Search results
Results from the WOW.Com Content Network
Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...
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.
Given the affine group of an affine space A, the stabilizer of a point p is isomorphic to the general linear group of the same dimension (so the stabilizer of a point in Aff(2, R) is isomorphic to GL(2, R)); formally, it is the general linear group of the vector space (A, p): recall that if one fixes a point, an affine space becomes a vector space.
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 ...
Origins from Alice's and Bob's perspectives. Vector computation from Alice's perspective is in red, whereas that from Bob's is in blue. The following characterization may be easier to understand than the usual formal definition: an affine space is what is left of a vector space after one has forgotten which point is the origin (or, in the words of the French mathematician Marcel Berger, "An ...
In projective geometry, affine space means the complement of a hyperplane at infinity in a projective space. Affine space can also be viewed as a vector space whose operations are limited to those linear combinations whose coefficients sum to one, for example 2x − y, x − y + z, (x + y + z)/3, ix + (1 − i)y, etc.
A constant function is also considered linear in this context, as it is a polynomial of degree zero or is the zero polynomial. Its graph, when there is only one variable, is a horizontal line. In this context, a function that is also a linear map (the other meaning) may be referred to as a homogeneous linear function or a linear form.
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 ...