NationMaster - Encyclopedia: Projection (mathematics)

In mathematics, a projection is any one of several different types of functions, mappings, operations, or transformations, for example, the following:

A set-theoretic operation typified by the j^th projection map, written $proj_{j}!$ , that takes an element $vec{x} = (x_1, ldots, x_j, ldots, x_k)$ of the cartesian product $(X_1 times cdots times X_j times cdots times X_k)$ to the value $proj_{j}(vec{x}) = x_j$ .

A linear transformation that remains unchanged if applied twice (p(u) = p(p(u))), in other words, an idempotent operator. For example, the mapping that takes a point (x, y, z) in three dimensions to the point (x, y, 0) in the plane is a projection. This type of projection naturally generalizes to any number of dimensions n for the source and k ≤ n for the target of the mapping. See orthogonal projection, projection (linear algebra), projection operator.

A mapping that takes an element to its equivalence class under a given equivalence relation.

An operation that maps a function f to the value f (x) for a fixed x.

Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, the Cartesian product (or direct product) X Y of two sets X and Y is the set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y. This concept is named after Ren Descartes. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ... In geometry, an orthogonal projection of a k-dimensional object onto a d-dimensional hyperplane (d < k) is obtained by intersections of (k − d)- dimensional hyperplanes drawn through the points of an object orthogonally to the d-hyperplane. ... In linear algebra, a projection is a linear transformation P such that P2 = P, i. ... In mathematics, a projection operator on a vector space is an idempotent linear transformation. ... In mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a: [a] = { x âˆˆ X | x ~ a } The notion of equivalence classes is useful for constructing sets out... In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ...