In Linear Algebra Done Right by Axler, the author defines the orthogonal complement as follows:
If $U$ is a subset of a vector space $V$, then the orthogonal complement of $U$, denoted by $U^\perp$, is the set of all vectors in $V$ that are orthogonal to every vector in $U$: $$U^{\perp}=\{v\in V : \langle v, u\rangle = 0 \quad\forall u\in U\}$$
Then, he says "for example, if $U$ is a line in $\mathbb{R}^3$, then $U^\perp$ is the plane containing the origin that is perpendicular to $U$". Below this comment, there is a theorem that says that for any subset $U$ of $V$, $U\cap U^\perp \subset \{0\}$.
I have some questions and confusions.
- Is it common to define the orthogonal complement for any subset of $V$ instead of a subspace of $V$?
- From the example he mentioned, if $U$ is a line not going through the origin and $U^\perp$ is the plane containing the origin that is perpendicular to $U$, I would expect that there would be an intersection of $U$ and $U^\perp$ which is not the origin. But the theorem implies this cannot happen.
- Graphically, how could one picture $U$ and $U^\perp$ such that $U\cap U^\perp=\emptyset$?
All of these questions will be answered if we define the orthogonal complement only for a subspace, not for any set. I wonder what his intension was to define it for an arbitrary subset.
