Determine whether the given set forms a basis for the indicated subspace Determine whether the given set forms a basis for the indicated subspace:
$\{(1,-1,0),(0,1,-1)\}$ for the subspace of $\mathbb{R}^3$ consisting of all $(x,y,z)$ such that $x+y+z=0$.
I know linear independence and span is involved but I'd like to see how you properly show it if possible.
I feel that because we are dealing with a subspace of $\mathbb{R}^3$, three vectors are needed to span it and because there are only two vectors then it can't span the subspace and thus isn't a basis for the subspace.

Firstly, note that you are not dealing with entire $\mathbb{R}^3$. You're concerned with only  part of it, namely the null space of a linear functional.
This space has dimension $3-1=2$ and any 2 linearly independent vectors in $\mathbb{R}^3$ works fine as a basis. Consequently see that your 2 vectors are linearly independent. (Hint: They have $0$ in different coordinates.)
Then you're done. 
