The definition of flat excludes non-straight curves and non-planar surfaces, which are subspaces having different notions of distance: arc length and geodesic length, respectively.
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.
Two parallel flats of the same dimension either coincide or do not intersect; they can be described by two systems of linear equations which differ only in their right-hand sides.
These two operations (referred to as meet and join) make the set of all flats in the Euclidean n-space a lattice and can build systematic coordinates for flats in any dimension, leading to Grassmann coordinates or dual Grassmann coordinates.