Definition 2

The space $P_{n,\ p}$ which is the quotient of the topological space $L_{n+1,\ p+1}$ by the equivalence relation $\Delta_{n,\ p}$ is called the space of projective linear varieties of $p \geq 0$ dimensions in real projective space $P_n$.

We shall use the following notation: given a matrix $X$ of $p + 1$ rows and $n + 1$ columns, and any strictly increasing sequence

of $p + 1$ indices belonging to the interval $[0, n]$ of $\mathbf{N}$, we denote by $X_\sigma$ the square submatrix of $X$ formed by the columns whose indices are $i_1, i_2, \ldots, i_{p+1}$. We denote by $A_\sigma$ the subset of $L_{p+1,\ p+1}$ consisting of these matrices $X$ such that $X_\sigma$ is non-singular. By Proposition 6 of $\S \ 1$, no. 6, $A_\sigma$ is a dense open set in $M_{p+1,\ n+1}$, and the function $X \to X_\sigma^{-1}$ is continuous on $A_\sigma$.

A geometrical interpretation of the set $A_\sigma$ is as follows: let $E_\sigma$ be the vector subspace of $\mathbf{R}^{n+1}$ generated by the vectors $e_i$ of the canonical basis such that $i \in \sigma$, and let $E'_\sigma$ be the complementary subspace generated by the $e_i$ such that $i \notin \sigma$; then to say that a matrix $x$ belongs to $A_\sigma$ means that the projections on $E_\sigma$ of its $p + 1$ rows of $x_k$ form a free system, or again that the vector subspace generated by the $x_k$ is a complement of $E'_\sigma$ (or that its intersection with $E'_\sigma$ consists only of 0).