Affine subspace

An affine subspace is a translate of a linear subspace within an affine space. It contains all affine combinations of its points but need not contain the origin.

Vector space: A set with vector addition and scalar multiplication satisfying the vector space axioms. Its linear subspaces supply the directions that an affine subspace translates.
Direction space: The linear subspace of displacement vectors between points in an affine subspace. It records the unchanged directions after the subspace is translated.
Linear system: A collection of linear equations imposed on the same variables. A consistent linear system has an affine subspace as its full solution set.
Affine transformation: A map that preserves affine combinations of points. Affine transformations send affine subspaces to affine subspaces.
Linear subspace: A subset of a vector space that contains zero and is closed under vector addition and scalar multiplication. It becomes an affine subspace when shifted, and only stays linear if the shift preserves zero.
Dimension: The number of coordinates needed to specify a point in a space or the size of a basis for a vector space. An affine subspace has the dimension of its direction space.
Least squares: A method for estimating unknowns by minimizing the sum of squared residuals. Its fitted values lie in the column space, while residuals are measured from that linear subspace.
Projective subspace: A subset of projective space defined by a linear subspace of homogeneous coordinates. Projective subspaces belong to projective geometry, where parallel affine flats meet at infinity.
Affine space: A set of points equipped with a consistent notion of displacement vectors, but with no distinguished origin. Affine subspaces are the flat subsets defined within this origin-free setting.
Affine hull: The smallest affine subspace containing a given set of points. It constructs the least affine subspace that contains specified points.

Linked from 7 pages