coset

A coset of a subspace U is a shifted copy of U: pick a vector v and slide the entire subspace over by it to get v + U = {v + u : u in U}. If U is a plane through the origin, its cosets are all the planes parallel to it — most of which do NOT pass through the origin. A coset is a subspace that has been picked up and translated; it carries U's shape but not its anchoring at zero.

Cosets are precisely the individual elements of the quotient space V/U; the quotient is the collection of all these parallel copies. The same coset has many names: v + U equals w + U exactly when v - w lies in U. Choosing v is choosing a representative of the coset, and the coset itself is indifferent to which one you pick — every point inside it is an equally valid label.

The clean intuition is 'affine flat'. A subspace must contain 0 and is closed under addition; a coset relaxes the first requirement — it is a flat (a point, line, plane, ...) that need not pass through the origin. Cosets are the linear-algebra meeting point with affine geometry, and they are exactly the right objects for describing parallel families and offsets that the origin-fixated language of subspaces cannot.

The most concrete place cosets appear is the solution set of a linear system A*x = b. If x_p is any particular solution and U = null(A) is the kernel, then the full solution set is the coset x_p + U. That is why an inhomogeneous system's solutions form a flat parallel to the homogeneous solutions but shifted off the origin by x_p — a coset, not a subspace, unless b happens to be zero.