Here's the problem:
For each k in {0, 1, 2, 3, 4} find a subspace of R^{4} of dimension k.
I don't know where to start with this, any help would be appreciated!!
Do you understand what a subspace is? If not, please read your text/consult your notes before even thinking about this question.
Otherwise, let $\displaystyle e_1,e_2,e_3,e_4$ be your basis vectors (standard or otherwise) for $\displaystyle \mathbb{R}^4$
What is the dimension of the subspace spanned by $\displaystyle e_1$?
What is the dimension of the subspace spanned by $\displaystyle e_1$ and $\displaystyle e_2$?
What about the subspaces spanned by $\displaystyle e_1,e_2,e_3$ or $\displaystyle e_1,e_2,e_3,e_4$?