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 $$v \;=\; \xi_1b_1\!+\ldots+\!\xi_nb_n$$ with $\xi_1,\,\ldots,\,\xi_n \in K$ . The $n$ -tuplet $(\xi_1,\,\ldots,\,\xi_n)$ 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 $K^n$ .