= Solution
The external <direct sum of vector spaces> consists of pairs $(v_1,v_2)$ with componentwise operations. For subspaces of one ambient space, the <sum of vector subspaces> is $V_1+V_2=\{v_1+v_2:v_i\in V_i\}$. The addition map from the external direct sum onto this subspace is surjective, with kernel $\{(w,-w):w\in V_1\cap V_2\}$. This kernel is isomorphic to the intersection. The <rank-nullity theorem>, together with $\dim(V_1\oplus V_2)=\dim V_1+\dim V_2$, proves the <dimension formula for a sum of subspaces>.
For the supplied bases, solving for annihilating linear forms gives
$$
V_1=\{x:x_1+x_2-x_3+x_4=0\},\qquad
V_2=\{x:2x_1-3x_3+x_4=0\}.
$$
Each <basis> has rank three, so these hyperplanes are the whole subspaces. The two independent equations make the intersection two-dimensional. Imposing the requested zero components yields
$$
\boxed{v_1=(0,-2,1,3),\qquad v_2=(-2,0,-1,1).}
$$
Both satisfy the two equations and are independent, since their first components differ while $v_1\ne0$. Hence they form the required <basis>.
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