Answer
$v$ can not be written as a linear combination of the vectors $u_1,u_2,u_3$.
Work Step by Step
Suppose the linear combination $v= au_1+bu_2+cu_3$, then we have
$$(0,5,3,0)=a(1,1,2,2)+b(2,3,5,6)+c(-3,1,-4,2), \quad a,b,c\in R.$$
Which yields the following system of equations
\begin{align*}
a+2b-3c&=0\\
a+3b+c&=5\\
2a+5b-4c&=3\\
2a+6b+2c&=0.
\end{align*}
The augmented matrix is given by
$$\left[ \begin {array}{cccc} 1&2&-3&0\\ 1&3&1&5
\\ 2&5&-4&3\\ 2&6&2&0\end {array}
\right].
$$
Using Gauss-eleminition, we get
$$\left[ \begin {array}{cccc} 1&2&-3&0\\ 0&1&4&5
\\ 0&0&-2&-2\\ &0&0&-10
\end {array} \right]
$$
which shows that the system has no solution. , So $v$ can not be written as a linear combination of the vectors $u_1,u_2,u_3$.
