaces===
{{Main|Topological vector space}}
Convergence questions are treated by considering vector spaces <math>V</math> carrying a compatible [[topological space|topology]], a structure that allows one to talk about elements being [[neighborhood (topology)|close to each other]].{{sfnm
 | 1a1 = Treves | 1y = 1967 
 | 2a1 = Bourbaki | 2y = 1987
}} Compatible here means that addition and scalar multiplication have to be [[continuous map]]s. Roughly, if <math>\mathbf{x}</math> and <math>\mathbf{y}</math> in <math>V</math>, and <math>a</math> in <math>F</math> vary by a bounded amount, then so do <math>\mathbf{x} + \mathbf{y}</math> and <math>a \mathbf{x}.</math><ref group=nb>This requirement implies that the topology gives rise to a [[uniform structure]], {{harvtxt|Bourbaki|1989}}, loc = ch. II.</ref> To make sense of specifying the amount a scalar changes, the field <math>F</math> also has to carry a topology in this context; a common choice is the reals or the complex numbers.

In such ''topological vector spaces'' one can consider [[series (mathematics)|series]] of vectors. The [[infinite sum]]
<math display=block>\sum_{i=1}^\infty f_i ~=~ \lim_{n \to \infty} f_1 + \cdots + f_n</math>
denotes the [[limit of a sequence|limit]] of the corresponding finite partial sums of the sequence <math>f_1, f_2, \ldots</math> of elements of <math>V.</math> For example, the <math>f_i</math> could be (real or complex) functions belonging to some [[function space]] <math>V,</math> in which case the series is a [[function series]]. The [[modes of convergence|mode of convergence]] of the series depends on the topology imposed on the function space. In such cases, [[pointwise convergence]] and [[uniform convergence]] are two prominent examples.{{sfn|Schaefer|Wolff|1999|loc=p. 7}}

[[Image:Vector norms2.svg|thumb|right|250px|[[Unit ball|Unit "spheres"]] in <math>\mathbf{R}^2</math> consist of plane vectors of norm 1.  Depicted are the unit spheres in different [[Lp norm|<math>p</math>-norm]]s, for <math>p = 1, 2,</math> and <math>\infty.</math> The bigger diamond depicts points of 1-norm equal to 2.]]
A way to ensure the existence of limits of certain infinite series is to restrict attention to spaces where any [[Cauchy sequence]] has a limit; such a vector space is called [[Completeness (topology)|complete]]. Roughly, a vector space is complete provided that it contains all necessary limits. For example, the vector space of polynomials on the unit interval <math>[0, 1],</math> equipped with the [[topology of uniform convergence]] is not complete because any continuous function on <math>[0, 1]</math> can be uniformly approximated by a sequence of polynomials, by the [[Weierstrass approximation theorem]].<ref>{{harvnb|Kreyszig|1989|loc=§4.11-5}}</ref>  In contrast, the space of ''all'' continuous functions on <math>[0, 1]</math> with the same topology is complete.<ref>{{harvnb|Kreyszig|1989|loc=§1.5-5}}</ref> A norm gives rise to a topology by defining that a sequence of vectors <math>\mathbf{v}_n</math> converges to <math>\mathbf{v}</math> if and only if
<math display=block>\lim_{n \to \infty} |\mathbf v_n - \mathbf v| = 0.</math>
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]]—focuses on infinite-dimensional vector spaces, since all norms on finite-dimensional topological vector spaces give rise to the same notion of convergence.{{sfn|Choquet|1966|loc=Proposition III.7.2}} The image at the right shows the equivalence of the <math>1</math>-norm and <math>\infty</math>-norm on <math>\mathbf{R}^2:</math> as the unit "balls" enclose each other, a sequence converges to zero in one norm if and only if it so does in the other norm. In the infinite-dimensional case, however, there will generally be inequivalent topologies, which makes the study of topological vector spaces richer than that of vector spaces without additional data.

