Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics I / 1 / a / iii

Put $w_1=v_1$ and $w_j=v_j-v_1$ for $j\ge 2$. Since $v_1=w_1$ and $v_j=w_j+w_1$, the two families have the same linear span. Moreover,If the $v_j$ are linearly independent, a vanishing expression forces $a_j=0$ for $j\ge 2$ and then $a_1=0$. Conversely,so linear independence of the $w_j$ forces all $b_j=0$. Thus both the spanning set condition and linear independence are equivalent for the two families, proving the desired equivalence of bases.