Euclidean Subspace

AKA: Flat, Linear Manifold, Linear Variety.
See: Geometry, n-Dimensional Space, Congruence (Geometry), Euclidean Space, Dimension, Point (Mathematics), Line (Mathematics), Three-Dimensional Space, Plane (Mathematics), Hyperplane, Linear Subspace.

References

(Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/Flat_(geometry) Retrieved:2017-7-16.
- In geometry, a flat is a subset of n-dimensional space that is congruent to a Euclidean space of lower dimension. The flats in two-dimensional space are points and lines, and the flats in three-dimensional space are points, lines, and planes.
  In n-dimensional space, there are flats of every dimension from 0 to $n - 1$ . ^[1] Flats of dimension $n - 1$ are called hyperplanes.
  Flats are similar to linear subspaces, except that they need not pass through the origin. If Euclidean space is considered as an affine space, the flats are precisely the affine subspaces. Flats are important in linear algebra, where they provide a geometric realization of the solution set for a system of linear equations.
  A flat is also called a linear manifold or linear variety.

↑ In addition, all of -dimensional space is sometimes considered an n-dimensional flat as a subset of itself.

(Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/Euclidean_space#Lines Retrieved:2017-7-16.
- The simplest (after points) objects in Euclidean space are flats, or Euclidean subspaces of lesser dimension. Points are 0-dimensional flats, 1-dimensional flats are called (straight) lines, and 2-dimensional flats are planes. $(n - 1)$ -dimensional flats are called hyperplanes.
  Any two distinct points lie on exactly one line. Any line and a point outside it lie on exactly one plane. More generally, the properties of flats and their incidence of Euclidean space are shared with affine geometry, whereas the affine geometry is devoid of distances and angles.