Editing Vector space (section)

==Related structures==

===Vector bundles===
{{Main|Vector bundle|Tangent bundle}}
[[Image:Mobius strip illus.svg|class=skin-invert-image|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|class=skin-invert-image|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.