Answer
$$(0,0,0,0)=-2t(1,2,3,4)+t(1,0,1,2)+t(1,4,5,6) .$$
$$(1,4,5,6)=2(1,2,3,4)-(1,0,1,2) .$$
Work Step by Step
Consider the combination
$$a(1,2,3,4)+b(1,0,1,2)+c(1,4,5,6) =0, \quad a,b,c \in R.$$
Which yields the following system of equations
\begin{align*}
a+b+c&=0\\
2a+4c &=0\\
3a+b+5c&=0\\
4a+2b+6c&=0.
\end{align*}
The augmented matrix is given by
$$\left[\begin {array}{cccc} 1&1&1\\ 2&0&4
\\ 3&1&5\\ 4&2&6
\end {array}
\right]$$
The reduced row echelon form can be calculated as follows
$$\left[\begin {array}{cccc}1&0&2\\ 0&1&-1
\\ 0&0&0\\ 0&0&0
\end {array}
\right],$$
then we have the solution
$$a=-2t,\quad b=t, \quad c= t.$$
Hence, we have the combinations
$$(0,0,0,0)=-2t(1,2,3,4)+t(1,0,1,2)+t(1,4,5,6) .$$
$$(1,4,5,6)=2(1,2,3,4)-(1,0,1,2) .$$
