Proposition 7

If $(F_i)$ is a finite family of subspaces of a vector space E, then

If $F = \bigcap_i F_i$, $E/F$ is isomorphic to a subspace of the direct sum of the $E/F_i$ (§ 1, no. 7, formula (27)).

Vector subspaces of dimension 1 (resp. dimension 2) of a vector space E are often called lines passing through 0 (resp. planes passing through 0) (or simply lines (resp. planes) if no confusion arises (cf. § 9, no. 3)), by analogy with the language of Classical Geometry; a subspace of E is called a hyperplane passing through 0 (or simply a hyperplane) if it is of codimension 1. Hyperplanes can also be defined as the maximal elements of the set $\mathscr{S}$ of vector subspaces of E distinct from E, ordered by inclusion. There is a one-to-one correspondence between the subspaces of E containing a subspace H and the subspaces of $E/H$ (I, § 4, no. 7, Theorem 4); if E is of dimension $\geq 1$, $\mathscr{S}$ is non-empty and to say that H is maximal in $\mathscr{S}$ means that $E/H$ contains no subspace distinct from $\{0\}$ and $E/H$, which implies that $E/H$ is generated by any of its elements $\neq 0$, in other words it is of dimension 1.

In a vector space of finite dimension $n \geq 1$, the hyperplanes are the subspaces of dimension $n - 1$ by formula (3).