Exercise 12

Prove that if $V=W_{1}\oplus\dots\oplus W_{k}$ then $W_{i}\cap W_{j}$ is trivial whenever $i\neq j$. This shows that the first half of the proof of Lemma 4.15 extends to the case of more than two subspaces. (Example 4.19 shows that this implication does not reverse; the other half does not extend.)