If $X=\{(x,0) ; x \in \Bbb R\}$ find infinitely many subspaces $Z_i$ such that $\Bbb R^2=X \oplus Z_i$.
I thought anwser would be $Y = \{(0,x) ; x \in \Bbb R\}$ so that $X\cap Y=\{0\}$. However the anwser is;
"Every line L,of non-zero gradient that passes through the origin is a subspace of $\Bbb R^2$ such that $X \oplus L=\Bbb R^2$"
Your example is fine, but the question was to find infinitely many examples. In fact, from the requirement $\Bbb R^2=X \oplus Z_i$ you know that $Z_i$ has to have dimension $1$ and intersect $X$ trivially. Therefore any line fits these requirements, excepts for $X$ itself.