= Solution
A finite <root system> in a real <Euclidean normed vector space>[Euclidean vector space] $E$ is a finite spanning set $\Phi\subset E\setminus\{0\}$ such that, for every $\alpha\in\Phi$, the <root reflection>
$$
s_\alpha(v)=v-\frac{2(v,\alpha)}{(\alpha,\alpha)}\alpha
$$
preserves $\Phi$, and the <Cartan integer> $2(\beta,\alpha)/(\alpha,\alpha)$ is an integer for all $\alpha,\beta\in\Phi$. It is <reduced root system>[reduced] when the only scalar multiples of $\alpha$ in $\Phi$ are $\alpha$ and $-\alpha$. Its <Weyl group> is the subgroup of the <orthogonal group> generated by the reflections $s_\alpha$. A <base of a root system> $\Delta$ is a <basis> of $E$ such that every root is an integer combination of elements of $\Delta$ whose nonzero coefficients all have the same sign.
The <coroot> of $\alpha$ is
$$
\alpha^\vee=\frac{2\alpha}{(\alpha,\alpha)}.
$$
Let $C$ be the <Weyl chamber> determined by $\Delta$:
$$
C=\{v\in E:(v,\alpha)>0\text{ for every }\alpha\in\Delta\}.
$$
The roots $\alpha$ and $\alpha^\vee$ are positive scalar multiples, so their reflecting hyperplanes and their positive half-spaces are identical. The same chamber $C$ therefore defines positivity in the <root system>[coroot system] $\Phi^\vee$. Its walls correspond exactly to the rays $\mathbb R_{>0}\alpha^\vee$ for $\alpha\in\Delta$. Hence its simple roots are
$$
\boxed{\Delta^\vee=\{\alpha^\vee:\alpha\in\Delta\},}
$$
which is therefore a base of $\Phi^\vee$.
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