Basis/Unique representation/Coordinates/Bijection/Remark

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

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

Here, the uniquely determined elements ${}s_{i}\in K$ (scalars) are called the coordinates of ${}u$ with respect to the given basis. This means that for a given basis, there is a correspondence between vectors and coordinate tuples ${}(s_{1},s_{2},\ldots ,s_{n})\in K^{n}$. We say that a basis determines a linear coordinate system of ${}V$. To paraphrase, a basis gives, in particular, a bijective mapping

$\Psi _{\mathfrak {v}}\colon K^{n}\longrightarrow V,{\begin{pmatrix}s_{1}\\\vdots \\s_{n}\end{pmatrix}}\longmapsto s_{1}v_{1}+s_{2}v_{2}+\cdots +s_{n}v_{n}.$

The inverse mapping

${\left(\Psi _{\mathfrak {v}}\right)}^{-1}\colon V\longrightarrow K^{n}$

is also called the coordinate mapping.