Just as a background information: I have absolutely no experience with topics in algebra. So please be patient, if I have done something terrible or not used the right terminology.
I have two vectorspaces containing vectors from a vectorspace as elements. The first vectorspace $A = \{v_2,v_3\}$ is a nullspace and the second vectorspace is $B = \{v_1,v_4\}$ and I need to evaluate the direct sum $A \oplus B=\{v_2,v_3\}\oplus\{v_1,v_4\}$.
This is what I have as a result (just by looking at other examples), but I am not sure if this is correct. It would be nice if someone more experienced could tell me if I did it right.
$$A \oplus B=\{v_2,v_3,v_2+\mu_1 v_3,v_1+\mu_1v_2+\mu_2v_3,v_4+\mu_1v_2+\mu_2v_3\}$$ where $\mu_1, \mu_2$ arbitrary numbers.
EDIT: The example that I have is from Ovsiannikov 1982 in which $\{v_4\}$ is the nullspace:
$$\{v_4\} \oplus \{v_1,v_2,v_1+v_3\}=\{v_1+\mu v_4,v_2+\mu v_4,v_1+v_3+\mu v_4,v_4\}$$