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| {{Merge from | Combining dimensions|discuss=Talk:Projection (mathematics)#Proposed Merge|date=August 2012}}
| | Hi there. Let me begin by introducing the author, her name is Sophia. For a while I've been in Alaska but I will have to transfer in a year or two. As a lady what she really likes is fashion and she's been performing it for quite a while. I am an invoicing officer and I'll be promoted quickly.<br><br>Here is my blog post; [http://www.khuplaza.com/dent/14869889 tarot card readings] |
| {{Merge to | Projection (linear algebra)|discuss=Talk:Projection (mathematics)#Merge to linear algebra|date=June 2013}}
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| {{Expert-subject|Mathematics|date=February 2013}}
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| In [[mathematics]], a '''projection''' is a mapping of a [[Set (mathematics)|set]] (or other [[mathematical structure]]) into a subset (or sub-structure), which is equal to its square for [[function composition|mapping composition]] (or, in other words, which is [[idempotence|idempotent]]). The [[restriction (mathematics)|restriction]] to a subspace of a projection is also called a ''projection'', even if the idempotence property is lost.
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| An everyday example of a projection is the casting of shadows onto a plane (paper sheet). The projection of a point is its shadow on the paper sheet. The shadow of a point of the paper sheet is the point itself (idempotence). The shadow of a three dimensional sphere is a circle. Originally, the notion of projection was introduced in [[Euclidean geometry]] to denote the projection of the [[Euclidean space]] of three dimensions onto a plane in it, like the shadow example. The two main projections of this kind are:
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| * {{anchor|Central projection}}The '''projection from a point onto a plane''' or '''central projection''': If ''C'' is the point, called '''center of projection''', the projection of a point ''P'' different from ''C'' is the intersection with the plane of the line ''CP''. The point ''C'' and the points ''P'' such that the line ''CP'' is parallel to the plane do not have any image by the projection.
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| * The '''projection parallel to a direction D, onto a plane''': The image of a point ''P'' is the intersection with the plane of the line parallel to ''D'' passing through ''P''.
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| The concept of '''projection''' in [[mathematics]] is a very old one, most likely having its roots in the phenomenon of the shadows cast by real world objects on the ground. This rudimentary idea was refined and abstracted, first in a geometric context and later in other branches of mathematics. Over time differing versions of the concept developed, but today, in a sufficiently abstract setting, we can unify these variations.
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| In [[cartography]], a [[map projection]] is a map of a part of the surface of the Earth onto a plane, which, in some cases, but not always, is the restriction of a projection in the above meaning. The [[3D projection]]s are also at the basis of the theory of [[perspective (graphical)|perspective]].
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| The need for unifying the two kinds of projections and of defining the image by a central projection of any point different of the center of projection are at the origin of [[projective geometry]]. However, a [[projective transformation]] is a [[bijection]] of a projective space, a property ''not'' shared with the ''projections'' of this article.
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| == Definition ==
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| [[Image:proj-map.png|thumb|right|Commutativity of this diagram is the universality of projection π, for any map ''f'' and set X.]]
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| In an abstract setting we can generally say that a ''projection'' is a mapping of a [[Set (mathematics)|set]] (or of a [[mathematical structure]]) which is [[idempotent]], which means that a projection is equal to its [[Function composition|composition]] with itself. A '''projection''' may also refer to a mapping which has a left inverse. Both notions are strongly related, as follows. Let ''p'' be an idempotent [[map (mathematics)|map]] from a set ''E'' into itself (thus ''p''∘''p'' = ''p'') and ''F'' = ''p''(''E'') be the image of ''p''. If we denote by π the map ''p'' viewed as a map from ''E'' onto ''F'' and by ''i'' the [[Injective function|injection]] of ''F'' into ''E'', then we have ''i''∘π = Id<sub>''F''</sub>. Conversely, ''i''∘π = Id<sub>''F''</sub> implies that π∘''i'' is idempotent.
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| == Applications ==
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| The original notion of projection has been extended or generalized to various mathematical situations, frequently, but not always, related to geometry, for example:
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| * In [[set theory]]:
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| ** An operation typified by the ''j'' <sup>th</sup> [[projection (set theory)|projection map]], written proj<sub>''j'' </sub>, that takes an element '''x''' = (''x''<sub>1</sub>, ..., ''x''<sub>''j'' </sub>, ..., ''x''<sub>''k''</sub>) of the [[cartesian product]] ''X''<sub>1</sub> × … × ''X''<sub>''j''</sub> × … × ''X''<sub>''k''</sub> to the value proj<sub>''j'' </sub>('''x''') = ''x''<sub>''j'' </sub>. This map is always [[surjective]].
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| ** A mapping that takes an element to its [[equivalence class]] under a given [[equivalence relation]] is known as the canonical projection.
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| ** The evaluation map sends a function ''f'' to the value ''f''(''x'') for a fixed ''x''. The space of functions ''Y''<sup>''X''</sup> can be identified with the cartesian product <math>\prod_{i\in X}Y_i</math>, and the evaluation map is a projection map from the cartesian product.
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| * In [[category theory]], the above notion of cartesian product of sets can be generalized to arbitrary [[category (mathematics)|categories]]. The [[product (category theory)|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 [[set (mathematics)|sets]], the [[product topology]] of [[topological space]]s (which is always surjective and [[open map|open]]), or from the [[direct product of groups|direct product]] of [[group (mathematics)|groups]], etc. Although these morphisms are often [[epimorphism]]s and even surjective, they do not have to be.
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| * 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.
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| * 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.
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| * In [[topology]], a [[retract]] is a continuous map ''r'': ''X'' → ''X'' which restricts to the identity map on its image. This satisfies a similar idempotency condition ''r''<sup>2</sup> = ''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.
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| * The [[scalar resolute|scalar projection]] (or resolute) of one [[vector (geometric)|vector]] onto another.
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| ==References==
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| * [[Thomas Craig (mathematician)|Thomas Craig]] (1882) [http://quod.lib.umich.edu/u/umhistmath/ABR2552.0001.001?rgn=works;view=toc;rgn1=author;q1=Craig A Treatise on Projections] from [[University of Michigan]] Historical Math Collection.
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| {{DEFAULTSORT:Projection (Mathematics)}}
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| [[Category:Mathematical terminology]]
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| [[pl:Rzut (matematyka)]]
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Hi there. Let me begin by introducing the author, her name is Sophia. For a while I've been in Alaska but I will have to transfer in a year or two. As a lady what she really likes is fashion and she's been performing it for quite a while. I am an invoicing officer and I'll be promoted quickly.
Here is my blog post; tarot card readings