Let $X,Y \subset \mathbb{C}^3$ with $X=\text{span}\{\begin{bmatrix} 1\\ 2 \\ 3\\ \end{bmatrix}\}$ and $Y=\text{span}\{\begin{bmatrix} 1 \\ 1 \\ 2\\ \end{bmatrix}\}$. Does there exist a subspace $Z\subset \mathbb{C}^3$ such that $\mathbb{C}^3 = X \oplus Z$ and $\mathbb{C}^3 = Y \oplus Z$. If so, describe this subspace. If not, explain why.
I don't know how to approach this question. My intuition tells me such a subspace $Z$ cannot exist but I don't know how to prove this. I have played a bit with random vectors and I've tried proving that $X \cap Z \neq Y \cap Z = 0$ but I wasn't able to do this. Could someone help me out.
Any $2$-dimensional subspace of $\mathbf C^3$ which contains neither $X$ nor $Y$ will do the trick.