In linear algebra, does $A \perp B$ mean the same thing as $A = B^\perp$?

Question

I'm having a hard time wrapping my head around orthogonal complements. I think my brain just rejects the notation, for whatever reason. If I could write $A \perp B$ and read it as, "The subset $A$ is orthogonal to the subset $B$," I think that would help, at least for the period of time where I'm still feeling uncomfortable with these ideas. So my specific questions are: Is there something wrong with this alternative notation? Does, "...is orthogonal to..." mean something different from, "...is the orthogonal complement of..."? If so, what is the difference? If not, then is $A \perp B$ already an established alternative to $A = B^\perp$?

Answer 1

Not necessarily. Consider $\mathbb{R}^3$ with its standard norm and the sets $A = \{(1,0,0)\}$, $B = \{(0,1,0)\}$. Then, clearly $A \perp B$ and $(0,0,1) \in B^{\perp}$ but $(0,0,1) \notin A$.

Answer 2

You can read $B^\perp$ as the maximal (i.e. largest) subspace $A$ so that $A\perp B$. The existance of such a space is clear, as with $A\perp B$ for all $A\in\mathcal F$ we also have $\langle \bigcup_{A\in\mathcal F} A\rangle\perp B$.

One set being orthogonal to the other simply means that any selection of two vectors is orthogonal.