Editing Vector space (section)

===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.