Hyperplane

In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space. If a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dimensional lines. This notion can be used in any general space in which the concept of the dimension of a subspace is defined. In machine learning, hyperplanes are a key tool to create support vector machines for such tasks as computer vision and natural language processing. In different settings, the objects which are hyperplanes may have different properties. For instance, a hyperplane of an n-dimensional affine space is a flat subset with dimension n − 1. By its nature, it separates the space into two half spaces. A hyperplane of an n-dimensional projective space does not have this property. The difference in dimension between a subspace S {displaystyle S} and its ambient space X {displaystyle X} is known as the codimension of S {displaystyle S} with respect to X {displaystyle X} . Therefore, a necessary condition for S {displaystyle S} to be a hyperplane in X {displaystyle X} is for S {displaystyle S} to have codimension one in X {displaystyle X} . In geometry, a hyperplane of an n-dimensional space V is a subspace of dimension n − 1, or equivalently, of codimension 1 in V. The space V may be a Euclidean space or more generally an affine space, or a vector space or a projective space, and the notion of hyperplane varies correspondingly since the definition of subspace differs in these settings; in all cases however, any hyperplane can be given in coordinates as the solution of a single (due to the 'codimension 1' constraint) algebraic equation of degree 1. If V is a vector space, one distinguishes 'vector hyperplanes' (which are linear subspaces, and therefore must pass through the origin) and 'affine hyperplanes' (which need not pass through the origin; they can be obtained by translation of a vector hyperplane). A hyperplane in a Euclidean space separates that space into two half spaces, and defines a reflection that fixes the hyperplane and interchanges those two half spaces.