|
In linear algebra, a (linear) cone is a subset of a vector space that is closed under multiplication by positive scalars. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations in finite dimensions. ...
A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. Every set is a subset of itself. ...
A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ...
In mathematics, multiplication is an arithmetic operation which is the inverse of division, and in elementary arithmetic, can be interpreted as repeated addition. ...
The term scalar is used in mathematics, physics, and computing basically for quantities that are characterized by a single numeric value and/or do not involve the concept of direction. ...
Definition
A subset C of a real vector space V is a (linear) cone iff λx belongs to C for any x in C and any positive scalar λ of V.
The condition can be written more succintly as "λC = C for any positive scalar λ of V".
The definition makes sense for any vector space V which allows the notion of "positive scalar", such as spaces over the rational, algebraic, or (more commonly) real numbers . The concept can also be extended for any vector space V whose scalar field is a superset of those fields (such as the complex numbers, quaternions, etc.), to the extent that such a space can be viewed as a real vector space of higher dimension. In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...
In mathematics, an algebraic number relative to a field is any element of a given field containing such that is a solution of a polynomial equation of the form: anxn + an−1xn−1 + ··· + a1x + a0 = 0 where n is a positive integer called the degree of the polynomial, every coefficient...
In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for the square root of negative one (−1), which cannot be represented by any real number. ...
In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...
Boolean, additive and linear closure Linear cones are closed under Boolean operations (set intersection, union, and complement). They are also closed under addition (if C and D are cones, so is C + D) and arbitrary linear maps. In particular, if Cis a cone, so is its opposite cone -C. The term intersection can mean: a road junction, where two roads intersect each other, such as a roundabout intersection; in mathematics, the set in which two or more other sets intersect each other; see intersection (set theory); a movie; see Intersection (movie). ...
In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ...
In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...
Pointed and blunt cones A cone C is said to be pointed if it includes the null vector (origin) 0 of the vector space; otherwise C is said to be blunt. Note that a pointed cone is closed under multiplication by arbitrary non-negative (not just positive) scalars. In linear algebra and related areas of mathematics, the null vector or zero vector in a vector space is the uniquely-determined vector, usually written 0, that is the identity element for vector addition. ...
The origin of something (from the Latin origo, beginning) is where it came from, in the sense of a physical location or a metaphysical source. ...
The cone of a set The (linear) cone of an arbitrary subset X of V is the set X* of all vectors λx where x belongs to X and λ is a positive real number.
With this definition, the cone of X is pointed or blunt depending on whether X owns the origin 0 or not. If "positive" is replaced by "non-negative" in the defitions, the cone X* will be always pointed.
Salient cone A cone X is salient if it does not contain any pair of opposite nonzero vectors; that is, iff C (-C)  {0}.
Spherical section and projection Let |·| be any norm for V, with the property that the norm of any vector is a scalar of V. By definition, a nonzero vector x belongs to a cone C of V if and only if the unit-norm vector x/|x| belongs to C. Therefore, a blunt (or pointed) cone C is completely specified by its central projection onto the sphere S; that is, by the set In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...
 It follows that there is a one-to-one correspondence between blunt (or pointed) cones and subsets of the unit-norm sphere of V, the set In mathematics and related technical fields, the term map or mapping is often a synonym for function. ...
A sphere is a perfectly symmetrical geometrical object. ...
 Indeed, the central projection C' is simply the spherical section of C, the set CS of its unit-norm elements.
A cone C is closed with respect to the norm |·| if it is a closed set in the topology induced by that norm. That is the case if and only if C is pointed and its spherical section is a closed subset of S. In topology and related branches of mathematics, a closed set is a set whose complement is open. ...
Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ...
Note that the cone C is salient iff its spherical section does not contain two opposite vectors; that is, C' (-C' ) = {}.
Convex cone A convex cone is a cone that is closed under convex combinations, i.e. iff αx + βy belongs to C for any non-negative scalars α, β with α + β = 1. A convex combination is a linear combination of data points (which can be vectors or scalars) where all coefficients are positive and sum up to 1. ...
Affine cone If C - v is a cone for some v in V, then C is said to be an (affine) cone with vertex v.
Proper cone The term proper cone is variously defined, depending on the context. It often means a salient and convex cone, or a cone that is contained in an open halfspace of V.
See also A cone is a basic geometrical shape: see cone (solid). ...
In common usage and elementary geometry, a cone (Greek: κώνος) is a solid object obtained by rotating a right triangle around one of its two short sides, the cones axis. ...
In topology, especially algebraic topology, the cone CX of a topological space X is the quotient space: of the product of X with the unit interval I = [0, 1]. Intuitively we make X into a cylinder and collapse one end of the cylinder to a point. ...
References |
Feeds |
Contact