coordinate vector

Let $V$ be a vector space of dimension $n$ over a field $K$.  If  $(b_1,\mathrm{\dots },b_n)$  is a basis of $V$, then any element $v$ of $V$ can be uniquely expressed in the form

$$v={\xi }_1{b}_1+\mathrm{\dots }+{\xi }_n{b}_n$$

with  ${\xi }_1,\mathrm{\dots },{\xi }_n\in K$.  The $n$-tuplet (http://planetmath.org/OrderedTuplet)  $({\xi }_1,\mathrm{\dots },{\xi }_n)$ is called the coordinate vector of $v$ with respect to the basis in question.  The scalars ${\xi }_i$ are the coordinates (or the components of $v$).

It’s evident that the correspondence

$$v\mapsto ({\xi }_1,\mathrm{\dots },{\xi }_n)$$

provides a linear isomorphism between the vector space $V$ and the vector space formed by the Cartesian product $K^n$.