6. Hint: You need to show two implications. For one of them, see Exercise 15 in Section 5.2. For the other, start with vectors ${\text{v}}_1\in S_1$ and ${\text{v}}_2\in S_2$ and show that if $S_1\cap S_2=\left\{0\right\}$, then ${\text{v}}_1={\text{v}}_2=0$.

7. Hint: If $V=S_1\oplus S_2$, the representation

is unique. Make use of the uniqueness to show that the vectors in B must be linearly independent.

9. Hint: $L\left(L^{k\text{â}1}\left(\mathbf{\text{v}}\right)\right)=L^k\left(\mathbf{\text{v}}\right)$.