In mathematics, an affine combination of x1, ..., xn is a linear combination
such that
Here, x1, ..., xn can be elements (vectors) of a vector space over a field K, and the coefficients 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 are elements of K (or 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 affine space 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
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 space (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.
and 27 Related for: Affine combination information
In mathematics, an affinecombination 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...
affine transformation is an automorphism of an affine space (Euclidean spaces are specific affine spaces), that is, a function which maps an affine space...
combinations, one can define the related concepts of affinecombination, conical combination, and convex combination, and the associated notions of sets closed...
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent...
to: Affine, a relative by marriage in law and anthropology Affine cipher, a special case of the more general substitution cipher Affinecombination, a...
algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all...
Otherwise, the set is called affinely independent. Any affinecombination is a linear combination; therefore every affinely dependent set is linearly dependent...
affine hull or affine span of a set S in Euclidean space Rn is the smallest affine set containing S, or equivalently, the intersection of all affine sets...
sets. Such an affinecombination is called a convex combination of u1, ..., ur. The collection of convex subsets of a vector space, an affine space, or a...
(x-{\tilde {b}})=0} . Affine hyperplanes are used to define decision boundaries in many machine learning algorithms such as linear-combination (oblique) decision...
relation that affinecombination in A agrees with affinecombination in F(A). Via this construction, the affine structure of the affine space A can be...
the convex conical hull of S is a closed set. Affinecombination Convex combination Linear combination Convex Analysis and Minimization Algorithms by...
function between vector spaces that preserves affine combinations. Affinecombination A linear combination in which the sum of the coefficients is 1. Basis...
function, then the value of the filtered signal will instead be an affinecombination of the input values and may fall outside of the minimum and maximum...
P(Class(X_{i})=Class(X_{j}))=p_{ij}} . Thus the predicted class is an affinecombination of the classes of every other point, weighted by the softmax function...
are each special case of improper rotation. Any improper rotation is an affine transformation and, in cases that keep the coordinate origin fixed, a linear...
not distinct) in the complex affine space. Therefore, most of algebraic geometry is built in complex affine spaces and affine spaces over algebraically closed...
mathematical field of differential geometry, the affine geometry of curves is the study of curves in an affine space, and specifically the properties of such...
coordinates and, more generally, to affine coordinates (see Affine space § Relationship between barycentric and affine coordinates). Barycentric coordinates...
continuous functions is a convex cone. An affine convex cone is the set resulting from applying an affine transformation to a convex cone. A common example...
In the fields of computer vision and image analysis, the Harris affine region detector belongs to the category of feature detection. Feature detection...
rotation, reflection, and scaling), and the centroid is equivariant under affine transformations. The same function may be an invariant for one group of...
D_{n},E_{6},E_{7},E_{8},F_{4},G_{2}} ), while affine type indecomposable matrices classify the affine Lie algebras (say over some algebraically closed...
topology, the line at infinity is a projective line that is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from...
Euclidean group of symmetries, is, therefore, a specialisation of affine geometry. All affine theorems apply. The origin of Euclidean geometry allows definition...
mimicking affine constructions like the de Casteljau algorithm for Bézier curves. Since the sphere is not an affine space, familiar properties of affine constructions...
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