coordinate vector

Let $V$ be a vector space of dimension $n$ over a field $K$.  If  $(b_{1},\,\ldots,\,b_{n})$  is a basis of $V$, then any element $v$ of $V$ can be uniquely expressed in the form

 $v\;=\;\xi_{1}b_{1}\!+\ldots+\!\xi_{n}b_{n}$

with  $\xi_{1},\,\ldots,\,\xi_{n}\in K$.  The $n$-tuplet (http://planetmath.org/OrderedTuplet)  $(\xi_{1},\,\ldots,\,\xi_{n})$ 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

 $v\;\mapsto\;(\xi_{1},\,\ldots,\,\xi_{n})$

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

Title coordinate vector CoordinateVector 2013-03-22 19:02:16 2013-03-22 19:02:16 pahio (2872) pahio (2872) 6 pahio (2872) Definition msc 15A03 ListVector coordinates components