In geometry, a flat is an affine subspace, i.e. a subset of an affine space that is itself an affine space. Particularly, in the case the parent space is Euclidean, a flat is a Euclidean subspace which inherits the notion of distance from its parent space.

In an n-dimensional space, there are k-flats of every dimension k from 0 to n; flats one dimension lower than the parent space, (n − 1)-flats, are called hyperplanes.

The flats in a plane (two-dimensional space) are points, lines, and the plane itself; the flats in three-dimensional space are points, lines, planes, and the space itself. 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.

Flats occur in linear algebra, as geometric realizations of solution sets of systems of linear equations.

A flat is a manifold and an algebraic variety, and is sometimes called a linear manifold or linear variety to distinguish it from other manifolds or varieties.

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.

A flat can also be described by a system of linear parametric equations. A line can be described by equations involving one parameter: