Exercise 11

(Refer to Example 4.19. This exercise shows that the requirement that pairwise intersections be trivial is genuinely stronger than the requirement only that the intersection of all of the subspaces be trivial.) Give a vector space and three subspaces $W_{1}$, $W_{2}$, and $W_{3}$ such that the space is the sum of the subspaces, the intersection of all three subspaces $W_{1}\cap W_{2}\cap W_{3}$ is trivial, but the pairwise intersections $W_{1}\cap W_{2}$, $W_{1}\cap W_{3}$, and $W_{2}\cap W_{3}$ are nontrivial.