Express span $\{(1,1,0),(0,1,1)\}$ in explicit set notation form I don't really get this. It's a $3$ coordinate system right?
So how do i express it in set notation.
Followup: Does $(2,3,4)$ belong to the span $\{(1,0,-1),(0,-1,1)\}$
 A: For your title question: Denote a vector in $\mathbb R^3$ by $v$. $$\text{Span }\left\{(1, 1, 0), (0, 1, 1)\right\}  = \{v \mid \exists c_1, c_2\in \mathbb R\,\text{ such that } v = c_1(1, 1, 0) + c_2(0, 1, 1) \}$$
For the follow up: you need to determine whether there exist constants $c_1, c_2 \in \mathbb R$ such that $$(2, 3, 4) = c_1(1, 0, -1) + c_2(0, -1, 1).$$ 
If yes, then the vector is in the span of the given vectors.
A: The span is just the set of all linear combinations of vectors in your set. For you, an arbitrary vector in the span looks like
$$\alpha_1(1, 1, 0) + \alpha_2(0, 1,1) = (\alpha_1, \alpha_1, 0) + (0, \alpha_2, \alpha_2)$$ where $\alpha_1$ and $\alpha_2$ are arbitrary members of your field.
So your span is simply the set
$$\{ (\alpha_1, \alpha_1 + \alpha_2, \alpha_2) \mid \alpha_1, \alpha_2 \in \mathbb{F}\}.$$
To answer your follow up question, you need to decide whether there exist scalars $\alpha_1$ and $\alpha_2$ that make $(2, 3, 4)$ fit the above form.
A: I assume we are working over the field of real numbers.
$\mathbb{R}-span\{(1,1,0),(0,1,1)\}=\{(x,y,z)\mid  x,z\in\mathbb{R}\text{ and } y=x+z\}$
For the second part, do there exist two real numbers $a$ and $b$ (again if your field is real) such that $(2,3,4)=a(1,0,-1)+b(0,-1,1)$?