From a conceptual point of view, all notions related to topological vector spaces should match the topology. For example, instead of considering all linear maps (also called [[functional (mathematics)|functional]]s) <math>V \to W,</math> maps between topological vector spaces are required to be continuous.{{sfn|Treves|1967|loc=p. 34–36}} In particular, the <span id=label2>(topological) dual space <math>V^*</math> consists of continuous functionals <math>V \to \mathbf{R}</math> (or to <math>\mathbf{C}</math>). The fundamental [[Hahn–Banach theorem]] is concerned with separating subspaces of appropriate topological vector spaces by continuous functionals.{{sfn|Lang|1983|loc=Cor. 4.1.2, p. 69}}</span>

====Banach spaces====
{{Main|Banach space}}
''[[Banach space]]s'', introduced by [[Stefan Banach]], are complete normed vector spaces.{{sfn|Treves|1967|loc=ch. 11}}

A first example is [[Lp space|the vector space <math>\ell^p</math>]] consisting of infinite vectors with real entries <math>\mathbf{x} = \left(x_1, x_2, \ldots, x_n, \ldots\right)</math>
whose [[p-norm|<math>p</math>-norm]] <math>(1 \leq p \leq \infty)</math> given by 
<math display=block>\|\mathbf{x}\|_\infty := \sup_i |x_i| \qquad \text{ for } p = \infty, \text{ and }</math>
<math display=block>\|\mathbf{x}\|_p := \left(\sum_i |x_i|^p\right)^\frac{1}{p} \qquad \text{ for } p < \infty.</math>
<!---- "is finite." - ?!  ---->

The topologies on the infinite-dimensional space <math>\ell^p</math> are inequivalent for different <math>p.</math> For example, the sequence of vectors <math>\mathbf{x}_n = \left(2^{-n}, 2^{-n}, \ldots, 2^{-n}, 0, 0, \ldots\right),</math> in which the first <math>2^n</math> components are <math>2^{-n}</math> and the following ones are <math>0,</math> converges to the [[zero vector]] for <math>p = \infty,</math> but does not for <math>p = 1:</math>
<math display=block>\|\mathbf{x}_n\|_\infty = \sup (2^{-n}, 0) = 2^{-n} \to 0,</math> 
but 
<math display=block>\|\mathbf{x}_n\|_1 = \sum_{i=1}^{2^n} 2^{-n} = 2^n \cdot 2^{-n} = 1.</math>

More generally than sequences of real numbers, functions <math>f : \Omega \to \Reals</math> are endowed with a norm that replaces the above sum by the [[Lebesgue integral]] 
<math display=block>\|f\|_p := \left(\int_{\Omega} |f(x)|^p \, {d\mu(x)}\right)^\frac{1}{p}.</math>

The space of [[integrable function]]s on a given [[domain of a function|domain]] <math>\Omega</math> (for example an interval) satisfying <math>\|f\|_p < \infty,</math> and equipped with this norm are called [[Lp space|Lebesgue spaces]], denoted <math>L^{\;\!p}(\Omega).</math><ref group="nb">The [[triangle inequality]] for <math>\|f + g\|_p \leq \|f\|_p + \|g\|_p</math> is provided by the [[Minkowski inequality]]. For technical reasons, in the context of functions one has to identify functions that agree [[almost everywhere]] to get a norm, and not only a [[seminorm]].</ref>

These spaces are complete.{{sfn|Treves|1967|loc=Theorem 11.2, p. 102}} (If one uses the [[Riemann integral]] instead, the space is {{em|not}} complete, which may be seen as a justification for Lebesgue's integration theory.<ref group="nb">
"Many functions in <math>L^2</math> 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 <math>L^2</math> norm, and the orthogonal decomposition would not apply to them. This shows one of the advantages of Lebesgue integration.", {{harvtxt|Dudley|1989}}, §5.3, p. 125.</ref>) Concretely this means that for any sequence of Lebesgue-integrable functions <math>f_1, f_2, \ldots, f_n, \ldots</math> with <math>\|f_n\|_p < \infty,</math> satisfying the condition
<math display=block>\lim_{k,\ n \to \infty} \int_{\Omega} \left|f_k(x) - f_n(x)\right|^p \, {d\mu(x)} = 0</math>
there exists a function <math>f(x)</math> belonging to the vector space <math>L^{\;\!p}(\Omega)</math> such that
<math display=block>\lim_{k \to \infty} \int_{\Omega} \left|f(x) - f_k(x)\right|^p \, {d\mu(x)} = 0.</math>

Imposing boundedness conditions not only on the function, but also on its [[derivative]]s leads to [[Sobolev space]]s.{{sfn|Evans|1998|loc = ch. 5}}
{{Clear}}

