Editing Vector space (section)

{{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|class=skin-invert-image|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>