Yes/No : Is $U_i$ 's are linearly independent in $\mathbb{R}^4$? let $U_i = ( U_i^{(1)} , U_i^{(2)} , U_i^{(3)} , U_i^{(4)}) ,$ for  $i = 1,2 , 3, 4$ be  four vectors in $\mathbb{R}^4$  such that  $\sum_{i}^{4} U_i^{(j)} = 0$, for  each $j = 1, 2 , 3, 4$
My question is that is  $U_i$ 's  are linearly independent in $\mathbb{R}^4$ ?
My attempt : i think yes  , because $C\sum_{i}^{4} U_i^{(j)} = C(U_1^1 + U_1^2+ U_1^3 + U_1^ 4 + U_2^1 + U_2^2+ U_2^3 + U_2^ 4 +U_3^1 + U_3^2+ U_3^3 + U_3^ 4 +U_4^1 + U_4^2+ U_4^3 + U_4^ 4)=0$ Here  $C= 0 $ where $C= C_1 =C_2 \dots C_n=0$
so $U_i$ is linearly independent
Is its true ?

Consider $U_1=(1,1,1,1)=U_2$ and $U_3=(-1,-1,-1,-1)=U_4$. Then for each $j$,
$$\sum_{i=1}^4 U_i^{(j)}=1+1-1-1=0$$
but the $U_i$ are not linearly independent.
Or you could let $U_i=(0,0,0,0)$ for all $i$. 
