Exercise 29

Because ‘span of’ is an operation on sets we naturally consider how it interacts with the usual set operations.

• If $S\subseteq T$ are subsets of a vector space, is $[S]\subseteq[T]$? Always? Sometimes? Never?

• If $S,T$ are subsets of a vector space, is $[S\cup T]=[S]\cup[T]$?

• If $S,T$ are subsets of a vector space, is $[S\cap T]=[S]\cap[T]$?

• Is the span of the complement equal to the complement of the span?