The Euclidean space is often presented as the standard Euclidean space of dimension . This is motivated by the fact that every Euclidean space of dimension is isomorphic to the Euclidean space More precisely, given such a Euclidean space, one may choose any point as an origin. By Gram–Schmidt process, one may also find an orthonormal basis of the associated vector space (a basis such that the inner product of two basis vectors is 0 if they are different and 1 if they are equal). This defines Cartesian coordinates of any point of the space, as the coordinates on this basis of the vector These choices define an isomorphism of the given Euclidean space onto by mapping any point to the -tuple of its Cartesian coordinates, and every vector to its coordinate vector.
The Euclidean space is often presented as the standard Euclidean space of dimension . This is motivated by the fact that every Euclidean space of dimension is isomorphic to the Euclidean space More precisely given such a Euclidean space, one may choose any point as an origin. By Gram–Schmidt process, one may also find an orthonormal basis of the associated vector space (a basis such that the inner product of two basis vectors is 0 if they are different and 1 if they are equal). This defines Cartesian coordinates of any point of the space, as the coordinates on this basis of th,e vector These choices define an isomorphism of the given Euclidean space onto by mapping any point to the -tuple of its Cartesian coordinates, and every vector to its coordinate vector.