====Hilbert spaces====
{{Main|Hilbert space}}
[[Image:Periodic identity function.gif|right|thumb|400px|The succeeding snapshots show summation of 1 to 5 terms in approximating a periodic function (blue) by finite sum of sine functions (red).]]
Complete inner product spaces are known as ''Hilbert spaces'', in honor of [[David Hilbert]].{{sfn|Treves|1967|loc=ch. 12}} The Hilbert space <math>L^2(\Omega),</math> with inner product given by
<math display=block>\langle f\ , \ g \rangle = \int_\Omega f(x) \overline{g(x)} \, dx,</math>
where <math>\overline{g(x)}</math> denotes the [[complex conjugate]] of <math>g(x),</math>{{sfn|Dennery|Krzywicki|1996|loc = p.190}}<ref group=nb>For <math>p \neq 2,</math> <math>L^p(\Omega)</math> is not a Hilbert space.</ref> is a key case.

By definition, in a Hilbert space, any Cauchy sequence converges to a limit. Conversely, finding a sequence of functions <math>f_n</math> with desirable properties that approximate a given limit function is equally crucial. Early analysis, in the guise of the [[Taylor approximation]], established an approximation of [[differentiable function]]s <math>f</math> by polynomials.{{sfn|Lang|1993|loc = Th. XIII.6, p. 349}} By the [[Stone–Weierstrass theorem]], every continuous function on <math>[a, b]</math> can be approximated as closely as desired by a polynomial.{{sfn|Lang|1993|loc = Th. III.1.1}} A similar approximation technique by [[trigonometric function]]s is commonly called [[Fourier expansion]], and is much applied in engineering. 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 <math>H,</math> in the sense that the ''[[closure (topology)|closure]]'' of their span (that is, finite linear combinations and limits of those) is the whole space. Such a set of functions is called a ''basis'' of <math>H,</math> its cardinality is known as the [[Hilbert space dimension]].<ref group=nb>A basis of a Hilbert space is not the same thing as a basis of a linear algebra. For distinction, a linear algebra basis for a Hilbert space is called a [[Hamel basis]].</ref> Not only does the theorem exhibit suitable basis functions as sufficient for approximation purposes, but also together with the [[Gram–Schmidt process]], it enables one to construct a [[orthogonal basis|basis of orthogonal vectors]].{{sfn|Choquet|1966|loc = Lemma III.16.11}} Such orthogonal bases are the Hilbert space generalization of the coordinate axes in finite-dimensional [[Euclidean space]].

The solutions to various [[differential equation]]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.{{sfn|Kreyszig|1999|loc=Chapter 11}} As an example from physics, the time-dependent [[Schrödinger equation]] in [[quantum mechanics]] describes the change of physical properties in time by means of a [[partial differential equation]], whose solutions are called [[wavefunction]]s.{{sfn|Griffiths|1995|loc=Chapter 1}} Definite values for physical properties such as energy, or momentum, correspond to [[eigenvalue]]s of a certain (linear) [[differential operator]] and the associated wavefunctions are called [[eigenstate]]s. The <span id=labelSpectralTheorem>[[spectral theorem]] decomposes a linear [[compact operator]] acting on functions in terms of these eigenfunctions and their eigenvalues.</span>{{sfn|Lang|1993|loc =ch. XVII.3}}

===Algebras over fields===
{{Main|Algebra over a field|Lie algebra}}
[[Image:Rectangular hyperbola.svg|right|thumb|250px|A [[hyperbola]], given by the equation <math>x \cdot y = 1.</math> The [[coordinate ring]] of functions on this hyperbola is given by <math>\mathbf{R}[x, y] / (x \cdot y - 1),</math> an infinite-dimensional vector space over <math>\mathbf{R}.</math>]]
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'' (or ''F''-algebra if the field ''F'' is specified).{{sfn|Lang|2002|loc=ch. III.1, p. 121}}

For example, the set of all [[polynomial]]s <math>p(t)</math> forms an algebra known as the [[polynomial ring]]: using that the sum of two polynomials is a polynomial, they form a vector space; they form an algebra since the product of two polynomials is again a polynomial. Rings of polynomials (in several variables) and their [[quotient ring|quotients]] form the basis of [[algebraic geometry]], because they are [[coordinate ring|rings of functions of algebraic geometric objects]].{{sfn|Eisenbud|1995|loc=ch. 1.6}}

Another crucial example are ''[[Lie algebra]]s'', which are neither commutative nor associative, but the failure to be so is limited by the constraints (<math>[x, y]</math> denotes the product of <math>x</math> and <math>y</math>):
* <math>[x, y] = - [y, x]</math> ([[anticommutativity]]), and
* <math>[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0</math> ([[Jacobi identity]]).{{sfn|Varadarajan|1974}}
Examples include the vector space of <math>n</math>-by-<math>n</math> matrices, with <math>[x, y] = x y - y x,</math> the [[commutator]] of two matrices, and <math>\mathbf{R}^3,</math> endowed with the [[cross product]].

