Question-9

Select all true statements.

• The intersection of two subspaces of a vector space is a subspace.
• The union of two subspaces of a vector space is a subspace.
• The empty set is a subspace of every vector space.
• Every subset of a vector space need not be a subspace.

Answer

• The intersection of two subspaces of a vector space is a subspace.
• The union of two subspaces of a vector space is a subspace.
• The empty set is a subspace of every vector space.
• Every subset of a vector space need not be a subspace.

Solution

As a counter example for the second option, consider two subspaces for \(\mathbb{R}^{2}\): the x-axis and the y-axis. Their union is not a subspace as it is not closed under vector addition. The third option is incorrect as every vector space ought to have at least the zero vector.