Here is an example of a solution:
v=<2,5,3>; u1=<1,1,0>; u2=<0,1,1>; u3=<1,0,1>.
Let v=au1+bu2+cu3, where a, b and c are scalars. We can write the following system of equations:
a+c=2
a+b=5
b+c=3
Add the first two equations together: 2a+b+c=7, 2a+3=7 (because b+c=3), a=2⇒c=0, b=3.
v=2u1+3u2, a unique solution.