Exercise 37, § 7

(a) Let $E$ be a left vector space (finite-dimensional or otherwise) and $A_i$ ($1 \leq i \leq n$) vector subspaces of $E$. Suppose that, for every sequence $(a_i)_{1 \leq i \leq n}$ of points of $E$ such that $a_i \in A_i$ for all $i$, the vector subspace of $E$ generated by the $a_i$ is of dimension $\leq m$ ($m$ an integer $\leq n$). Then prove that there exists a subspace $W$ of $E$ of dimension $h \leq m$, containing $h + (n - m)$ of the subspaces $A_i$. (Argue by induction on $n$. Prove first that (for fixed $n$) attention may be confined to the case where $m < n$ and $\dim(A_i) \leq m$ for all $i$; it may also be assumed that $A_i \neq \{0\}$ for all $i$ and that the $A_i$ are not all of dimension 1. Then argue (for fixed $n$) by induction on $d = \sum_{i=1}^m \dim(A_i)$. If for example $\dim(A_n) \geq 2$, consider a 1-dimensional subspace $B_n$ of $A_n$ of dimension 1 and apply the induction hypothesis to $A_1, \ldots, A_{n-1}, B_n$; conclude that there exists a number $k$ such that $1 \leq k \leq m$ and a $k$-dimensional subspace $U$ of $E$ containing $k$ of the $A_i$, for example $A_1, \ldots, A_k$; replacing, if need be, $k$ by an integer $k'$ such that $1 \leq k' \leq k$, show also that it can be assumed that there exist vectors $b_i \in A_i$ for $1 \leq i \leq k$ such that $b_1, \ldots, b_k$ form a basis of $U$; then project onto a supplementary subspace of $U$ in $E$ and use the induction hypothesis.)

(b) Let $K$ be a finite field and let $E$ be an $N$-dimensional vector space over $K$. Let $U_i$ ($1 \leq i \leq n$) be vector subspaces of $E$ such that, for every subset $H$ of $\{1, n\}$, the intersection of the $U_i$ such that $i \in H$ has dimension $\leq N - \mathrm{Card}(H)$. Show that the union of the $U_i$ cannot be equal to $E$ (argue by duality using (a)).