Past exam of the mathematics course of the University of Cambridge / 2004 / ib / Paper 3 / 1E (Linear Algebra)

The dual space is $V^{\ast }={\mathrm{Hom}}_{\mathbb{R}}(V,\mathbb{R})$, the vector space of real-valued linear maps on $V$, with pointwise addition and scalar multiplication. Choose a basis $e_1,\dots ,e_n$ of $V$. Every $v$ has unique coordinates $v={\sum }_i{x}_i{e}_i$, so $e^i(v)=x_i$ defines a linear functional, with $e^i(e_j)={\delta }_{ij}$.

If ${\sum }_i{a}_i{e}^i=0$, evaluation on $e_j$ gives $a_j=0$, proving linear independence. For any $\ell \in V^{\ast }$, 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