Answer
$$v=\frac{9}{8} u_1+\frac{1}{8}u_2.$$
Work Step by Step
Suppose the following linear combination
$$(1,2,3,5)=a(1,2,3,4)+b(-1,-2,-3,4)+c(0,0,1,1), \quad a,b,c\in R.$$
Which yields the following system of equations
\begin{align*}
a-b&=1\\
2a-2b&=2\\
3a-3b+c&=3\\
4a+4b+c&=5.
\end{align*}
Solving the above system, we get the solution $a=\frac{9}{8}$, $b=\frac{1}{8}$, $c=0$. Hence,
$$v=\frac{9}{8} u_1+\frac{1}{8}u_2.$$
