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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.
is a K-linear transformation of this vector space into itself. The trace, Tr L/K (α), is defined as the trace (in the linear algebra sense) of this linear transformation. [1] For α in L, let σ 1 (α), ..., σ n (α) be the roots (counted with multiplicity) of the minimal polynomial of α over K (in some extension field of K). Then
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A squeeze mapping moves one purple hyperbolic sector to another with the same area. It also squeezes blue and green rectangles.. In 1688, long before abstract group theory, the squeeze mapping was described by Euclid Speidell in the terms of the day: "From a Square and an infinite company of Oblongs on a Superficies, each Equal to that square, how a curve is begotten which shall have the same ...
In linear algebra, particularly projective geometry, a semilinear map between vector spaces V and W over a field K is a function that is a linear map "up to a twist", hence semi-linear, where "twist" means "field automorphism of K". Explicitly, it is a function T : V → W that is:
Hahn–Banach dominated extension theorem [18] (Rudin 1991, Th. 3.2) — If : is a sublinear function, and : is a linear functional on a linear subspace which is dominated by p on M, then there exists a linear extension : of f to the whole space X that is dominated by p, i.e., there exists a linear functional F such that = for all , and | | for ...
A linear map : between two topological vector spaces is said to be compact if there exists a neighborhood of the origin in such that () is a relatively compact subset of . [ 3 ] Let X , Y {\displaystyle X,Y} be normed spaces and T : X → Y {\displaystyle T:X\to Y} a linear operator.
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