Exercise 26

Subspaces are subsets and so we naturally consider how ‘is a subspace of’ interacts with the usual set operations.

• If $A,B$ are subspaces of a vector space, must their intersection $A\cap B$ be a subspace? Always? Sometimes? Never?

• Must the union $A\cup B$ be a subspace?

• If $A$ is a subspace of some $V$, must its set complement $V-A$ be a subspace?

(Hint. Try some test subspaces from Example 2.19.)