Exercise 16, § 9

a) Let $E = \mathbf{P}(V)$ and $E' = \mathbf{P}(V')$ be two projective spaces of the same finite dimension $n \geq 2$ over two fields $K, K'$ respectively, and let $u$ be a bijective mapping of $E$ onto $E'$, transforming any three points on a straight line into three points on a straight line. Now prove that there exists an isomorphism $\sigma$ of $K$ onto $K'$ and a bijective semilinear mapping $v$ of $V$ onto $V'$ (relative to $\sigma$) such that $u$ is the mapping obtained by passing to quotients from $v$ (“fundamental theorem of projective geometry”; use exerc. 7 of II, p. 201). Suppose in addition $V' = V$ and $K$ commutative; for $u$ to be a projective mapping, it is necessary and sufficient that one have in addition $\begin{bmatrix} u(a) & u(b) \\ u(d) & u(c) \end{bmatrix} = \begin{bmatrix} a & b \\ d & c \end{bmatrix}$ for every quadruplet $(a, b, c, d)$ of distinct points on a straight line in $\mathbf{P}(V)$. b) Let $p$ be an integer such that $1 \leq p \leq n - 1$. Now prove that the first conclusion of a) remains valid when one supposes that the image by $u$ of every projective linear variety of dimension $p$ is contained in a projective linear variety of dimension $p$.