Prove The Orthogonal Complement of an Intersection is the Sum of Orthogonal Complements

Question

How does one prove that $(A∩B)^⊥=A^⊥+B^⊥$?

Seems a bit harder than proving $(A+B)^⊥=A^⊥∩B^⊥$.

Answer 1

Presumably, $A, B$ are subspaces of some (finite-dimensional) vector space $\Bbb V$, and $\perp$ is the orthogonal complement w.r.t. some inner product on $\Bbb V$.

Hint We can write the nominally easier second identity as $$(C + D)^{\perp} = C^{\perp} \cap D^{\perp}.$$ Then, set $C = A^{\perp}$ and $D = B^{\perp}$.