Affine Subspaces

Linear subspaces must pass through the origin. But solving $Ax=b$ often gives a solution set that doesn’t. This is an Affine Subspace.

Definition

An affine subspace $L$ is a “shifted” vector subspace. $L = \mathbf{v} + U$ Where $\mathbf{v}$ is a translation vector and $U$ is a subspace.

Connection to Linear Systems

The solution to a consistent system $Ax = b$ forms an affine subspace:

• $\mathbf{v}$ is a particular solution ($x_p$).
• $U$ is the solution to the homogeneous system ($Ax=0$).