Basis/Unique representation/Coordinates/Bijection/Remark

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

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

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

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

The inverse mapping

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

is also called the coordinate mapping.