Editing Basis (linear algebra) (section)

== Definition ==
A '''basis''' {{math|''B''}} of a [[vector space]] {{math|''V''}} over a [[field (mathematics)|field]] {{math|''F''}} (such as the [[real numbers]] {{math|'''R'''}} or the [[complex number]]s {{math|'''C'''}}) is a linearly independent [[subset]] of {{math|''V''}} that [[linear span|span]]s {{math|''V''}}. This means that a subset {{mvar|B}} of {{math|''V''}} is a basis if it satisfies the two following conditions:
;''linear independence''
: for every [[Finite set|finite]] subset <math>\{\mathbf v_1, \dotsc, \mathbf v_m\}</math> of {{mvar|B}}, if <math>c_1 \mathbf v_1 + \cdots + c_m \mathbf v_m = \mathbf 0</math> for some <math>c_1,\dotsc,c_m</math> in {{math|''F''}}, then {{nowrap|<math>c_1 = \cdots = c_m = 0</math>;}}
;''spanning property''
: for every vector {{math|'''v'''}} in {{math|''V''}}, one can choose <math>a_1,\dotsc,a_n</math> in {{math|''F''}} and <math>\mathbf v_1, \dotsc, \mathbf v_n</math> in {{mvar|B}} such that {{nowrap|<math>\mathbf v = a_1 \mathbf v_1 + \cdots + a_n \mathbf v_n</math>.}}

The [[scalar (mathematics)|scalar]]s <math>a_i</math> are called the coordinates of the vector {{math|'''v'''}} with respect to the basis {{math|''B''}}, and by the first property they are uniquely determined.

A vector space that has a [[finite set|finite]] basis is called [[Dimension (vector space)|finite-dimensional]]. In this case, the finite subset can be taken as {{math|''B''}} itself to check for linear independence in the above definition.

It is often convenient or even necessary to have an [[total order|ordering]] on the basis vectors, for example, when discussing [[orientation (vector space)|orientation]], or when one considers the scalar coefficients of a vector with respect to a basis without referring explicitly to the basis elements. In this case, the ordering is necessary for associating each coefficient to the corresponding basis element. This ordering can be done by numbering the basis elements. In order to emphasize that an order has been chosen, one speaks of an '''ordered basis''', which is therefore not simply an unstructured [[Set (mathematics)|set]], but a [[sequence]], an [[indexed family]], or similar; see {{slink||Ordered bases and coordinates}} below.