16 Facts About Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space.

Therefore, in many cases, it is possible to work with a specific Euclidean space, which is generally the real -space equipped with the dot product.

Euclidean space was introduced by ancient Greeks as an abstraction of our physical space.

One way to think of the Euclidean space plane is as a set of points satisfying certain relationships, expressible in terms of distance and angles.

One of the basic tenets of Euclidean space geometry is that two figures of the plane should be considered equivalent if one can be transformed into the other by some sequence of translations, rotations and reflections .

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Standard way to mathematically define a Euclidean space, as carried out in the remainder of this article, is as a set of points on which a real vector space acts, the space of translations which is equipped with an inner product.

Euclidean space is an affine space over the reals such that the associated vector space is a Euclidean vector space.

Flat, Euclidean subspace or affine subspace of is a subset of such that.

Linear subspaces are Euclidean subspaces and a Euclidean subspace is a linear subspace if and only if it contains the zero vector.

Vector space associated to a Euclidean space is an inner product space.

Two lines, and more generally two Euclidean subspaces are orthogonal if their direction are orthogonal.

An affine basis of a Euclidean space of dimension is a set of points that are not contained in a hyperplane.

An isometry of Euclidean vector spaces is a linear isomorphism.

When such a space is defined by geometrical axioms, embedding the space in a Euclidean space is a standard way for proving consistency of its definition, or, more precisely for proving that its theory is consistent, if Euclidean geometry is consistent .

Euclidean space is an affine space equipped with a metric.

An inner product of a real vector Euclidean space is a positive definite bilinear form, and so characterized by a positive definite quadratic form.