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Affine transformation (Euclidean geometry) Bäcklund transform; Bilinear transform; Box–Muller transform; Burrows–Wheeler transform (data compression) Chirplet transform; Distance transform; Fractal transform; Gelfand transform; Hadamard transform; Hough transform (digital image processing) Inverse scattering transform; Legendre ...
The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result. [1]
More precisely, a unitary transformation is an isometric isomorphism between two inner product spaces (such as Hilbert spaces). In other words, a unitary transformation is a bijective function U : H 1 → H 2 {\displaystyle U:H_{1}\to H_{2}}
Algebra is the branch of mathematics that studies certain abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication.
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems.Classically, it studies zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects.
[4] This use of variables entails use of algebraic notation and an understanding of the general rules of the operations introduced in arithmetic: addition, subtraction, multiplication, division, etc. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers.
The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra, as established by the influence and works of Emmy Noether. [36] Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics.
A nonzero element of the algebra does not necessarily have a multiplicative inverse. For example, if u {\displaystyle u} is a vector in V {\displaystyle V} such that u 2 = 1 {\displaystyle u^{2}=1} , the element 1 2 ( 1 + u ) {\displaystyle \textstyle {\frac {1}{2}}(1+u)} is both a nontrivial idempotent element and a nonzero zero ...
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