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In mathematics, a transformation, transform, or self-map [1] is a function f, usually with some geometrical underpinning, that maps a set X to itself, i.e. f: X → X. [ 2 ] [ 3 ] [ 4 ] Examples include linear transformations of vector spaces and geometric transformations , which include projective transformations , affine transformations , and ...
Pushforward measure and transformation formula. The pushforward measure in terms of a transformation , is defined as := (()). The change of variables formula for ...
In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space.For example, using the convention below, the matrix = [ ]
This formalism is used in geometric algebra and, more generally, in the Clifford algebra representation of Lie groups. In the case of a positive-definite Euclidean quadratic form, the double covering group of the isometry group S O ( n ) {\displaystyle \mathrm {SO} (n)} is known as the Spin group , S p i n ( n ) {\displaystyle \mathrm {Spin} (n)} .
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication.
In linear algebra, linear transformations can be represented by matrices.If is a linear transformation mapping to and is a column vector with entries, then there exists an matrix , called the transformation matrix of , [1] such that: = Note that has rows and columns, whereas the transformation is from to .
The equations defining the transformation in two dimensions, which rotates the xy axes counterclockwise through an angle into the x′y′ axes, are derived as follows. In the xy system, let the point P have polar coordinates ( r , α ) {\displaystyle (r,\alpha )} .
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 ...
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