Definitions for affine combination
affine com·bi·na·tion

WiktionaryRate this definition:0.0 / 0 votes

1. affine combinationnoun

   A linear combination (of vectors in Euclidean space) in which the coefficients all add up to one.

WikipediaRate this definition:0.0 / 0 votes

1. Affine combination

   In mathematics, an affine combination of x1, ..., xn is a linear combination ∑ i = 1 n α i ⋅ x i = α 1 x 1 + α 2 x 2 + ⋯ + α n x n , {\displaystyle \sum _{i=1}^{n}{\alpha _{i}\cdot x_{i}}=\alpha _{1}x_{1}+\alpha _{2}x_{2}+\cdots +\alpha _{n}x_{n},} such that ∑ i = 1 n α i = 1. {\displaystyle \sum _{i=1}^{n}{\alpha _{i}}=1.} Here, x1, ..., xn can be elements (vectors) of a vector space over a field K, and the coefficients α i {\displaystyle \alpha _{i}} are elements of K. The elements x1, ..., xn can also be points of a Euclidean space, and, more generally, of an affine space over a field K. In this case the α i {\displaystyle \alpha _{i}} are elements of K (or R {\displaystyle \mathbb {R} } for a Euclidean space), and the affine combination is also a point. See Affine space § Affine combinations and barycenter for the definition in this case. This concept is fundamental in Euclidean geometry and affine geometry, because the set of all affine combinations of a set of points forms the smallest subspace containing the points, exactly as the linear combinations of a set of vectors form their linear span. The affine combinations commute with any affine transformation T in the sense that T ∑ i = 1 n α i ⋅ x i = ∑ i = 1 n α i ⋅ T x i . {\displaystyle T\sum _{i=1}^{n}{\alpha _{i}\cdot x_{i}}=\sum _{i=1}^{n}{\alpha _{i}\cdot Tx_{i}}.} In particular, any affine combination of the fixed points of a given affine transformation T {\displaystyle T} is also a fixed point of T {\displaystyle T} , so the set of fixed points of T {\displaystyle T} forms an affine subspace (in 3D: a line or a plane, and the trivial cases, a point or the whole space). When a stochastic matrix, A, acts on a column vector, b→, the result is a column vector whose entries are affine combinations of b→ with coefficients from the rows in A.

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1. Affine combination

   In mathematics, an affine combination of vectors x1, ..., xn is a vector called a linear combination of x1, ..., xn, in which the sum of the coefficients is 1, thus: Here the vectors are elements of a given vector space V over a field K, and the coefficients are scalars in K. This concept is important, for example, in Euclidean geometry. The act of taking an affine combination commutes with any affine transformation T in the sense that In particular, any affine combination of the fixed points of a given affine transformation is also a fixed point of, so the set of fixed points of forms an affine subspace. When a stochastic matrix, A, acts on a column vector, B, the result is a column vector whose entries are affine combinations of B with coefficients from the rows in A.

How to pronounce affine combination?

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Numerology

1. Chaldean Numerology

   The numerical value of affine combination in Chaldean Numerology is: 5

2. Pythagorean Numerology

   The numerical value of affine combination in Pythagorean Numerology is: 3

References

1. ^ Wiktionary
   https://en.wiktionary.org/wiki/Affine_Combination
2. ^ Wikipedia
   https://en.wikipedia.org/wiki/Affine_Combination
3. ^ Wikidata
   https://www.wikidata.org/w/index.php?search=affine combination