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<p><b>New page</b></p><div>[[File:Perpendicular-coloured.svg|Perpendicular-coloured|thumb]] '''Orthogonality''' is a concept originating from [[geometry]] and [[mathematics]], with broad applications in fields such as [[computer science]], [[statistics]], [[systems engineering]], and [[signal processing]]. The term derives from the Greek ''orthogonios'', meaning "right-angled". In its most basic geometric sense, orthogonality refers to the concept of perpendicularity between lines or planes. However, the application and significance of orthogonality extend far beyond simple geometric interpretations.<br />
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==Definition==<br />
In [[geometry]], two lines or vectors in a two-dimensional plane or a three-dimensional space are considered orthogonal if they meet at a right angle (90 degrees). The concept can be generalized to higher dimensions in [[Euclidean space]]. In [[linear algebra]], two vectors are orthogonal if their [[dot product]] is zero. This definition extends the geometric concept of perpendicular lines to more abstract vector spaces, including infinite-dimensional spaces.<br />
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==Applications==<br />
===Mathematics and Physics===<br />
In mathematics, particularly in [[linear algebra]] and [[functional analysis]], orthogonality is crucial for defining [[orthonormal bases]], which simplify the analysis and solution of linear equations. In [[physics]], orthogonal coordinates simplify the mathematical description of physical systems, such as in the case of Cartesian, cylindrical, and spherical coordinate systems.<br />
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===Computer Science===<br />
In [[computer science]], orthogonality implies a design where operations change just one thing without affecting others. This concept is applied in [[software engineering]] and [[programming languages]] to enhance readability and maintainability. Orthogonal instruction sets in [[computer architecture]] allow for more efficient processing and simpler hardware design.<br />
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===Signal Processing===<br />
In [[signal processing]], orthogonality is used in the design of orthogonal functions, which are the basis of [[Fourier series]] and [[wavelets]]. Orthogonal functions allow for the efficient encoding and decoding of signals for transmission over communication channels, reducing interference and improving signal quality.<br />
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===Statistics===<br />
In [[statistics]], orthogonality pertains to the independence of variables or factors. Orthogonal factors in experimental design or [[regression analysis]] do not correlate with each other, simplifying the analysis and interpretation of data.<br />
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==Orthogonal Matrices and Transformations==<br />
An [[orthogonal matrix]] is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors). Orthogonal matrices are significant in linear algebra and related fields because they represent [[orthogonal transformations]], which preserve angles and lengths, making them isometries of Euclidean space.<br />
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==See Also==<br />
* [[Dot product]]<br />
* [[Vector space]]<br />
* [[Euclidean space]]<br />
* [[Linear algebra]]<br />
* [[Functional analysis]]<br />
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[[Category:Mathematics]]<br />
[[Category:Geometry]]<br />
[[Category:Linear Algebra]]<br />
[[Category:Computer Science]]<br />
[[Category:Signal Processing]]<br />
[[Category:Statistics]]<br />
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