$W$ a subset of $\mathbb{R}^5$ consisting of all vectors an odd number of the entries in which are equal to $0$. Is $W$ a subspace of $\mathbb{R}^5$? Let $W$ be the subset of $\mathbb{R}^5$ consisting of all vectors an odd number of the entries in which are equal to $0$. Is $W$ a subspace of $\mathbb{R}^5$?
I'm not sure how to do this. Any solutions or hints are greatly appreciated. I know that in order for anything to be a subspace of something the zero vector must be in it. How would I go about this? What exactly do we mean by subset here? Is it any $5$-tuple or could it be $1,2,3,4,5$-tuples?

Since $W$ is a subspace of $\mathbb{R}^{5}$ if and only if (i) $0_{\mathbb{R}^{5}} \in W$, (ii) $x,y \in W$ implies $x+y \in W$, and (iii) $c \in \mathbb{R}$ and $x \in W$ imply $cx \in W$, and since $(1,2,0,3,4) \in W$  and $(4,0,3,2,1) \in W$ but their sum is a vector having no zero components, so $W$ is not a subspace of $\mathbb{R}^{5}$.
I misunderstood your question. :) 
