Problem-6

subspaces

Question

Select all true statements.

  1. The intersection of two subspaces of a vector space is a subspace.

  2. The union of two subspaces of a vector space is a subspace.

  3. The empty set is a subspace of every vector space.

  4. Every subset of a vector space need not be a subspace

Solution

Options (a) and (d) are correct. 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 the zero element.