Answer
$$(0,0,0)=-t(1,1,1)+t(1,1,0)+0(0,1,1) +t(0,0,1) .$$
$$(1,1,1)=(1,1,0)+0(0,1,1) + (0,0,1) .$$
Work Step by Step
Consider the combination
$$a(1,1,1)+b(1,1,0)+c(0,1,1) +d(0,0,1)=0, \quad a,b,c ,d\in R.$$
Which yields the following system of equations
\begin{align*}
a+b&=0\\
a+b+c &=0\\
a+c+d&=0.
\end{align*}
The augmented matrix is given by
$$\left[\begin {array}{cccc} 1&1&0&0\\ 1&1&1&0
\\ 1&0&1&1
\end {array}
\right]$$
The reduced row echelon form can be calculated as follows
$$\left[\begin {array}{cccc}1&0&0&1\\ 0&1&0&-1
\\ 0&0&1&0
\end {array}
\right],$$
then we have the solution
$$a=-t,\quad b=t, \quad c=0,\quad d=t.$$
Hence, we have the combinations
$$(0,0,0)=-t(1,1,1)+t(1,1,0)+0(0,1,1) +t(0,0,1) .$$
$$(1,1,1)=(1,1,0)+0(0,1,1) + (0,0,1) .$$
