Projection (mathematics)

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

• In set theory:
  • An operation typified by the j ^th projection map, written proj_j , that takes an element x = (x₁, ..., x_j , ..., x_k) of the cartesian product X₁ × … × X_j × … × X_k to the value proj_j (x) = x_j . This map is always surjective.
  • A mapping that takes an element to its equivalence class under a given equivalence relation is known as the canonical projection.
  • The evaluation map sends a function f to the value f(x) for a fixed x. The space of functions Y^X can be identified with the cartesian product , and the evaluation map is a projection map from the cartesian product.
• In category theory, the above notion of cartesian product of sets can be generalized to arbitrary categories. The product of some objects has a canonical projection morphism to each factor. This projection will take many forms in different categories. The projection from the Cartesian product of sets, the product topology of topological spaces (which is always surjective and open), or from the direct product of groups, etc. Although these morphisms are often epimorphisms and even surjective, they do not have to be.
• In linear algebra, 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). In the case of orthogonal projections, the space admits a decomposition as a product, and the projection operator is a projection in that sense as well.
• In differential topology, any fiber bundle includes a projection map as part of its definition. Locally at least this map looks like a projection map in the sense of the product topology, and is therefore open and surjective.
• In topology, a retract is a continuous map r: X → X which restricts to the identity map on a subspace. This satisfies a similar idempotency condition r² = r and can be considered a generalization of the projection map. A retract which is homotopic to the identity is known as a deformation retract. This term is also used in category theory to refer to any split epimorphism.
• The scalar projection (or resolute) of one vector onto another.