Let $V$ be a real vector space. Define the dual vector space $V^{*}$ of $V$. If $U$ is a subspace of $V$, define the annihilator $U^{0}$ of $U$. If $x_{1}, x_{2}, \ldots, x_{n}$ is a basis for $V$, define its dual $x_{1}^{*}, x_{2}^{*}, \ldots, x_{n}^{*}$ and prove that it is a basis for $V^{*}$.

If $V$ has basis $x_{1}, x_{2}, x_{3}, x_{4}$ and $U$ is the subspace spanned by

$$x_{1}+2 x_{2}+3 x_{3}+4 x_{4} \quad \text { and } \quad 5 x_{1}+6 x_{2}+7 x_{3}+8 x_{4},$$

give a basis for $U^{0}$ in terms of the dual basis $x_{1}^{*}, x_{2}^{*}, x_{3}^{*}, x_{4}^{*}$.