Is the representation of a linear affine space unique?

Question

Suppose $S \subset \mathbb R^n$ is a linear affine subspace. I know picking any $s \in S$, $S- s =: U$ is a subspace and we can write $S = s + U$. Now consider writing $s = s_{U} + s_{U^{\perp}}$ where the subscripts denote the orthogonal projection. Then $S = s_{U^{\perp}} + U$. I would like to know whether this representation, i.e., $s_{U^{\perp}}$, is a uniquely defined vector.

Answer 1

Yes, it is.

The $U$ in the representation $S=s+U$ is the same for all $s\in S$, because it is defined as $U\{s-t:s,t\in S\}$.

As a consequence you have that $s_U^\perp = t_U^\perp$ for all $s,t\in S$, because $s-t=(s_U-t_U)+(s_U^\perp-t_U^\perp) \in U$ and $s_U-t_U\in U$ imply $s_U^\perp-t_U^\perp\in U$, so it has to be $0$.

Moreover this vector can be characterized as the projection of $0$ on $S$.