Real Affine Set Representations

Why

Since every affine set is a translate of a unique subspace, we can represent them by representing the vector and the subspace.

Definition

Recall that $M$ is affine means $M = S + a$ for some subspace $S$ and vector $a \in \R ^n$. The dimension of $M$ is the dimension of the subspace. Suppose $\dim S = k$, then there exists $Q \in \R ^{n \times k}$ with $Q^\top Q = I$, so that for any $x \in S$, there exists unique $z \in \R ^k$ with $x = Qz$. Since $M = S + a$, we have

\[ M = \Set{y \in \R ^n}{(\exists z \in \R ^k)(y = a + Qz)} \]