# Basis/Unique representation/Coordinates/Remark

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Let a basis of a vector space be given. Due to fact, this means that for every vector there exists a uniquely determined representation

The elementa
(scalars) are called the *coordinates* of with respect to the given basis. Thus, for a fixed basis, we have a (bijective) correspondence between the vectors from and the coordinate tuples
.
We express this by saying that a basis determines a *linear coordinate system*.^{[1]}

- ↑ Linear coordinates give a bijective relation between points and number tuples. Due to linearity, such a bijection respects the addition and the scalar multiplication. In many different contexts, also nonlinear (curvilinear) coordinates are important. Also these put points of space and number tuples into a bijective relation. Examples are polar coordinates, cylindrical coordinates and spherical coordinates. By choosing suitable coordinate, mathematical problems like the computation of volumes can be simplified.