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Concept
Flat (geometry)
Summary
In geometry, a flat or Euclidean subspace is a subset of a Euclidean space that is itself 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 a n-dimensional space, there are flats of every dimension from 0 to n − 1; flats of dimension n − 1 are called hyperplanes.
Flats are the affine subspaces of Euclidean spaces, which means that they are similar to linear subspaces, except that they need not pass through the origin. 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:
while the description of a plane would require two parameters:
In general, a parameterization of a flat of dimension k would require parameters t1, ..., tk.
An intersection of flats is either a flat or the empty set.
If each line from one flat is parallel to some line from another flat, then these two flats are parallel. 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.
Official source
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