What does affine mean in maths?

In geometry, an affine transformation or affine map (from the Latin, affinis, “connected with”) between two vector spaces consists of a linear transformation followed by a translation. In a geometric setting, these are precisely the functions that map straight lines to straight lines.

How do you use affine in a sentence?

Affine sentence example

  1. In Cryst, a Wyckoff position W is specified by such a representative affine subspace.
  2. Based on these centroids, the relationship between two different local spectra is characterized by an affine transformation.
  3. P.
  4. Note the equivalence of orthogonal arrays with affine resolvable designs.

What is the purpose of affine geometry?

Affine geometry provides the basis for Euclidean structure when perpendicular lines are defined, or the basis for Minkowski geometry through the notion of hyperbolic orthogonality.

What is meant by affine function?

An affine function is a function composed of a linear function + a constant and its graph is a straight line. The general equation for an affine function in 1D is: y = Ax + c. An affine function demonstrates an affine transformation which is equivalent to a linear transformation followed by a translation.

Is 0 an affine function?

A linear function in the French sense is an affine function that passes through the origin, that is a=0 and f(x)=bx for some number b independent of x.

What is affine independence?

A set X of points of an affine space is said to be affinely independent or, simply, independent, if the affine span of any strict subset of X is a strict subset of the affine span of X.

What does affine mean?

Definition of affine (Entry 2 of 2) : of, relating to, or being a transformation (such as a translation, a rotation, or a uniform stretching) that carries straight lines into straight lines and parallel lines into parallel lines but may alter distance between points and angles between lines affine geometry.

What is an affine set?

A set A is said to be an affine set if for any two distinct points, the line passing through these points lie in the set A. Note − S is an affine set if and only if it contains every affine combination of its points. Empty and singleton sets are both affine and convex set.

What is affine linear combination?

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

How do you prove affine subspace?

To see this, note that every element s ∈ S is uniquely expressible as s = v + w for some w ∈ S (namely w = s−v). So whenever v belongs to S, the affine subspace is a subspace; in fact it is just S. ⋆ (b) Any two affine subspaces of the form v + S and w + S (same S) are either equal or disjoint. u = v + v , u = w + w .

What is an affine hyperplane?

An affine hyperplane is an affine subspace of codimension 1 in an affine space. In Cartesian coordinates, such a hyperplane can be described with a single linear equation of the following form (where at least one of the ‘s is non-zero and is an arbitrary constant):

What are the axioms of affine geometry?

In geometry, an affine plane is a system of points and lines that satisfy the following axioms: Any two distinct points lie on a unique line. Each line has at least two points. Given any line and any point not on that line there is a unique line which contains the point and does not meet the given line.

How do you prove a function is affine?

Definition 4 We say a function A : <m → <n is affine if there is a linear function L : <m → <n and a vector b in <n such that A(x) = L(x) + b for all x in <m. In other words, an affine function is just a linear function plus a translation.

What is affine regression?

Max-affine regression refers to a model where the unknown regression function is modeled as a maximum of k unknown affine functions for a fixed k \\geq 1. This generalizes linear regression and (real) phase retrieval, and is closely related to convex regression.

What is affine Boolean function?

A Boolean function of algebraic degree at most unity is called an affine Boolean function, the general form for. n-variable affine function is. If the constant term of an affine function is zero then the function is called a linear Boolean function.

How can you tell if a transformation is affine?

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What is an affine subset?

An affine subset is defined (in Linear Algebra Done Right 3th edition) as a subset of vector space V, that can be expressed as v+U, where v∈V, U is a subspace of V.

What is the difference between affine and convex?

A set S is convex iff for every pair of points x,y∈S, the line segment ¯xy joining x to y is a subset of S. S is affine iff for every pair of points x,y∈S, the whole infinite line containing x and y is a subset of A.

What do you mean by Affinely independent and dependent?

A set of vectors is said to be affinely dependent if at least one of the vectors in the set can be defined as an affine combination of the others. Otherwise, the set is called affinely independent. Any affine combination is a linear combination; therefore every affinely dependent set is linearly dependent.

What is an affine space online?

The affine space is a space that preserves the angles of transformation. An affine structure is the generalized abstraction of a vector space – in that the affine space does not contain a unique element known as the “origin”. In other words, affine spaces are average combinations – differences between two points.

How do you pronounce affine transformation?

Phonetic spelling of affine transformation

  1. affine trans-for-ma-tion.
  2. affine transformation. Christian Heller.
  3. af-fine trans-form-a-tion. Samuel Henning.

Is affine a word?

a person related to one by marriage.

What is affine anthropology?

1. affine – (anthropology) kin by marriage. anthropology – the social science that studies the origins and social relationships of human beings. kin, kinsperson, family – a person having kinship with another or others; “he’s kin”; “he’s family”

Who are Affines in sociology?

noun. Two or more in-laws; parents-in-law, siblings-in-law, and other relatives by marriage; may refer to groups related to one another by marriage.

Is a single point an affine set?

A set is affine iff it contains all lines through any two points in the set (hence, as a trivial case, a set containing a single point is affine).

What operations are allowed in an affine space?

Affine Spaces Definition: A d-dimensional Affine Space consists of (i) a set of points, (ii) an associated d-dimensional vector space, and (iii) two operations: (iii. a) subtraction of two points in the set (which yields a vector in the associated vector space), and (iii.

How do I find affine combination?

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What is an affine sum of points?

1 Affine Combinations of Points Therefore P is the sum of a point and a vector which is again a point (see the notes on Points and Vectors. This point P represents, in the affine space of two-dimensional points and vectors, a point on the line that passes through P1 and P2.

Is a zero vector linearly independent?

Two vectors are linearly dependent if and only if they are collinear, i.e., one is a scalar multiple of the other. Any set containing the zero vector is linearly dependent.

Is the empty set an affine subspace?

The empty set ∅, every singleton {x}, and the entire space L are affine subspaces of L. A non-empty subset C of a linear space L is an affine subspace if and only if any affine combination of elements of C belongs to C.

What is the difference between vector space and affine space?

A vector space is an algebraic object with its characteristic operations, and an affine space is a group action on a set, specifically a vector space acting on a set faithfully and transitively.

What is the affine hull of two points?

The affine hull of a singleton (a set made of one single element) is the singleton itself. The affine hull of a set of two different points is the line through them. The affine hull of a set of three points not on one line is the plane going through them.

What is a coordinate hyperplane?

Each pair of axes defines a coordinate hyperplane. These hyperplanes divide space into eight trihedra, called octants. The octants are: | (+x,+y,+z) | (-x,+y,+z) | (+x,+y,-z) | (-x,+y,-z) | (+x,-y,+z) | (-x,-y,+z) | (+x,-y,-z) | (-x,-y,-z) |

What is a hyperplane and what is it used for?

A hyperplane is a plane of dimension one less than the dimension of data space, which divides the classes of data. SVM is a learning algorithm mainly used on classification problems, which considers the data as support vectors and generates a hyperplane to classify them.

What is hyperplane with example?

For example, in two-dimensional space a hyperplane is a straight line, and in three-dimensional space, a hyperplane is a two-dimensional subspace. Imagine a knife cutting through a piece of cheese that is in cubical shape and dividing it into two parts.

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