= Solution
The <dual space> is $V^*=\operatorname{Hom}_{\mathbb R}(V,\mathbb R)$, the vector space of real-valued <linear maps> on $V$, with pointwise addition and scalar multiplication. Choose a <basis> $e_1,\ldots,e_n$ of $V$. Every $v$ has unique coordinates $v=\sum_i x_ie_i$, so $e^i(v)=x_i$ defines a <linear functional>, with $e^i(e_j)=\delta_{ij}$.
If $\sum_i a_ie^i=0$, evaluation on $e_j$ gives $a_j=0$, proving linear independence. For any $\ell\in V^*$, linearity gives $\ell(v)=\sum_i x_i\ell(e_i)$, hence $\ell=\sum_i\ell(e_i)e^i$. Thus the <dual basis> spans the <dual space> as well, proving
$$
\boxed{\dim V^*=n=\dim V.}
$$
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