21 examples
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- In mathematics, a complex line is a one-dimensional affine subspace of a vector space over the complex numbers.
- Any affine subspace of the Euclidean n-dimensional space is also an example since the principal curvatures of any shape operator are zero.
- Near to any point, S can be approximated by its tangent plane at that point, which is an affine subspace of Euclidean space.
- If Euclidean space is considered as an affine space, the flats are precisely the affine subspaces.
- In finite dimensions, such an affine subspace corresponds to the solution set of an inhomogeneous linear system.
- This reflects the space along an -dimensional affine subspace.
- Though, its affine structure provides that concepts of line, plane and, generally, of an affine subspace (flat), can be used without modifications, as well as line segments.
- Such isometries have a set of fixed points (the "mirror") that is an affine subspace, but is possibly smaller than a hyperplane.
- This is a generalization of the fact that, in real Euclidean space, given a convex set and a point outside, there always exists an affine subspace separating the two.
- Approaches towards clustering in axis-parallel or arbitrarily oriented affine subspaces differ in how they interpret the overall goal, which is finding clusters in data with high dimensionality.
- The affine subspace manifests itself as the functional f. In the present context, the family of separable states is a convex set in the space of trace class operators.
- A conic optimization problem consists of minimizing a convex function over the intersection of an affine subspace and a convex cone.
- A linear subspace is an affine subspace containing the origin, or, equivalently, a subspace that is closed under linear combinations.
- In terms of geometric vectors, these affine subspaces are all the "lines" or "planes" parallel to the subspace, which is a line or plane going through the origin.
- More generally, given a converging series of vectors in a finite-dimensional real vector space E, the set of sums of converging rearranged series is an affine subspace of E.
- A system of coordinates on -dimensional space is defined by a (+1)-tuple of points not belonging to any affine subspace of a lesser dimension.
- An affine subspace (sometimes called a linear manifold, linear variety, or a flat) of a vector space is a subset closed under affine combinations of vectors in the space.
- Let V be a \kappa -dimensional affine space over a field F, Given a set define its closure to be its affine hull (i.e. the smallest affine subspace containing it).
- It is also possible to generalize the Radon transform still further by integrating instead over k-dimensional affine subspaces of R n, The X-ray transform is the most widely used special case of this construction, and is obtained by integrating over straight lines.
- For a vector space V, a subspace W, and a fixed vector a in V, the sets are called affine subspaces, and are cosets (both left and right, since the group is abelian).