The [[tensor algebra]] <math>\operatorname{T}(V)</math> is a formal way of adding products to any vector space <math>V</math> to obtain an algebra.{{sfn|Lang|2002|loc=ch. XVI.7}} As a vector space, it is spanned by symbols, called simple [[tensor]]s
<math display=block>\mathbf{v}_1 \otimes \mathbf{v}_2 \otimes \cdots \otimes \mathbf{v}_n,</math>
where the [[rank of a tensor|degree]] <math>n</math> 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 in the above section on [[#Tensor product|tensor products]]. In general, there are no relations between <math>\mathbf{v}_1 \otimes \mathbf{v}_2</math> and <math>\mathbf{v}_2 \otimes \mathbf{v}_1.</math> Forcing two such elements to be equal leads to the [[symmetric algebra]], whereas forcing <math>\mathbf{v}_1 \otimes \mathbf{v}_2 = - \mathbf{v}_2 \otimes \mathbf{v}_1</math> yields the [[exterior algebra]].{{sfn|Lang|2002|loc=ch. XVI.8}}

==Related structures==

===Vector bundles===
{{Main|Vector bundle|Tangent bundle}}
[[Image:Mobius strip illus.svg|thumb|249px|right|A Möbius strip. Locally, it [[homeomorphism|looks like]] {{math|''U'' × '''R'''}}.]]
A ''vector bundle'' is a family of vector spaces parametrized continuously by a [[topological space]] ''X''.{{sfn|Spivak|1999|loc = ch. 3}} More precisely, a vector bundle over ''X'' is a topological space ''E'' equipped with a continuous map 
<math display=block>\pi : E \to X</math>
such that for every ''x'' in ''X'', the [[fiber (mathematics)|fiber]] π<sup>−1</sup>(''x'') is a vector space. The case dim {{math|1=''V'' = 1}} is called a [[line bundle]]. For any vector space ''V'', the projection {{math|''X'' × ''V'' → ''X''}} makes the product {{math|''X'' × ''V''}} into a [[trivial bundle|"trivial" vector bundle]]. Vector bundles over ''X'' are required to be [[locally]] a product of ''X'' and some (fixed) vector space ''V'': for every ''x'' in ''X'', there is a [[neighborhood (topology)|neighborhood]] ''U'' of ''x'' such that the restriction of π to π<sup>−1</sup>(''U'') is isomorphic<ref group=nb>That is, there is a [[homeomorphism]] from π<sup>−1</sup>(''U'') to {{math|''V'' × ''U''}} which restricts to linear isomorphisms between fibers.</ref> to the trivial bundle {{math|''U'' × ''V'' → ''U''}}. 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 {{math|''X'' × ''V''}}). For example, the [[Möbius strip]] can be seen as a line bundle over the circle ''S''<sup>1</sup> (by [[homeomorphism#Examples|identifying open intervals with the real line]]). It is, however, different from the [[cylinder (geometry)|cylinder]] {{math|''S''<sup>1</sup> × '''R'''}}, because the latter is [[orientable manifold|orientable]] whereas the former is not.{{sfn|Kreyszig|1991|loc=§34, p. 108}}

Properties of certain vector bundles provide information about the underlying topological space. For example, the [[tangent bundle]] consists of the collection of [[tangent space]]s parametrized by the points of a differentiable manifold. The tangent bundle of the circle ''S''<sup>1</sup> is globally isomorphic to {{math|''S''<sup>1</sup> × '''R'''}}, since there is a global nonzero [[vector field]] on ''S''<sup>1</sup>.<ref group=nb>A line bundle, such as the tangent bundle of ''S''<sup>1</sup> is trivial if and only if there is a [[section (fiber bundle)|section]] that vanishes nowhere, see {{harvtxt|Husemoller|1994}}, Corollary 8.3. The sections of the tangent bundle are just [[vector field]]s.</ref> In contrast, by the [[hairy ball theorem]], there is no (tangent) vector field on the [[2-sphere]] ''S''<sup>2</sup> which is everywhere nonzero.{{sfn|Eisenberg|Guy|1979}} [[K-theory]] studies the isomorphism classes of all vector bundles over some topological space.{{sfn|Atiyah|1989}} In addition to deepening topological and geometrical insight, it has purely algebraic consequences, such as the classification of finite-dimensional real [[division algebra]]s: '''R''', '''C''', the [[quaternion]]s '''H''' and the [[octonion]]s '''O'''.

