Activity 3.5.2

We will look at some more subspaces of \(\real^2\text{.}\)

  1. Explain why a line that does not pass through the origin, as seen to the right, is not a subspace of \(\real^2\text{.}\)

  2. Explain why any subspace of \(\real^2\) must contain the zero vector \(\zerovec\text{.}\)

  3. Explain why the subset \(S\) of \(\real^2\) that consists of only the zero vector \(\zerovec\) is a subspace of \(\real^2\text{.}\)

  4. Explain why the subspace \(S=\real^2\) is itself a subspace of \(\real^2\text{.}\)

  5. If \(\vvec\) and \(\wvec\) are two vectors in a subspace \(S\text{,}\) explain why \(\span{\vvec,\wvec}\) is contained in the subspace \(S\) as well.

  6. Suppose that \(S\) is a subspace of \(\real^2\) containing two vectors \(\vvec\) and \(\wvec\) that are not scalar multiples of one another. What is the subspace \(S\) in this case?