The direct sum of the three subspaces $V_1, V_2, V_3$is defined as $V_1 \oplus V_2 \oplus V_3$ iff in their intersection lies only a zero vector. I want to find the subspace in $\Bbb R^3$ for which the direct sum exists and the subspace where the direct sum isn't defined.
1) Direct sum exists: For example $\{v_1,v_2,v_3\}=\{(1,-1,1), (-1,-1,1),(1,1,1)\}$ (the set is linearly independent)
2) Direct sum doesn't exist: For example $\{v_1,v_2,v_3\}=\{(1,-1,1), (2,-2,2), (1,1,1)\}$ (the set is linearly dependent)
PS: $v_i \in V_i$ and these vectors individually form the subspaces because they are closed under a multiplication by a scalar and under an addition.