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For any non-empty subset X of an affine space A, there is a smallest affine subspace that contains it, called the affine span of X. It is the intersection of all affine subspaces containing X, and its direction is the intersection of the directions of the affine subspaces that contain X.
More generally, an affine transformation is an automorphism of an affine space (Euclidean spaces are specific affine spaces), that is, a function which maps an affine space onto itself while preserving both the dimension of any affine subspaces (meaning that it sends points to points, lines to lines, planes to planes, and so on) and the ratios ...
The k-dimensional affine subspaces of R n are in one-to-one correspondence with the (k+1)-dimensional linear subspaces of R n+1 that are in general position with respect to the plane x n+1 = 1. Indeed, a k-dimensional affine subspace of R n is the locus of solutions of a rank n − k system of affine equations
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.
Affine subspace, a geometric structure that generalizes the affine properties of a flat; Projective subspace, a geometric structure that generalizes a linear subspace of a vector space; Multilinear subspace in multilinear algebra, a subset of a tensor space that is closed under addition and scalar multiplication
In mathematics, the affine group or general affine group of any affine space is the group of all invertible affine transformations from the space into itself. In the case of a Euclidean space (where the associated field of scalars is the real numbers), the affine group consists of those functions from the space to itself such that the image of every line is a line.
A set of subspaces is independent when the only intersection between any pair of subspaces is the trivial subspace. The direct sum is the sum of independent subspaces, written as . An equivalent restatement is that a direct sum is a subspace sum under the condition that every subspace contributes to the span of the sum.
More generally, a k-dimensional affine subspace of A is the intersection of A with a (k+1)-dimensional linear subspace of L that intersects A. Every point of the affine subspace A is the intersection of A with a one-dimensional linear subspace of L. However, some one-dimensional subspaces of L are parallel to A; in some sense, they intersect A at