# What would be the proper notation for this vector space?

Let $A = \operatorname{span}\left\{ a_1, a_2, \ldots , a_m\right\}$ given a set of basis vectors $a_1, a_2, \ldots , a_m$ in $\mathbb{C}^{n}$, and likewise let $B = \operatorname{span}\left\{ b_1, b_2, \ldots , b_m\right\}$ for a set of basis vectors $b_1, b_2, \ldots , b_m$.

Let C be the sum of the two vector spaces, i.e. $$C = \left\{ c_1 a + c_2 b; \;\; a \in A,\, b\in B; \; c_1, c_2 \in \mathbb{C}\right\}$$

and let $D$ be the subspace of $C$ orthogonal to everything in $A$, i.e. $$D = \left\{ c \in C \;\;\text{s.t}\;\;c\cdot a = 0\;\;\forall \;a \in A\right\}$$ Is there an accepted compact notation for $D$ in terms of $A$ and $B$? For example, would $$(A \oplus B) \mod A$$ or $$B \ominus A$$ generally be comprehensible to a reader knowing that $A$ and $B$ are both vector spaces in $\mathbb{C}^n$?

I'm working this into a physics paper where space is extremely tight, and I'd ideally like to describe $D$ as compactly as possible.

Thanks for any assistance.

• Note that $C=A+B.$ Thus $D=(A+B)\cap A^{\perp}=B\cap A^{\perp}.$ – mfl Jul 4 '17 at 18:07
• Why not just say, "let $E$ be the subspace of $C$ whose elements are orthogonal to everything in $A$"? I think that this is the clearest and briefest way to refer to the space you have. – Alex Ortiz Jul 4 '17 at 18:07
• @mfl: Wouldn't it be $B\cap A^{\perp}$? In which case, this is exactly what I need. – COTO Jul 4 '17 at 18:10
• Yes. I have fixed the typo. – mfl Jul 4 '17 at 18:10
• It is not true that $(A+B)\cap A^\perp=B\cap A^\perp$. For instance, if $A$ is spanned by $(1,0)$ and $B$ is spanned by $(1,1)$, then $B\cap A^\perp=0$ but $(A+B)\cap A^\perp$ is spanned by $(0,1)$. – Eric Wofsey Jul 4 '17 at 18:25

Neither of your proposed notations is standard. The most compact standard way to write this that I can think of is $D=(A+B)\cap A^\perp$. Here $A+B$ is a notation for what you call $C$ and $A^\perp$ is the set of all vectors which are orthogonal to every element of $A$.