Exercise 10, § 9

In a projective space $P(V)$ of dimension n over a field K, one calls a projective frame a set S of $n + 2$ points, any $n + 1$ 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 two arbitrary projective frames of $P(V)$, now prove that there exists a transform $f \in PGL(V)$ such that $f(a_i) = a'_i$ for $0 \leq i \leq n + 1$. For this transform to be unique, it is necessary and sufficient that K be commutative. (Reduce to the case where $a'_i = a_i$ for all i, and observe that one can always write (with the notation of II, p. 133) $a_i = \pi(b_i)$, 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 where there exists an infinite set T of points of $P(V)$,