Define the dual space $V^{*}$ of a vector space $V$. Given a basis $\left\{x_{1}, \ldots, x_{n}\right\}$ of $V$ define its dual and show it is a basis of $V^{*}$.

Let $V$ be a 3-dimensional vector space over $\mathbb{R}$ and let $\left\{\zeta_{1}, \zeta_{2}, \zeta_{3}\right\}$ be the basis of $V^{*}$ dual to the basis $\left\{x_{1}, x_{2}, x_{3}\right\}$ for $V$. Determine, in terms of the $\zeta_{i}$, the bases dual to each of the following: (a) $\left\{x_{1}+x_{2}, x_{2}+x_{3}, x_{3}\right\}$, (b) $\left\{x_{1}+x_{2}, x_{2}+x_{3}, x_{3}+x_{1}\right\}$.