Internal direct sum of vector spaces Let $U$ be the subspace of $\mathbb{R}^3$ spanned by $\{(1,1,0), (0,1,1)\}$. Find a subspace $W$ of $\Bbb R^3$ such that $\mathbb{R}^3 = U \oplus W$.
As I am having an examination tomorrow, it would be really helpful if one could explain the methodology for doing this problem. I am mostly interested in the methodology, rather than the result. 
Thank you very much in advance.

Can you see that $W$ must be one-dimensional? So you are just looking for a single (non-zero) vector that's not in $U$ --- then let $W$ be the span of that vector. 
