Projection (mathematics)

<h2 id="definition">Definition</h2>

Generally, a mapping where the <a href="/facts/Domain_of_a_function/FQvoeUpX">domain</a> and <a href="/facts/Codomain/MGc43nwQ">codomain</a> are the same <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> (or <a href="/facts/Mathematical_structure/6CXReUNv">mathematical structure</a>) is a projection if the mapping is <a href="/facts/Idempotent/khbSpexv">idempotent</a>, which means that a projection is equal to its <a href="/facts/Function_composition/r5Pvnmth">composition</a> with itself. A projection may also refer to a mapping which has a <a href="/facts/Inverse_function/Tg5nnXlu">right inverse</a>. Both notions are strongly related, as follows. Let p be an idempotent mapping from a set A into itself (thus p ∘ p = p) and B = p(A) be the image of p. If we denote by π the map p viewed as a map from A onto B and by i the <a href="/facts/Injective_function/5ES6tqf5">injection</a> of B into A (so that p = i ∘ π), then we have π ∘ i = IdB (so that π has a right inverse). Conversely, if π has a right inverse i, then π ∘ i = IdB implies that i ∘ π ∘ i ∘ π = i ∘ IdB ∘ π = i ∘ π; that is, p = i ∘ π is idempotent.

<h2 id="applications">Applications</h2>
The original notion of projection has been extended or generalized to various mathematical situations, frequently, but not always, related to geometry, for example:

<ul><li>In <a href="/facts/Set_theory/1ptfsvUP">set theory</a>:
<ul><li>An operation typified by the j-th <a href="/facts/Projection_(set_theory)/10mwpNsM">projection map</a>, written projj, that takes an element x = (x1, ..., xj, ..., xn) of the <a href="/facts/Cartesian_product/BfqBWzdL">Cartesian product</a> X1 × ⋯ × Xj × ⋯ × Xn to the value projj(x) = xj.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> This map is always <a href="/facts/Surjective/KezhLpCb">surjective</a> and, when each space Xk has a <a href="/facts/Topological_space/v2ut7CbK">topology</a>, this map is also <a href="/facts/Continuity_(topology)/sKbl02pB">continuous</a> and <a href="/facts/Open_map/abHUKX7l">open</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a></li>
<li>A mapping that takes an element to its <a href="/facts/Equivalence_class/1d2sh0TB">equivalence class</a> under a given <a href="/facts/Equivalence_relation/1MQ5rtkW">equivalence relation</a> is known as the <a href="/facts/Canonical_projection/QxxC9XTp">canonical projection</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a></li>
<li>The evaluation map sends a function f to the value f(x) for a fixed x. The space of functions YX can be identified with the Cartesian product 
 
 
 
 
 ∏
 
 i
 ∈
 X
 
 
 Y
 
 
 {\textstyle \prod _{i\in X}Y}
 
, and the evaluation map is a projection map from the Cartesian product.</li></ul></li>
<li>For <a href="/facts/Relational_database/pEKGmJLy">relational databases</a> and <a href="/facts/Query_language/Sa6K4xXM">query languages</a>, the <a href="/facts/Projection_(relational_algebra)/YgQm2wgA">projection</a> is a <a href="/facts/Unary_operation/kb6CkgsX">unary operation</a> written as 
 
 
 
 
 Π
 
 
 a
 
 1
 
 
 ,
 …
 ,
 
 a
 
 n
 
 
 
 
 (
 R
 )
 
 
 {\displaystyle \Pi _{a_{1},\ldots ,a_{n}}(R)}
 
 where 
 
 
 
 
 a
 
 1
 
 
 ,
 …
 ,
 
 a
 
 n
 
 
 
 
 {\displaystyle a_{1},\ldots ,a_{n}}
 
 is a set of attribute names. The result of such projection is defined as the <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> that is obtained when all <a href="/facts/Tuple/BI1HIxL8">tuples</a> in R are restricted to the set 
 
 
 
