Exercise 22

A basis for a space consists of elements of that space. So we are naturally led to how the property ‘is a basis’ interacts with operations $\subseteq$ and $\cap$ and $\cup$. (Of course, a basis is actually a sequence that it is ordered, but there is a natural extension of these operations.)

• Consider first how bases might be related by $\subseteq$. Assume that $U,W$ are subspaces of some vector space and that $U\subseteq W$. Can there exist bases $B_{U}$ for $U$ and $B_{W}$ for $W$ such that $B_{U}\subseteq B_{W}$? Must such bases exist?

  For any basis $B_{U}$ for $U$, must there be a basis $B_{W}$ for $W$ such that $B_{U}\subseteq B_{W}$?

  For any basis $B_{W}$ for $W$, must there be a basis $B_{U}$ for $U$ such that $B_{U}\subseteq B_{W}$?

  For any bases $B_{U},B_{W}$ for $U$ and $W$, must $B_{U}$ be a subset of $B_{W}$?

• Is the $\cap$ of bases a basis? For what space?

• Is the $\cup$ of bases a basis? For what space?

• What about the complement operation?

(Hint. Test any conjectures against some subspaces of $\mathbb{R}^{3}$.)