Answer
$\{ v_1, v_2, v_3 \}$ do not span $\mathbb{R}$
Work Step by Step
We let $A = [ v_1, v_2, v_3 ] $
Therefore,
\[
A=
\left[ {\begin{array}{cc}
1 & 0 & 1 \\
0 & -1 & 0 \\
-1 & 0 & 0 \\
0 & 1 & 1 \\
\end{array} } \right]
\]
For $A = [ v_1, v_2, v_3 ]$, $A$ must have a pivot position in every row with 4 pivots in total. However, each column can have at most one pivot. With 3 columns, $A$ can have at most 3 pivots and needs 4 to span $\mathbb{R}$. Therefore, $\{ v_1, v_2, v_3 \}$ do not span $\mathbb{R}$