 {
 
 a
 
 1
 
 
 ,
 …
 ,
 
 a
 
 n
 
 
 }
 
 
 {\displaystyle \{a_{1},\ldots ,a_{n}\}}
 
.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a><a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>[verification needed] R is a <a href="/facts/Relation_(database)/Sj67AUyD">database-relation</a>.</li>
<li>In <a href="/facts/Spherical_geometry/WGLQb1Ys">spherical geometry</a>, projection of a sphere upon a plane was used by <a href="/facts/Ptolemy/htudGQTQ">Ptolemy</a> (~150) in his <a href="/facts/Planisphaerium/jQx56xyA">Planisphaerium</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> The method is called <a href="/facts/Stereographic_projection/oVyMrWqR">stereographic projection</a> and uses a plane <a href="/facts/Tangent/sF0jH3EI">tangent</a> to a sphere and a pole C diametrically opposite the point of tangency. Any point P on the sphere besides C determines a line CP intersecting the plane at the projected point for P.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> The correspondence makes the sphere a <a href="/facts/One-point_compactification/d3Xc0QKu">one-point compactification</a> for the plane when a <a href="/facts/Point_at_infinity/GAuS0wHQ">point at infinity</a> is included to correspond to C, which otherwise has no projection on the plane. A common instance is the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a> where the compactification corresponds to the <a href="/facts/Riemann_sphere/4XgBzplC">Riemann sphere</a>. Alternatively, a <a href="/facts/Sphere/4Ivb1cTj">hemisphere</a> is frequently projected onto a plane using the <a href="/facts/Gnomonic_projection/YHMYNmld">gnomonic projection</a>.</li>
<li>In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, a <a href="/facts/Linear_transformation/Q1PBKNVV">linear transformation</a> that remains unchanged if applied twice: p(u) = p(p(u)). In other words, an <a href="/facts/Idempotent/khbSpexv">idempotent</a> operator. For example, the mapping that takes a point (x, y, z) in three dimensions to the point (x, y, 0) is a projection. This type of projection naturally generalizes to any number of dimensions n for the domain and k ≤ n for the codomain of the mapping. See <a href="/facts/Orthogonal_projection/sElXGkxD">Orthogonal projection</a>, <a href="/facts/Projection_(linear_algebra)/sElXGkxD">Projection (linear algebra)</a>. 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.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a><a class="footnote-ref" id="fnref:10" href="#fn:10">10</a>[verification needed]</li>
<li>In <a href="/facts/Differential_topology/Y21eGQe0">differential topology</a>, any <a href="/facts/Fiber_bundle/SJ7Ek6cI">fiber bundle</a> includes a projection map as part of its definition. Locally at least this map looks like a projection map in the sense of the <a href="/facts/Product_topology/0hg02IaX">product topology</a> and is therefore <a href="/facts/Open_map/abHUKX7l">open</a> and surjective.</li>
<li>In <a href="/facts/Topology/2EqW47cU">topology</a>, a <a href="/facts/Retraction_(topology)/477Gm67f">retraction</a> is a <a href="/facts/Continuous_map_(topology)/sKbl02pB">continuous map</a> r: X → X which restricts to the <a href="/facts/Identity_map/9AtsP0LC">identity map</a> on its image.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> This satisfies a similar idempotency condition r2 = r and can be considered a generalization of the projection map. The image of a retraction is called a retract of the original space. A retraction which is <a href="/facts/Homotopic/1nWrPxdj">homotopic</a> to the identity is known as a <a href="/facts/Deformation_retraction/477Gm67f">deformation retraction</a>. This term is also used in <a href="/facts/Category_theory/ojfc36YN">category theory</a> to refer to any split epimorphism.</li>
<li>The <a href="/facts/Scalar_resolute/ENlzoaCx">scalar projection</a> (or resolute) of one <a href="/facts/Vector_(geometric)/bCLrdksy">vector</a> onto another.</li>
<li>In <a href="/facts/Category_theory/ojfc36YN">category theory</a>, the above notion of Cartesian product of sets can be generalized to arbitrary <a href="/facts/Category_(mathematics)/uTvF1odV">categories</a>. The <a href="/facts/Product_(category_theory)/evFecz0t">product</a> of some objects has a canonical projection <a href="/facts/Morphism/Kdpuyz3S">morphism</a> to each factor. Special cases include the projection from the <a href="/facts/Cartesian_product/BfqBWzdL">Cartesian product</a> of <a href="/facts/Set_(mathematics)/BfucfHMq">sets</a>, the <a href="/facts/Product_topology/0hg02IaX">product topology</a> of <a href="/facts/Topological_space/v2ut7CbK">topological spaces</a> (which is always surjective and <a href="/facts/Open_map/abHUKX7l">open</a>), or from the <a href="/facts/Direct_product_of_groups/4MWCxY0I">direct product</a> of <a href="/facts/Group_(mathematics)/IQTYqINt">groups</a>, etc. Although these morphisms are often <a href="/facts/Epimorphism/IrzonI01">epimorphisms</a> and even surjective, they do not have to be.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>[verification needed]</li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li><a href="/facts/Thomas_Craig_(mathematician)/WFWxgeJk">Craig, Thomas </a> (1882) <a href="http://quod.lib.umich.edu/u/umhistmath/ABR2552.0001.001?rgn=works;view=toc;rgn1=author;q1=Craig">A Treatise on Projections</a> from <a href="/facts/University_of_Michigan/6e1Ea5xh">University of Michigan</a> Historical Math Collection.</li>
<li>Henrici, Olaus Magnus Friedrich (1911). <a href="https://en.wikisource.org/wiki/1911_Encyclop%C3%A6dia_Britannica/Projection">"Projection" </a>. <a href="/facts/Encyclop%25C3%25A6dia_Britannica_Eleventh_Edition/mPElbCRY">Encyclopædia Britannica</a>. Vol. 22 (11th ed.). pp. 427–434.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">"Direct product - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2021-08-11. <a href="https://encyclopediaofmath.org/wiki/Direct_product" target="_blank">https://encyclopediaofmath.org/wiki/Direct_product</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Lee, John M. (2012). Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Vol. 218 (Second ed.). p. 606. doi:10.1007/978-1-4419-9982-5. ISBN 978-1-4419-9982-5. Exercise A.32. Suppose 
 
