Exercise 10, § 9

In an $n$-dimensional projective space $\mathbf{P}(V)$ over a field $K$, a projective basis is a set $S$ of $n + 2$ points any two of which form a projectively free system. If $S = (a_i)_{0 \leq i \leq n+1}$ and $S' = (a'_i)_{0 \leq i \leq n+1}$ are any two projective bases of $\mathbf{P}(V)$, show that there exists a transformation $f \in \mathbf{PGL}(V)$ such that $f(a_i) = a'_i$ for $0 \leq i \leq n + 1$. For this transformation to be unique, it is necessary and sufficient that $K$ be commutative. (Reduce it to the case where $a'_i = a_i$ for all $i$ and note that it is always possible to write (in the notation of no. 6) $a_i = \pi(b_i)$, where $(b_i)_{1 \leq i \leq n+1}$ is a basis of $V$ and $b_0 = b_1 + b_2 + \cdots + b_{n+1}$.) Give an example where $K$ is a field of quaternions and there exists an infinite set $T$ of points of $\mathbf{P}(V)$ any $n + 1$ of which form a projectively free system and a transformation $f \in \mathbf{PGL}(V)$, distinct from the identity, leaving invariant all the points of $T$.