Editing Vector space (section)

==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}}