affine subspace

An affine subspace is a subspace that has been translated off the origin. Take a genuine subspace U and shift it by a fixed vector p to get the set A = p + U = {p + u : u in U}. Lines and planes that do NOT pass through the origin are the everyday examples. An affine subspace has all the flat, parallel shape of a subspace but abandons the requirement that it contain 0 — it is exactly a coset of U viewed geometrically.

The defining algebraic property is closure under affine combinations rather than linear ones: if x and y are in A, then so is t*x + (1-t)*y for every scalar t (and more generally any combination whose coefficients sum to 1). The constraint 'coefficients sum to 1' is what keeps you on the flat without anchoring it at the origin. Drop that constraint and you recover an ordinary subspace; tighten t to lie in [0, 1] and you get a convex set — the affine notion sits exactly between the two.

The headline appearance is the solution set of an inhomogeneous linear system A*x = b. When b is nonzero this set is not a subspace (it misses 0), but it IS an affine subspace: x_p + null(A), one particular solution plus the homogeneous kernel. This is the structural reason every consistent linear system has a solution set shaped like a translated kernel — a flat parallel to null(A), offset by any particular solution.

The clean takeaway is the division of labor: subspaces are about directions through the origin, affine subspaces are about positioned flats anywhere in space. The 'direction subspace' U is uniquely recovered from A as the set of differences A - A = {x - y : x, y in A}, but the offset p is only defined up to a vector in U. Affine geometry is what you get when you keep the linear structure but forget where the origin is.