Editing Coordinate vector (section)

== Definition ==
Let ''V'' be a [[vector space]] of [[dimension (vector space)|dimension]] ''n'' over a [[field (mathematics)|field]] ''F'' and let 
:<math> B = \{ b_1, b_2, \ldots, b_n \} </math>
be an [[ordered basis]] for ''V''. Then for every <math> v \in V </math> there is a unique [[linear combination]] of the basis vectors that equals ''<math> v </math>'':
:<math> v = \alpha _1 b_1 + \alpha _2 b_2 + \cdots + \alpha _n b_n .</math>
The '''coordinate vector''' of ''<math> v </math>'' relative to ''B'' is the [[sequence]] of [[coordinates]]
:<math> [v]_B = (\alpha _1, \alpha _2, \ldots, \alpha _n) .</math>
This is also called the ''representation of <math> v </math> with respect to B'', or the ''B representation of <math> v </math>''. The <math> \alpha _1, \alpha _2, \ldots, \alpha _n</math> are called the ''coordinates of <math> v </math>''. The order of the basis becomes important here, since it determines the order in which the coefficients are listed in the coordinate vector.

Coordinate vectors of finite-dimensional vector spaces can be represented by [[matrix (mathematics)|matrices]] as [[column vector|column]] or [[row vector]]s. In the above notation, one can write
:<math> [v]_B = \begin{bmatrix} \alpha_1 \\ \vdots \\ \alpha_n \end{bmatrix}</math>
and
:<math>[v]_B^T = \begin{bmatrix} \alpha_1 & \alpha_2 & \cdots & \alpha_n \end{bmatrix}</math> 
where <math>[v]_B^T</math> is the [[transpose]] of the matrix <math>[v]_B</math>.