Flat (geometry)

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. 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 .

A flat can be described by a system of linear equations. For example, a line in two-dimensional space can be described by a single linear equation involving x and y:

In three-dimensional space, a single linear equation involving x, y, and z defines a plane, while a pair of linear equations can be used to describe a line. In general, a linear equation in n variables describes a hyperplane, and a system of linear equations describes the intersection of those hyperplanes. Assuming the equations are consistent and linearly independent, a system of k equations describes a flat of dimension n − k.

...
Wikipedia