The [[cotangent bundle]] of a differentiable manifold consists, at every point of the manifold, of the dual of the tangent space, the [[cotangent space]]. [[Section (fiber bundle)|Sections]] of that bundle are known as [[differential form|differential one-form]]s.

===Modules===
{{Main|Module (mathematics)|l1=Module}}
''Modules'' are to [[ring (mathematics)|rings]] what vector spaces are to fields: the same axioms, applied to a ring ''R'' instead of a field ''F'', yield modules.{{sfn|Artin|1991|loc=ch. 12}} The theory of modules, compared to that of vector spaces, is complicated by the presence of ring elements that do not have [[multiplicative inverse]]s. For example, modules need not have bases, as the '''Z'''-module (that is, [[abelian group]]) [[Modular arithmetic|'''Z'''/2'''Z''']] shows; those modules that do (including all vector spaces) are known as [[free module]]s. Nevertheless, a vector space can be compactly defined as a [[Module (mathematics)|module]] over a [[Ring (mathematics)|ring]] which is a [[Field (mathematics)|field]], with the elements being called vectors.  Some authors use the term ''vector space'' to mean modules over a [[division ring]].{{sfn|Grillet|2007}} The algebro-geometric interpretation of commutative rings via their [[spectrum of a ring|spectrum]] allows the development of concepts such as [[locally free module]]s, the algebraic counterpart to vector bundles.

===Affine and projective spaces===
{{Main|Affine space|Projective space}}
[[Image:Affine subspace.svg|thumb|right|200px|An [[affine space|affine plane]] (light blue) in '''R'''<sup>3</sup>. It is a two-dimensional subspace shifted by a vector '''x''' (red).]] 
Roughly, ''affine spaces'' are vector spaces whose origins are not specified.{{sfn|Meyer|2000|loc=Example 5.13.5, p. 436}} More precisely, an affine space is a set with a [[transitive group action|free transitive]] vector space [[Group action (mathematics)|action]]. In particular, a vector space is an affine space over itself, by the map
<math display=block>V \times V \to W, \; (\mathbf{v}, \mathbf{a}) \mapsto \mathbf{a} + \mathbf{v}.</math>
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 {{math|'''x''' ∈ ''W''}}; this space is denoted by {{math|'''x''' + ''V''}} (it is a [[coset]] of ''V'' in ''W'') and consists of all vectors of the form {{math|'''x''' + '''v'''}} for {{math|'''v''' ∈ ''V''.}} An important example is the space of solutions of a system of inhomogeneous linear equations
<math display=block>A \mathbf{v} = \mathbf{b}</math>
generalizing the homogeneous case discussed in the [[#equation3|above section]] on linear equations, which can be found by setting <math>\mathbf{b} = \mathbf{0}</math> in this equation.{{sfn|Meyer|2000|loc=Exercise 5.13.15–17, p. 442}} The space of solutions is the affine subspace {{math|'''x''' + ''V''}} where '''x''' is a particular solution of the equation, and ''V'' is the space of solutions of the homogeneous equation (the [[nullspace]] of ''A'').

The set of one-dimensional subspaces of a fixed finite-dimensional vector space ''V'' is known as ''projective space''; it may be used to formalize the idea of [[parallel (geometry)|parallel]] lines intersecting at infinity.{{sfn|Coxeter|1987}} [[Grassmannian manifold|Grassmannians]] and [[flag manifold]]s generalize this by parametrizing linear subspaces of fixed dimension ''k'' and [[flag (linear algebra)|flags]] of subspaces, respectively.

==Notes==
{{Reflist|group=nb|3}}

==Citations==
{{Reflist|20em}}

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* {{Weibel IHA}}
{{refend}}

==External links==
{{Wikibooks|Linear Algebra|Definition and Examples of Vector Spaces|Real vector spaces}}
{{Wikibooks|Linear Algebra|Vector spaces}}
* {{springer|title=Vector space|id=p/v096520}}

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