 
 
 
 X
 
 1
 
 
 ,
 …
 ,
 
 X
 
 k
 
 
 
 
 {\displaystyle X_{1},\ldots ,X_{k}}
 
 are topological spaces. Show that each projection 
 
 
 
 
 π
 
 i
 
 
 :
 
 X
 
 1
 
 
 ×
 ⋯
 ×
 
 X
 
 k
 
 
 →
 
 X
 
 i
 
 
 
 
 {\displaystyle \pi _{i}:X_{1}\times \cdots \times X_{k}\to X_{i}}
 
 is an open map. <a href="978-1-4419-9982-5" target="_blank">978-1-4419-9982-5</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Brown, Arlen; Pearcy, Carl (1994-12-16). An Introduction to Analysis. Springer Science & Business Media. ISBN 978-0-387-94369-5. <a href="978-0-387-94369-5" target="_blank">978-0-387-94369-5</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Alagic, Suad (2012-12-06). Relational Database Technology. Springer Science & Business Media. ISBN 978-1-4612-4922-1. <a href="978-1-4612-4922-1" target="_blank">978-1-4612-4922-1</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Date, C. J. (2006-08-28). The Relational Database Dictionary: A Comprehensive Glossary of Relational Terms and Concepts, with Illustrative Examples. "O'Reilly Media, Inc.". ISBN 978-1-4493-9115-7. <a href="978-1-4493-9115-7" target="_blank">978-1-4493-9115-7</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">"Relational Algebra". www.cs.rochester.edu. Archived from the original on 30 January 2004. Retrieved 29 August 2021. <a href="https://web.archive.org/web/20040130014938/https://www.cs.rochester.edu/~nelson/courses/csc_173/relations/algebra.html" target="_blank">https://web.archive.org/web/20040130014938/https://www.cs.rochester.edu/~nelson/courses/csc_173/relations/algebra.html</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Sidoli, Nathan; Berggren, J. L. (2007). "The Arabic version of Ptolemy's Planisphere or Flattening the Surface of the Sphere: Text, Translation, Commentary" (PDF). Sciamvs. 8. Retrieved 11 August 2021. <a href="http://individual.utoronto.ca/acephalous/Sidoli_Berggren_2007.pdf" target="_blank">http://individual.utoronto.ca/acephalous/Sidoli_Berggren_2007.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">"Stereographic projection - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2021-08-11. <a href="https://encyclopediaofmath.org/wiki/Stereographic_projection" target="_blank">https://encyclopediaofmath.org/wiki/Stereographic_projection</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">"Projection - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2021-08-11. <a href="https://encyclopediaofmath.org/wiki/Projection" target="_blank">https://encyclopediaofmath.org/wiki/Projection</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Roman, Steven (2007-09-20). Advanced Linear Algebra. Springer Science & Business Media. ISBN 978-0-387-72831-5. <a href="978-0-387-72831-5" target="_blank">978-0-387-72831-5</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">"Retraction - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2021-08-11. <a href="https://encyclopediaofmath.org/wiki/Retraction" target="_blank">https://encyclopediaofmath.org/wiki/Retraction</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">"Product of a family of objects in a category - Encyclopedia of Mathematics". encyclopediaofmath.org. Retrieved 2021-08-11. <a href="https://encyclopediaofmath.org/wiki/Product_of_a_family_of_objects_in_a_category" target="_blank">https://encyclopediaofmath.org/wiki/Product_of_a_family_of_objects_in_a_category</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
</ol>

Projection (mathematics) open-in-new

Projection (mathematics)