I was assigned the following problem:
Let $W$ and $Z$ be two subspaces of $V$ such that $W \cap Z = \emptyset$ and let $K = W + Z$. Prove that every element of $K$ can be written uniquely as the sum of an element of $W$ and an element of $Z$.
I did actually manage to prove this. That is not my question. What seems odd to me about this problem though, is the nature of $W$ and $Z$. I don't understand how $W \cap Z$ could ever equal $\emptyset$ given that the zero vector of vector space $V$ will certainly always be in $W \cap Z$. ($W$ and $Z$ are subspaces; closed under scalar mult.; just multiply any element by $0$ in each). Am I missing something? Or is this situation just impossible?
Thanks, and sorry for the poor symbol formatting.