Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics I / 2 / b / ii

Put $L=\mathrm{span}\{v\}$. Every two-dimensional vector subspace containing $v$ is $W=\mathrm{span}\{v,w\}$ for some $w\notin L$. There are $p^n-p$ choices for $w$ in $V$. Within any one such $W$, exactly $p^2-p$ choices of $w\in W\setminus L$ yield that same $W$: each forms a linearly independent pair with $v$, and that pair is a basis of $W$.

These sets of choices partition $V\setminus L$, because $w$ determines $\mathrm{span}\{v,w\}$ uniquely. Thus the number isEquivalently these vector subspaces correspond to the one-dimensional vector subspaces of the quotient vector space $V/L$. For $n=1$ there are none and the formula gives zero.