# Specific Linear Algebra Question

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.

• Yes, that is a sloppy typo. The hypothesis should have been that $W$ and $Z$ have "trivial intersection," i.e., precisely $\{0\}$. P.S. Welcome to MSE. Work on writing more informative titles for your questions :) – Ted Shifrin Feb 18 at 1:17
• You are correct that any subspace must contain the zero vector – J. W. Tanner Feb 18 at 1:19
• Which textbook are you using? It’s certainly sloppy notation, but I wonder if the author makes some statement at the beginning of the book clarifying that the notation “means” trivial intersection. – Clayton Feb 18 at 1:21
• Thanks all, looks like this was just a typo. The question was professor-generated (not sure if from a book) and I do indeed remember my prof explicitly clarifying that {zero vector} is not the same as {∅}. And I will work on my titling in the future :) – Albert Elling Feb 18 at 1:48

Notice that the definition of direct sum determines that given a vector space $$V$$, and two sub-vector spaces of $$V$$, let them be $$U,W$$, then:
$$V = U\oplus W \iff \forall v\in V \exists u\in U, w\in W: v=u+w$$
Also, for every $$v\in V$$ there is exactly \one $$u\in U, w\in W$$ such that $$v=u+w$$ is unique.