Basis/Unique representation/Coordinates/Remark

Let a basis ${}v_{1},\ldots ,v_{n}$ of a ${}K$-vector space ${}V$ be given. Due to fact  (3), this means that for every vector ${}u\in V$, there exists a uniquely determined representation

${}u=s_{1}v_{1}+s_{2}v_{2}+\cdots +s_{n}v_{n}\,.$

The elements ${}s_{i}\in K$ (scalars) are called the coordinates of ${}u$ with respect to the given basis. Thus, for a fixed basis, we have a (bijective) correspondence between the vectors from ${}V$, and the coordinate tuples ${}(s_{1},s_{2},\ldots ,s_{n})\in K^{n}$. We express this by saying that a basis determines a linear coordinate system.